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Готові списки джерел за темами / Degenerate elliptic operators

Добірка наукової літератури з теми "Degenerate elliptic operators"

Автор: Grafiati

Опубліковано: 7 липня 2024

Оновлено: 7 липня 2024

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Статті в журналах
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Книги
Частини книг

Статті в журналах з теми "Degenerate elliptic operators":

1

Shakhmurov, Veli B. "Degenerate Differential Operators with Parameters." Abstract and Applied Analysis 2007 (2007): 1–27. http://dx.doi.org/10.1155/2007/51410.

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The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valuedLp−spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

2

Duc, Duong Minh. "A class of strongly degenerate elliptic operators." Bulletin of the Australian Mathematical Society 39, no. 2 (April 1989): 177–200. http://dx.doi.org/10.1017/s0004972700002665.

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Using a weighted Poincaré inequality, we study (ω1,…,ωn)-elliptic operators. This method is applied to solve singular elliptic equations with boundary conditions in W1,2. We also obtain a result about the regularity of solutions of singular elliptic equations. An application to (ω1,…,ωn)-parabolic equations is given.

3

Igisinov, S. Zh, L. D. Zhumaliyeva, A. O. Suleimbekova, and Ye N. Bayandiyev. "Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 107, no. 3 (September 30, 2022): 51–58. http://dx.doi.org/10.31489/2022m3/51-58.

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In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.

4

Robinson, Derek W., and Adam Sikora. "L1-uniqueness of degenerate elliptic operators." Studia Mathematica 203, no. 1 (2011): 79–103. http://dx.doi.org/10.4064/sm203-1-5.

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5

Morimoto, Yoshinori. "Non-hypoellipticity for degenerate elliptic operators." Publications of the Research Institute for Mathematical Sciences 22, no. 1 (1986): 25–30. http://dx.doi.org/10.2977/prims/1195178369.

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6

Hua, Chen, and Chen Hongge. "Eigenvalue problem of degenerate elliptic operators." SCIENTIA SINICA Mathematica 51, no. 6 (March 8, 2021): 833. http://dx.doi.org/10.1360/ssm-2020-0219.

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7

Robinson, Derek W., and Adam Sikora. "Degenerate elliptic operators in one dimension." Journal of Evolution Equations 10, no. 4 (April 23, 2010): 731–59. http://dx.doi.org/10.1007/s00028-010-0068-9.

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8

Ouhabaz, El Maati, and Derek W. Robinson. "Uniqueness properties of degenerate elliptic operators." Journal of Evolution Equations 12, no. 3 (May 11, 2012): 647–73. http://dx.doi.org/10.1007/s00028-012-0148-0.

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9

Levendorskiĭ, S. Z. "ON TYPES OF DEGENERATE ELLIPTIC OPERATORS." Mathematics of the USSR-Sbornik 66, no. 2 (February 28, 1990): 523–40. http://dx.doi.org/10.1070/sm1990v066n02abeh001183.

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10

Morimoto, Yoshinori. "Estimates for degenerate Schrödinger operators and hypoellipticity for infinitely degenerate elliptic operators." Journal of Mathematics of Kