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Tepoyan, L. "The mixed problem for a degenerate operator equation." Universität Potsdam, 2008. http://opus.kobv.de/ubp/volltexte/2009/3033/.
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We consider a mixed problem for a degenerate differentialoperator equation of higher order. We establish some embedding theorems in weighted Sobolev spaces and show existence and uniqueness of the generalized solution of this problem. We also give a description of the spectrum for the corresponding operator.
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Brinkschulte, Judith. "The Cauchy-Riemann equation with support conditions in domains with Levi degenerate boundaries." [S.l.] : [s.n.], 2002. http://dochost.rz.hu-berlin.de/dissertationen/brinkschulte-judith-2002-04-19.
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Picard, Sebastien. "A priori estimates of the degenerate Monge-Ampère equation on compact Kähler manifolds." Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=119756.
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The regularity theory of the degenerate complex Monge-Ampère equation is studied. First, the equation is considered on a compact Kahler manifold without boundary. Accordingly, some background information on Kahler geometry is presented. Given a solution of the degenerate complex Monge-Ampère equation, it is shown that its oscillation and gradient can be bounded. The Laplacian of the solution is also estimated. There is a slight improvement from the literature on the conditions required in order to obtain the estimate on the Laplacian of the solution, however the estimates developed only hold in the case of manifolds with non-negative bisectional curvature. As an application, a Dirichlet problem in complex space is considered. The obtained estimates are used to show existence and uniqueness of pluri-subharmonic solutions to the degenerate complex Monge-Ampere equation in a domain in complex space.La question de la régularité des solutions de l'équation complexe Monge-Ampère dégénérée est étudiée. Premièrement, l'équation est considérée sur une variété compacte Kahler sans frontière. Une revue des concepts clés de la géométrie Kahler est présentée. Étant donné une solution de l'équation complexe Monge-Ampère dégénérée, il est démontré que la différence entre la borne supérieure et la borne inférieure de la solution est sous contrôle, et ainsi pour le gradient de la solution. Le Laplacien de la solution est également bornée. Cette borne du Laplacien est une amélioration de ce qui a été établi dans la littérature jusqu'à présent, mais par contre, l'argument tient seulement sous la condition que la variété a une courbure non-négative. Les résultats sont appliqués à un problème de Dirichlet dans l'espace complexe. L'existence et l'unicité d'une solution pluri-subharmonique de l'équation complexe Monge-Ampère dégénérée dans un domaine contenu dans l'espace complexe est démontré.
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Brinkschulte, Judith. "The Cauchy-Riemann equation with support conditions on domains with Levi-degenerate boundaries." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2002. http://dx.doi.org/10.18452/14734.
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In einem ersten Teil betrachten wir ein relativ kompaktes Gebiet Omega einer n-dimensionalen Kähler-Mannigfaltigkeit, mit Lipschitz-Rand, welches eine gewisse "log delta"-Pseudokonvexität besitzt. Wir zeigen, dass die Cauchy-Riemann Gleichung mit exaktem Träger in Omega für alle Bigrade (p,q) mit 0< q< n-1 eine Lösung besitzt. Ausserdem ist das Bild des Cauchy-Riemann Operators auf glatten (p,n-1)-Formen mit exaktem Träger in Omega abgeschlossen. Wir geben Anwendungen für die Lösbarkeit der tangentialen Cauchy-Riemann Gleichungen für glatte Formen und Ströme auf Rändern von schwach pseudokonvexen Gebieten Steinscher Mannigfaltigkeiten und für die Lösbarkeit der tangentialen Cauchy-Riemann Gleichungen für Ströme auf Levi-flachen CR Mannigfaltigkeiten beliebiger Kodimension. In einem zweiten Teil untersuchen wir die Cauchy-Riemann Gleichung mit Randbedingung Null entlang einer Hyperfläche mit konstanter Signatur. Wir geben Anwendungen für die Lösbarkeit der tangentialen Cauchy-Riemann Gleichung für glatte Formen mit kompaktem Träger und für Ströme auf der Hyperfläche. Wir zeigen auch, dass das Hartogs-Phänomen in schwach 2-konvex-konkaven Hyperflächen mit konstanter Signatur Steinscher Mannigfaltigkeiten gilt.In a first part, we consider a domain Omega with Lipschitz boundary, which is relatively compact in an n-dimensional Kaehler manifold and satisfies some "log delta-pseudoconvexity" condition. We show that the Cauchy-Riemann equation with exact support in Omega admits a solution in bidegrees (p,q), 1 < q < n. Moreover, the range of the Cauchy-Riemann operator acting on smooth (p,n-1)-forms with exact support in Omega is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi-flat CR manifolds of arbitrary codimension. In a second part, we study the Cauchy-Riemann equation with zero Cauchy data along a hypersurface with constant signature. Applications to the solvability of the tangential Cauchy-Riemann equations for smooth forms with compact support and currents on the hypersurface are given. We also prove that the Hartogs phenomenon holds in weakly 2-convex-concave hypersurfaces with constant signature of Stein manifolds.
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Watling, K. D. "Formulae for solutions to (possibly degenerate) diffusion equations exhibiting semi-classical and small time asymptotics." Thesis, University of Warwick, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.380277.
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ROCCHETTI, DARIO. "Generation of analytic semigroups for a class of degenerate elliptic operators." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2009. http://hdl.handle.net/2108/749.
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Questa tesi è suddivisa in due capitoli. Nel primo si da un risultato di buona positura
per una classe di problemi parabolici degeneri. I risultati ottenuti, validi in dimensione 2,
garantiscono che le soluzioni di tali problemi supportano l'integrazione per parti.
Nel secondo capitolo, si studia la controllabilità allo zero per una classe di operatori
parabolici degeneri in forma non-divergenza. In particolare, i coefficienti del termine
del secondo ordine possono degenerare al bordo del dominio spaziale.
A questo scopo si giunge previo una disuguaglianza di osservabilità per il problema aggiunto
usando opportune stime di Carleman.This thesis is composed by two chapters. The first one is devoted to the generation of analytic semigroups in the L^2 topology by second order elliptic operators in divergence form, that may degenerate at the boundary of the space domain. Our results, that hold in two space dimension, guarantee that the solutions of the corresponding evolution problems support integration by parts. So, this paper provides the basis for deriving Carleman type estimates for degenerate parabolic operators. In the second chapter we give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.
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Rao, Arvind Satya. "Weak solutions to a Monge-Ampère type equation on Kähler surfaces." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/582.
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In the context of moment maps and diffeomorphisms of Kähler manifolds, Donaldson introduced a fully nonlinear Monge-Ampère type equation. Among the conjectures he made about this equation is that the existence of solutions is equivalent to a positivity condition on the initial data. Weinkove later affirmed Donaldson's conjecture using a gradient flow for the equation in the space of Kähler potentials of the initial data. The topic of this thesis is the case when the initial data is merely semipositive and the domain is a closed Kähler surface. Regularity techniques for degenerate Monge-Ampère equations, specifically those coming from pluripotential theory, are used to prove the existence of a bounded, unique, weak solution. With the aid of a Nakai criterion, due to Lamari and Buchdahl, it is shown that this solution is smooth away from some curves of negative self-intersection.
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Boiger, Wolfgang Josef. "Stabilised finite element approximation for degenerate convex minimisation problems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16790.
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Infimalfolgen nichtkonvexer Variationsprobleme haben aufgrund feiner Oszillationen häufig keinen starken Grenzwert in Sobolevräumen. Diese Oszillationen haben eine physikalische Bedeutung; Finite-Element-Approximationen können sie jedoch im Allgemeinen nicht auflösen. Relaxationsmethoden ersetzen die nichtkonvexe Energie durch ihre (semi)konvexe Hülle. Das entstehende makroskopische Modell ist degeneriert: es ist nicht strikt konvex und hat eventuell mehrere Minimalstellen. Die fehlende Kontrolle der primalen Variablen führt zu Schwierigkeiten bei der a priori und a posteriori Fehlerschätzung, wie der Zuverlässigkeits- Effizienz-Lücke und fehlender starker Konvergenz. Zur Überwindung dieser Schwierigkeiten erweitern Stabilisierungstechniken die relaxierte Energie um einen diskreten, positiv definiten Term. Bartels et al. (IFB, 2004) wenden Stabilisierung auf zweidimensionale Probleme an und beweisen dabei starke Konvergenz der Gradienten. Dieses Ergebnis ist auf glatte Lösungen und quasi-uniforme Netze beschränkt, was adaptive Netzverfeinerungen ausschließt. Die vorliegende Arbeit behandelt einen modifizierten Stabilisierungsterm und beweist auf unstrukturierten Netzen sowohl Konvergenz der Spannungstensoren, als auch starke Konvergenz der Gradienten für glatte Lösungen. Ferner wird der sogenannte Fluss-Fehlerschätzer hergeleitet und dessen Zuverlässigkeit und Effizienz gezeigt. Für Interface-Probleme mit stückweise glatter Lösung wird eine Verfeinerung des Fehlerschätzers entwickelt, die den Fehler der primalen Variablen und ihres Gradienten beschränkt und so starke Konvergenz der Gradienten sichert. Der verfeinerte Fehlerschätzer konvergiert schneller als der Fluss- Fehlerschätzer, und verringert so die Zuverlässigkeits-Effizienz-Lücke. Numerische Experimente mit fünf Benchmark-Tests der Mikrostruktursimulation und Topologieoptimierung ergänzen und bestätigen die theoretischen Ergebnisse.Infimising sequences of nonconvex variational problems often do not converge strongly in Sobolev spaces due to fine oscillations. These oscillations are physically meaningful; finite element approximations, however, fail to resolve them in general. Relaxation methods replace the nonconvex energy with its (semi)convex hull. This leads to a macroscopic model which is degenerate in the sense that it is not strictly convex and possibly admits multiple minimisers. The lack of control on the primal variable leads to difficulties in the a priori and a posteriori finite element error analysis, such as the reliability-efficiency gap and no strong convergence. To overcome these difficulties, stabilisation techniques add a discrete positive definite term to the relaxed energy. Bartels et al. (IFB, 2004) apply stabilisation to two-dimensional problems and thereby prove strong convergence of gradients. This result is restricted to smooth solutions and quasi-uniform meshes, which prohibit adaptive mesh refinements. This thesis concerns a modified stabilisation term and proves convergence of the stress and, for smooth solutions, strong convergence of gradients, even on unstructured meshes. Furthermore, the thesis derives the so-called flux error estimator and proves its reliability and efficiency. For interface problems with piecewise smooth solutions, a refined version of this error estimator is developed, which provides control of the error of the primal variable and its gradient and thus yields strong convergence of gradients. The refined error estimator converges faster than the flux error estimator and therefore narrows the reliability-efficiency gap. Numerical experiments with five benchmark examples from computational microstructure and topology optimisation complement and confirm the theoretical results.
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Čakaitė, Inga. "Dalinių išvestinių sistemos su kvazireguliariuoju išsigimimu sprendimas." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2006. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2006~D_20060609_122917-49917.
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The system of the four partial fluxions of the primary row of differential equations the row of which dwindles at the points of plane has been analysed. The systems of the expressions of families of the detached solutions have been derived by converging degree rows at the environment of malformation rows through the technique of summation of degree rows. The solutions at the malformation points are particular for having degree particularities. Still, the particularities depend on the other to variables, in conformity to which there are no system malformation weigh. The effect is not evident in the analytical theory of malformed vulgar differential equation.
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Mumcu, Gokhan. "EM Characterization of Magnetic Photonic / Degenerate Band Edge Crystals and Related Antenna Realizations." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1221860344.
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Civin, Damon. "Stability of charged rotating black holes for linear scalar perturbations." Thesis, University of Cambridge, 2015. https://www.repository.cam.ac.uk/handle/1810/247397.
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In this thesis, the stability of the family of subextremal Kerr-Newman space- times is studied in the case of linear scalar perturbations. That is, nondegenerate energy bounds (NEB) and integrated local energy decay (ILED) results are proved for solutions of the wave equation on the domain of outer communications. The main obstacles to the proof of these results are superradiance, trapping and their interaction. These difficulties are surmounted by localising solutions of the wave equation in phase space and applying the vector field method. Miraculously, as in the Kerr case, superradiance and trapping occur in disjoint regions of phase space and can be dealt with individually. Trapping is a high frequency obstruction to the proof whereas superradiance occurs at both high and low frequencies. The construction of energy currents for superradiant frequencies gives rise to an unfavourable boundary term. In the high frequency regime, this boundary term is controlled by exploiting the presence of a large parameter. For low superradiant frequencies, no such parameter is available. This difficulty is overcome by proving quantitative versions of mode stability type results. The mode stability result on the real axis is then applied to prove integrated local energy decay for solutions of the wave equation restricted to a bounded frequency regime. The (ILED) statement is necessarily degenerate due to the trapping effect. This implies that a nondegenerate (ILED) statement must lose differentiability. If one uses an (ILED) result that loses differentiability to prove (NEB), this loss is passed onto the (NEB) statement as well. Here, the geometry of the subextremal Kerr-Newman background is exploited to obtain the (NEB) statement directly from the degenerate (ILED) with no loss of differentiability.
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Bredies, Kristian. "Optimal control of degenerate parabolic equations in image processing analysis of evolution equations with variable degeneracy and associated minimization problems." Berlin Logos-Verl, 2007. http://d-nb.info/987598511/04.
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Bredies, Kristian. "Optimal control of degenerate parabolic equations in image processing : analysis of evolution equations with variable degeneracy and associated minimization problems /." Berlin : Logos-Verl, 2008. http://deposit.d-nb.de/cgi-bin/dokserv?id=3071675&prov=M&dok_var=1&dok_ext=htm.
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Moyano, Garcia Iván. "Controllability of of some kinetic equations, of parabolic degenerated equations and of the Schrödinger equation via domain transformation." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX062/document.
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Ce mémoire présente les travaux réalisés au cours de ma thèse dans le but d'étudier la contrôlabilité de quelques équations aux dérivées partielles. La première partie de cette thèse est consacrée à l'étude de la contrôlabilité de quelques équations cinétiques en différents régimes. Dans un régime collisionnel, nous étudions la contrôlabilité de l'équation de Kolmogorov, un modèle de type Fokker-Planck cinétique, posée dans l'espace de phases $R^d times R^d$. Nous obtenons la contrôlabilité à zéro de cette équation grâce à l'utilisation d'une inégalité spectrale associée à l'opérateur Laplacien dans tout l'espace. Dans un régime non-collisionnel, nous étudions la contrôlabilité de deux systèmes de couplage fluide-cinétique, les systèmes de Vlasov-Stokes et de Vlasov-Navier-Stokes, comportant des non-linéarités dues au terme de couplage. Dans ces cas, l'approche repose sur la méthode du retour.Dans la deuxième partie nous étudions la contrôlabilité d'une famille d'équations paraboliques dégénérées 1-D par la méthode de platitude, qui permet la constructions de contrôles explicites. La troisième partie porte sur le problème de la contrôlabilité de l'équation de Schrödinger par la forme du domaine, c'est-à-dire, en utilisant le domaine comme variable de contrôle. Nous obtenons un résultat de ce type dans le cas du disque unité bidimensionnel. Nos méthodes sont basées sur un résultat de contrôle exact local autour d'une certaine trajectoire, obtenu grâce au théorème d'inversion localeThis memoir presents the results obtained during my PhD, whose goal is the study of the controllability of some Partial Differential Equations.The first part of this thesis is concerned with the study of the controllability of some kinetic equations undergoing different regimes. Under a collisional regime, we study the controllability of the Kolmogorov equation, a particular case of kinetic Fokker-Planck equation, in the phase space $R^d times R^d$. We obtain the null-controllability of this equation thanks to the use of a spectral inequality associated to the Laplace operator in the whole space. Under a non-collisional regime, we study the controllability of two fluid-kinetic models, the Vlasov-Stokes system and the Vlasov-Navier-Stokes system, which exhibe nonlinearities due to the coupling terms. In those cases, the strategy relies on the Return method.In the second part, we study the controllability of a family of 1-D degenerate parabolic equations by the flatness method, which allows the construction of explicit controls.The third part is focused on the problem of the controllability of the Schrödinger equation via domain deformations, i.e., using the domain as a control. We obtain a result of this kind in the case of the two-dimensional unit disk, for radial data. Our methods are based on a local exact controllability result around a certain trajectory, obtained thanks to the Inverse Mapping theorem
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Figalli, Alessio. "Optimal transportation and action-minimizing measures." Doctoral thesis, Scuola Normale Superiore, 2007. http://hdl.handle.net/11384/85683.
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Hofmanová, Martina. "Degenerate parabolic stochastic partial differential equations." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.
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In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Dekkers, Sophia Antonia Janna. "Degenerate parabolic equations on Riemannanian manifolds." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.405755.
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MARINO, LORENZO. "Regolarizzazione debole attraverso rumore di Lévy degenere e sue applicazioni." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2021. http://hdl.handle.net/10281/330542.
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Dopo un'introduzione generale sul fenomeno della regolarizzazione attraverso rumore in un contesto degenere, la prima parte di questa tesi si concentra nello stabilire le stime di Schauder, un strumento analitico utile per dimostrare anche il carattere ben posto di equazioni differenziali stocastiche (EDS), per due classi di equazioni di Kolmogorov sotto una condizione di tipo Hörmander debole, i cui coefficienti giacciono in opportuni spazi di Hölder anisotropi con multi-indici di regolarità.
La prima classe considera un sistema non lineare controllato da un operatore simmetrico ⍺-stabile che agisce solo su alcune componenti. Il nostro metodo di dimostrazione si basa su un approccio perturbativo basato su espansioni della parametrice progressiva tramite formule di tipo Duhamel. A causa delle scarse proprietà regolarizzanti date dal contesto degenere, sfruttiamo anche alcuni controlli sulle norme di Besov, per trattare la perturbazione non lineare.
Come estensione del primo modello, presentiamo anche delle stime di Schauder associate a un operatore di Ornstein-Uhlenbeck degenere guidato da una classe più ampia di operatori di tipo quasi-stabile, come quello stabile relativistico o quello di Lamperti. La dimostrazione di questo risultato si basa invece su un'analisi precisa del comportamento del semigruppo di Markov corrispondente tra spazi di Hölder anisotropici e alcune tecniche di interpolazione.
Sfruttando un approccio della parametrice retrograda, la seconda parte di questa tesi
cerca di stabilire il carattere ben posto in senso debole per una catena degenere di EDS guidate dalla stessa classe di processi quasi-stabili, sotto le assunzioni di regolarità di Hölder minime per i coefficienti. Come corollario del nostro metodo, presentiamo anche stime di tipo Krylov di interesse indipendente per il processo canonico sottostante. Infine, sottolineiamo attraverso opportuni controesempi che esiste effettivamente una soglia (quasi) ottimale sugli esponenti di regolarità che garantiscono il carattere ben posto debole per l'EDS.
In relazione ad alcune applicazioni meccaniche per delle dinamiche cinetiche con attrito, concludiamo studiando la stabilità delle perturbazioni del secondo ordine per operatori degeneri di Kolmogorov nelle norme Lp e Hölder.After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness of stochastic differential equations (SDEs), for two different classes of Kolmogorov equations under a weak Hörmander-like condition, whose coefficients lie in suitable anisotropic Hölder spaces with multi-indices of regularity. The first class considers a nonlinear system controlled by a symmetric ⍺-stable operator acting only on some components. Our method of proof relies on a perturbative approach based on forward parametrix expansions through Duhamel-type formulas. Due to the low regularizing properties given by the degenerate setting, we also exploit some controls on Besov norms, in order to deal with the non-linear perturbation. As an extension of the first one, we also present Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of ⍺-stable-like operators, like the relativistic or the Lamperti stable one. The proof of this result relies instead on a precise analysis of the behaviour of the associated Markov semigroup between anisotropic Hölder spaces and some interpolation techniques. Exploiting a backward parametrix approach, the second part of this thesis aims at establishing the well-posedness in a weak sense of a degenerate chain of SDEs driven by the same class of ⍺-stable-like processes, under the assumptions of the minimal Hölder regularity on the coefficients. As a by-product of our method, we also present Krylov-type estimates of independent interest for the associated canonical process. Finally, we emphasize through suitable counter-examples that there exists indeed an (almost) sharp threshold on the regularity exponents ensuring the weak well-posedness for the SDE. In connection with some mechanical applications for kinetic dynamics with friction, we conclude by investigating the stability of second-order perturbations for degenerate Kolmogorov operators in Lp and Hölder norms.
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Zhang, Zhou Ph D. Massachusetts Institute of Technology. "Degenerate Monge-Ampere equations over projective manifolds." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/34685.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 253-257).
In this thesis, we study degenerate Monge-Ampere equations over projective manifolds. The main degeneration is on the cohomology class which is Kähler in classic cases. Our main results concern the case when this class is semi-ample and big with certain generalization to more general cases. Two kinds of arguments are applied to study this problem. One is maximum principle type of argument. The other one makes use of pluripotential theory. So this article mainly consists of three parts. In the first two parts, we apply these two kinds of arguments separately and get some results. In the last part, we try to combine the results and arguments to achieve better understanding about interesting geometric objects. Some interesting problems are also mentioned in the last part for future consideration. The generalization of classic pluripotential theory in the second part may be of some interest by itself.
by Zhou Zhang.
Ph.D.
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Tepoyan, Liparit. "Degenerated operator equations of higher order." Universität Potsdam, 2000. http://opus.kobv.de/ubp/volltexte/2008/2588/.
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Pereira, Liliana Angélica Costa Matos. "Approximation of degenerate partial differential equations arising in finance." Doctoral thesis, Instituto Superior de Economia e Gestão, 2018. http://hdl.handle.net/10400.5/15849.
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Doutoramento em Matemática Aplicada à Economia e GestãoEsta dissertação estuda a discretização de equações diferenciais parciais degeneradas com aplicações às Finanças. Em particular, o problema de Cauchy para uma equação diferencial parcial linear de segunda ordem é discretizado nas variáveis espaciais para os casos de coeficientes limitados e ilimitados. A semi-discretização é considerada para a versão multidimensional da EDP e também para o caso particular de uma dimensão espacial. A aproximação à solução da EDP é obtida com recurso a métodos básicos de diferenças finitas em versões discretas de espaços de Sobolev e de Sobolev ponderados. São deduzidos resultados de existência e unicidade para a solução generalizada do problema semi-discretizado. Finalmente, é dada uma estimativa para a taxa de convergência da solução do problema semi-discretizado para a solução do problema exacto correspondente. São obtidos resultados mais fortes para o caso especial de uma dimensão no espaço.
This thesis focuses on the discretization of degenerate partial differential equa- tions arising in Finance. In particular, the Cauchy problem for a second order linear parabolic PDE is discretized in the spatial variables for both the bounded and unbounded coefficient cases. The semi-discretization is considered for the general multi-dimensional version of the PDE and also for the particular one-dimensional case. The approximation to the PDE problem solution is obtained by using basic finite difference methods in discrete Sobolev and weighted Sobolev spaces. Existence and uniqueness results for the generalized solution to the semi- discretized problem are deduced. Finally, we give an estimate for the rate of convergence of the solution of the semi-discretized problem to the solution of corresponding the exact problem. Stronger results are deduced for the special case of one dimension on space.
info:eu-repo/semantics/publishedVersion
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Zhan, Yi. "Viscosity solutions of nonlinear degenerate parabolic equations and several applications." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/NQ49931.pdf.
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Pang, Ho Cheung. "Analysis on stochastic anisotropic degenerate parabolic-hyperbolic mixed-type equations." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:4364a7ef-07fa-458a-bd59-3de79f092144.
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This dissertation consists chiefly of three parts, which tell different facets in the development of one topic. The first part is an exploration of continuous dependence estimates of stochastically driven degenerate parabolic equations. The second is an extension of work done by Debussche and Vovelle on first order stochastic conservation laws - we extend their results to degenerate parabolic-hyperbolic conservation laws with additive noise, and derive results on the existence and uniqueness of invariant measures. In the third part we explore the long time behaviour of solutions to stochastic degenerate parabolic-hyperbolic conservation laws with multiplicative noise, depending non-linearly on the solution itself.
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Maizurna, Isna. "Semigroup methods for degenerate cauchy problems and stochastic evolution equations /." Title page, abstract and contents only, 1999. http://web4.library.adelaide.edu.au/theses/09PH/09phm2328.pdf.
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Voldman, Aleksandr. "Topological classification of non-degenerate quadratic system." CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/1192.
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Götmark, Elin, and Kaj Nyström. "Boundary behaviour of non-negative solutions to degenerate sub-elliptic equations." Uppsala universitet, Analys och tillämpad matematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-164532.
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Let X = {X-1, ..., X-m} be a system of C-infinity vector fields in R-n satisfying Hormander's finite rank condition and let Omega be a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. We study the boundary behavior of non-negative solutions to the equation Lu = Sigma(i, j -1) X-i*(a(ij)X(j)u) = Sigma X-i, j=1(i)*(x)(aij(x)X-j(x)u(x)) = 0 for some constant beta >= 1 and for some non-negative and real-valued function lambda = lambda(x). Concerning kappa we assume that lambda defines an A(2)-weight with respect to the metric introduced by the system of vector fields X =, {X-1,..., X-m}. Our main results include a proof of the doubling property of the associated elliptic measure and the Holder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes et al. (1982, 1983) [18-20] (m = n, {X-(1), ..., X-m} = {partial derivative(x1), ...., partial derivative x(n)}, A is an A(2)-weight) and Capogna and Garofalo (1998) [6] (X = {X-1,..., X-m} satisfies Hormander's finite rank condition and X(x) equivalent to lambda A for some constant lambda). One motivation for this study is the ambition to generalize, as far as possible, the results in Lewis and Nystrom (2007, 2010, 2008) [35-38], Lewis et al. (2008) [34] concerning the boundary behavior of non-negative solutions to (Euclidean) quasi-linear equations of p-Laplace type, to non-negative solutions, to certain sub-elliptic quasi-linear equations of p-Laplace type.
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Schneider, Mathias. "Finite element approximation of some degenerate/singular elliptic and parabolic equations." Thesis, Imperial College London, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.265861.
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SaÌnchez, Garduño Faustino. "Travelling waves in one-dimensional degenerate non-linear reaction-diffussion equations." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.334929.
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Agueh, Martial Marie-Paul. "Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29180.
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Abedin, Farhan. "Harnack Inequality for a class of Degenerate Elliptic Equations in Non-Divergence Form." Diss., Temple University Libraries, 2018. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/523174.
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MathematicsPh.D.
We provide two proofs of an invariant Harnack inequality in small balls for a class of second order elliptic operators in non-divergence form, structured on Heisenberg vector fields. We assume that the coefficient matrix is uniformly positive definite, continuous, and symplectic. The first proof emulates a method of E. M. Landis, and is based on the so-called growth lemma, which establishes a quantitative decay of oscillation for subsolutions. The second proof consists in establishing a critical density property for non-negative supersolutions, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez and Lanconelli to obtain Harnack’s inequality.
Temple University--Theses
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CIBELLI, Gennaro. "Sharp Estimates for Fundamental Solutions of some degenerate Kolmogorov equations arising in Finance." Doctoral thesis, Università degli studi di Ferrara, 2017. http://hdl.handle.net/11392/2488066.
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The thesis is devoted to the study of a degenerate parabolic partial differential equation which arises in models for the pricing of Arithmetic Average Asian Options in Finance in the framework introduced by Black, Scholes and Merton.
The aim of the work is to prove optimal estimates for the fundamental solution of the related operator.
The interest in this result is in that an expression of the fundamental solution is not available, whereas explicit information on its asymptotic behaviour are provided in the work.
The problem of proving upper and lower estimates for the fundamental solution of a second order partial evolution operator has long history and it has been considered by many authors in the study of PDE's.
The methodology used involves several techniques belonging to the theory of Partial Differential Equations, Stochastic Processes and Optimal Control Theory, and can be applied to several different problems.
In particular, the proof of the lower bound relies on the repeated application of the Harnack inequality for positive solution of along a suitable chain of points, combined with an optimization procedure. Such procedure lead us to naturally consider an optimal control problem which will be explicitly solved.
For the upper bound, we combine analytical results with elementary tools belonging to Optimal Control Theory. In particular we use an analogous results to the Moser iteration and the Hamilton-Jacobi-Bellman equation for the value function related to a relevant optimal control problem.La tesi è dedicata allo studio di una equazione differenziale alle derivate parziali parabolica degenere che interviene in modelli per il pricing di Opzioni asiatiche a media aritmetica in Finanza nel setting introdotto da Black, Scholes e Merton. Lo scopo del lavoro è quello di ricavare stime ottimali per la soluzione fondamentale del relativo operatore. L'interesse in questo risultato risiede nel fatto che un' espressione della soluzione fondamentale non è disponibile, mentre informazioni esplicite sul suo comportamento asintotico sono fornite nel lavoro di Tesi. Il problema di dimostrare stime ottimali dall'alto e dal basso per la soluzione fondamentale di un operatore di evoluzione del secondo ordine ha lunga storia ed è stato considerato da molti autori nello studio delle PDE's. La metodologia utilizzata coinvonlge diverse tecniche appartenenti all' Analisi Matematica, Processi stocastici e Teoria del controllo ottimo e può essere applicata a diversi problemi. In particolare, la prova del limite inferiore si basa sulla ripetuta applicazione della disuguaglianza di Harnack per soluzioni positive lungo un'opportuna catena di punti, combinata con una procedura di ottimizzazione. Tale procedura ci conduce a considerare naturalmente un problema di controllo ottimo che sarà risolto in modo esplicito. Per il limite superiore, si combinano alcuni risultati di PDE con strumenti elementari appartenenti alla teoria del controllo ottimo. In particolare, si usano risultati analoghi all' iterazione Moser e l'equazione di Hamilton-Jacobi-Bellman per la funzione valore relativo ad un pertinente problema di controllo ottimo.
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Iwaki, Kohei. "On WKB theoretic transformations for Painleve transcendents on degenerate Stokes segments." 京都大学 (Kyoto University), 2014. http://hdl.handle.net/2433/188457.
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Pesce, Antonello. "Stochastic fundamental solutions for a class of degenerate SPDEs." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/14559/.
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In this thesis, we look for a fundamental solution for a broad, possibly degenerate class of stochastic partial differential equations (SPDEs), whose deterministic part is a Kolmogorov equation with coefficients measurable in the time variable. We use a version of the It\^o-Wentzell formula to reduce the SPDE to a PDE, for which we extend the classic Levi's parametrix method to find a fundamental solution.
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Chen, Hua, and Ke Li. "The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations." Universität Potsdam, 2007. http://opus.kobv.de/ubp/volltexte/2009/3024/.
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Let X = (X1,.....,Xm) be an infinitely degenerate system of vector fields, we study the existence and regularity of multiple solutions of Dirichelt problem for a class of semi-linear infinitely degenerate elliptic operators associated
with the sum of square operator Δx = ∑m(j=1) Xj* Xj.
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Alhojilan, Yazid Yousef M. "Higher-order numerical scheme for solving stochastic differential equations." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/15973.
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We present a new pathwise approximation method for stochastic differential equations driven by Brownian motion which does not require simulation of the stochastic integrals. The method is developed to give Wasserstein bounds O(h3/2) and O(h2) which are better than the Euler and Milstein strong error rates O(√h) and O(h) respectively, where h is the step-size. It assumes nondegeneracy of the diffusion matrix. We have used the Taylor expansion but generate an approximation to the expansion as a whole rather than generating individual terms. We replace the iterated stochastic integrals in the method by random variables with the same moments conditional on the linear term. We use a version of perturbation method and a technique from optimal transport theory to find a coupling which gives a good approximation in Lp sense. This new method is a Runge-Kutta method or so-called derivative-free method. We have implemented this new method in MATLAB. The performance of the method has been studied for degenerate matrices. We have given the details of proof for order h3/2 and the outline of the proof for order h2.
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Figalli, Alessio. "Optimal transportation and action-minimizing measures." Doctoral thesis, Lyon, École normale supérieure (sciences), 2007. http://www.theses.fr/2007ENSL0422.
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Lucertini, Giacomo. "Parametrix technique for a class of degenerate parabolic operators with measurable coefficients under the weak Hörmander condition." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021.
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In this thesis we study a wide class of differential operators of Kolmogorov type characterized by two structural hypotheses: it is satisfied the weak Hörmander condition and the coefficients are merely measurable in time variable. The main purpose of this work is to adapt the classical Levi parametrix method to this framework, in order to construct a fundamental solution of the operator. In particular, we will use a time-dependent parametrix.
This problem is strongly related to the study of stochastic differential equations (SDEs), since the backward Kolmogorov operators associated to some linear SDEs take the form of the operator in the class we have considered. Therefore, the obtained fundamental solution can be seen as the transition density of the solution of a SDE.
Choosing coefficients that are merely measurable in time, these results may be applied in the study of stochastic partial differential equations (SPDEs), which naturally appear in applications with rough coefficients.
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MORALES, DANIA GONZALEZ. "TWO TOPICS IN DEGENERATE ELLIPTIC EQUATIONS INVOLVING A GRADIENT TERM: EXISTENCE OF SOLUTIONS AND A PRIORI ESTIMATES." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2018. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=36440@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
PROGRAMA DE EXCELENCIA ACADEMICA
Esta tese tem o intuito do estudo da existência, não existência e estimativas a priori de soluções não negativas de alguns tipos de problemas elípticos degenerados coercivos e não coercivos com um termo adicional dependendo do gradiente. Dentre outras coisas, obtemos condições integrais generalizadas tipo Keller-Osserman para a existência e não existência de soluções. Também mostramos que condições adicionais e diferentes são necessárias quando p é maior ou igual à 2 ou p é menor ou igual à 2, devido ao caráter degenerado do operador. As estimativas a priori são obtidas para super-soluções e soluções de EDPs elípticas superlineares o sistemas de tais tipos de equações em forma divergente com diferentes operadores e não linearidades. Além do mais, obtemos extensões até a fronteira de algumas desigualdades de Harnack fracas e lemas quantitativos de Hopf para operadores elípticos como o p-Laplaciano.
This thesis concerns the study of existence, nonexistence and a priori estimates of nonnegative solutions of some types of degenerate coercive and non coercive elliptic problems involving an additional term which depends on the gradient. Among other things, we obtain generalized integral conditions of Keller-Osserman type for the existence and nonexistence of solutions. Also, we show that different conditions are needed when p is higher or equal to 2 or p is less than or equal to 2, due to the degeneracy of the operator. The uniform a priori estimates are obtained for supersolutions and solutions of superlinear elliptic PDE or systems of such PDE in divergence form that can contain different operators and nonlinearities. We also give full boundary extensions to some half Harnack inequalities and quantitative Hopf lemmas, for degenerate elliptic operators like the p-Laplacian.
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Floridia, Giuseppe. "Approximate multiplicative controllability for degenerate parabolic problems and regularity properties of elliptic and parabolic systems." Doctoral thesis, Università di Catania, 2012. http://hdl.handle.net/10761/1051.
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This thesis consists of two parts, both related to the theory of parabolic equations and systems.
The first part is devoted to control theory which studies the possibility of influencing
the evolution of a given system by an external action called control. Here we address approximate controllability problems via multiplicative controls, motivated by our interest in some differential models for the study of climatology.
In the second part of the thesis we address regularity issues on the local differentiability and H\"older regularity for weak solutions of nonlinear systems in divergence form.
In order to improve readability, the two parts have been organized as completely independent chapters, with
two separate introductions and bibliographies.
All the new results of this thesis have been presented at conferences and workshops, and
most of them appeared or are to appear as research articles in international journals.
Related directions for future research are also outlined in body of the work.
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Tarhini, Rana. "Équation de films minces fractionnaire pour les fractures hydrauliques." Thesis, Paris Est, 2018. http://www.theses.fr/2018PESC1061/document.
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Ces travaux concernent deux équations paraboliques, dégénérées et non-locales. La première équation est une équation de films minces fractionnaire et la deuxième est une équation des milieux poreux fractionnaire. La présentation des problèmes, les résultats existants dans la littérature, ainsi que le résumé de nos résultats font l'objet de l'introduction. Le deuxième chapitre est consacré à la présentation de la méthode de De Giorgi utilisée pour montrer la régularité Hölder des solutions des équations elliptiques. On présente de plus les résultats utilisant cette approche dans les cas paraboliques local et non-local. Dans le troisième chapitre, on montre l'existence de solutions faibles d'une équation des films minces fractionnaire. C'est une équation parabolique, dégénérée, non-locale d'ordre $alpha+2$ où $0 < alpha < 2$. C'est une généralisation d'une équation étudiée par Imbert et Mellet en 2011 pour $alpha = 1$. Pour construire les solutions, on passe par un problème régularisé. En utilisant les injections de Sobolev, on passe à la limite pour trouver des solutions faibles. Vu la différence des injections de Sobolev, on distingue deux cas $0 In this thesis, we study two degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove Hölder regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local degenerate parabolic equation of order $alpha + 2$ where $0 < alpha < 2$. It is a generalization of an equation studied by Imbert and Mellet in 2011 for $alpha = 1$. To construct these solutions, we consider a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$. We also prove that the solution is positive if the initial condition is so. The fourth chapter is dedicated for a fractional porous medium equation. We prove Hölder regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$ because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove Hölder regularity of solutions using the scaling property
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Krüger, Matthias [Verfasser], Ingo [Akademischer Betreuer] Witt, Ingo [Gutachter] Witt, and Dorothea [Gutachter] Bahns. "On the Cauchy problem for a class of degenerate hyperbolic equations / Matthias Krüger ; Gutachter: Ingo Witt, Dorothea Bahns ; Betreuer: Ingo Witt." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2018. http://d-nb.info/1166399818/34.
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Leobacher, Gunther, and Michaela Szölgyenyi. "Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient." Springer Nature, 2018. http://dx.doi.org/10.1007/s00211-017-0903-9.
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We prove strong convergence of order 1/4 - E for arbitrarily small E > 0 of
the Euler-Maruyama method for multidimensional stochastic differential equations
(SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is
based on estimating the difference between the Euler-Maruyama scheme and another
numerical method, which is constructed by applying the Euler-Maruyama scheme to
a transformation of the SDE we aim to solve.
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De, Zan Cecilia. "Some new results on reaction-diffusion equations and geometric flows." Doctoral thesis, Università degli studi di Padova, 2012. http://hdl.handle.net/11577/3422529.
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In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion equations in the unbounded domain Rn × (0 + ∞), in the cases when such a behavior is described in terms of moving interfaces.
As first class of asymptotic problems we consider the singular limit of bistable reaction-diffusion equations in the case when the velocity of the traveling wave equation depends on the space variable, i.e. cε = cε(x), and it satisfies, in some suitable sense, cε/ετ → α, as ε → 0+, where α is a discontinuous function and τ is an integer that can be equal to 0 or 1. The second part of the thesis concerns semilinear reaction-diffusion equations with diffusion term of type tr(Aε(x)D2uε), where tr denotes the trace operator, Aε = σεσtε for some matrix map σε : Rn → Rn×(m+n) and Aε converges to a degenerate matrix.
In order to establish such results rigorously, we modify and adapt to our problems the ”geometric approach” introduced by G. Barles and P. E. Souganidis for solving problems in Rn, and then partially revisited by G. Barles and F. Da Lio for reaction-diffusion equations in bounded domains. When it is possible we always consider the question of the well posedness of the Cauchy problems governing the motion of the fronts that describe the asymptotics we considerIn questa tesi discutiamo il comportamento asintotico delle soluzioni di equazioni di reazione-diffusione nel dominio illimitato Rn × (0,+∞) nei casi in cui tale comportamento sia descritto da un’interfaccia in movimento. Come primo tipo di problemi asintotici consideriamo il limite singolare di equazioni di reazione-diffusione bistabili nel caso in cui la velocità dell’onda viaggiante dipenda dalla variabile di stato, cioè cε = cε(x), e sia soddisfatto, al tendere di ε a zero e in qualche modo opportuno, cε/ετ → α, laddove α è una funzione discontinua e τ è un intero che può essere uguale a 0 o a 1. La seconda parte della tesi riguarda equazioni di reazione-diffusione semilineari e aventi termini di diffusione del tipo tr(Aε(x)D2uε), laddove tr denota l’operatore traccia, Aε = σεσtε per qualche funzione σε : Rn → Rn×(m+n) e Aε converge ad una matrice degenere. Al fine di provare tali risultati in modo rigoroso, abbiamo modificato e adattato "l’approccio geometrico" introdotto da G. Barles e P. E. Souganidis per risolvere problemi in Rn e in seguito parzialmente rivisto dallo stesso G. Barles assieme a F. Da Lio per equazioni di reazione-diffusione in domini limitati. Laddove possibile abbiamo sempre considerato la questione della buona posizione dei problemi di Cauchy che governano il moto dei fronti che descrivono le asintotiche da noi considerate
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Liu, Weian, Yin Yang, and Gang Lu. "Viscosity solutions of fully nonlinear parabolic systems." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2621/.
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In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
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Sande, Olow. "Boundary Estimates for Solutions to Parabolic Equations." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-281451.
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This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.
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Leahy, James-Michael. "On parabolic stochastic integro-differential equations : existence, regularity and numerics." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/10569.
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In this thesis, we study the existence, uniqueness, and regularity of systems of degenerate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted coefficients in the whole space. We also investigate explicit and implicit finite difference schemes for SIDEs with non-degenerate diffusion. The class of equations we consider arise in non-linear filtering of semimartingales with jumps. In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by Lévy driven stochastic differential equations (SDEs) with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of linear parabolic second order stochastic partial differential equations (SPDEs) by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions of degenerate SIDEs using the method of stochastic characteristics. More precisely, we use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of stochastic flows generated by SDEs with jumps to construct solutions. In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application, we establish the existence and uniqueness of solutions of degenerate linear stochastic integro-differential equations in the L2-Sobolev scale. Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5. Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear SIDEs and show that the rate is of order one in space and order one-half in time.
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Hall, Eric Joseph. "Accelerated numerical schemes for deterministic and stochastic partial differential equations of parabolic type." Thesis, University of Edinburgh, 2013. http://hdl.handle.net/1842/8038.
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First we consider implicit finite difference schemes on uniform grids in time and space for second order linear stochastic partial differential equations of parabolic type. Under sufficient regularity conditions, we prove the existence of an appropriate asymptotic expansion in powers of the the spatial mesh and hence we apply Richardson's method to accelerate the convergence with respect to the spatial approximation to an arbitrarily high order. Then we extend these results to equations where the parabolicity condition is allowed to degenerate. Finally, we consider implicit finite difference approximations for deterministic linear second order partial differential equations of parabolic type and give sufficient conditions under which the approximations in space and time can be simultaneously accelerated to an arbitrarily high order.
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Şanlı, Zafer Çöken Abdilkadir Ceylan. "Dejenere helisler üzerine /." Isparta : SDÜ Fen Bilimleri Enstitüsü, 2009. http://tez.sdu.edu.tr/Tezler/TF01302.pdf.
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Stilgenbauer, Patrik [Verfasser]. "The Stochastic Analysis of Fiber Lay-Down Models : An Interplay between Pure and Applied Mathematics involving Langevin Processes on Manifolds, Ergodicity for Degenerate Kolmogorov Equations and Hypocoercivity [[Elektronische Ressource]] / Patrik Stilgenbauer." München : Verlag Dr. Hut, 2014. http://d-nb.info/1050331729/34.
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Brenner, Konstantin. "Méthodes de volumes finis sur maillages quelconques pour des systèmes d'évolution non linéaires." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00647336.
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Les travaux de cette thèse portent sur des méthodes de volumes finis sur maillages quelconque pour la discrétisation de problèmes d'évolution non linéaires modélisant le transport de contaminants en milieu poreux et les écoulements diphasiques.Au Chapitre 1, nous étudions une famille de schémas numériques pour la discrétisation d'une équation parabolique dégénérée de convection-reaction-diffusion modélisant le transport de contaminants dans un milieu poreux qui peut être hétérogène et anisotrope. La discrétisation du terme de diffusion est basée sur une famille de méthodes qui regroupe les schémas de volumes finis hybrides, de différences finies mimétiques et de volumes finis mixtes. Le terme de convection est traité à l'aide d'une famille de méthodes qui s'appuient sur les inconnues hybrides associées aux interfaces du maillage. Cette famille contient à la fois les schémas centré et amont. Les schémas que nous étudions permettent une discrétisation localement conservative des termes d'ordre un et d'ordre deux sur des maillages arbitraires en dimensions d'espace deux et trois. Nous démontrons qu'il existe une solution unique du problème discret qui converge vers la solution du problème continu et nous présentons des résultats numériques en dimensions d'espace deux et trois, en nous appuyant sur des maillages adaptatifs.Au Chapitre 2, nous proposons un schéma de volumes finis hybrides pour la discrétisation d'un problème d'écoulement diphasique incompressible et immiscible en milieu poreux. On suppose que ce problème a la forme d'une équation parabolique dégénérée de convection-diffusion en saturation couplée à une équation uniformément elliptique en pression. On considère un schéma implicite en temps, où les flux diffusifs sont discrétisés par la méthode des volumes finis hybride, ce qui permet de pouvoir traiter le cas d'un tenseur de perméabilité anisotrope et hétérogène sur un maillage très général, et l'on s'appuie sur un schéma de Godunov pour la discrétisation des flux convectifs, qui peuvent être non monotones et discontinus par rapport aux variables spatiales. On démontre l'existence d'une solution discrète, dont une sous-suite converge vers une solution faible du problème continu. On présente finalement des cas test bidimensionnels.Le Chapitre 3 porte sur un problème d'écoulement diphasique, dans lequel la courbe de pression capillaire admet des discontinuité spatiales. Plus précisément on suppose que l'écoulement prend place dans deux régions du sol aux propriétés très différentes, et l'on suppose que la loi de pression capillaire est discontinue en espace à la frontière entre les deux régions, si bien que la saturation de l'huile et la pression globale sont discontinues à travers cette frontière avec des conditions de raccord non linéaires à l'interface. On discrétise le problème à l'aide d'un schéma, qui coïncide avec un schéma de volumes finis standard dans chacune des deux régions, et on démontre la convergence d'une solution approchée vers une solution faible du problème continu. Les test numériques présentés à la fin du chapitre montrent que le schéma permet de reproduire le phénomène de piégeage de la phase huile.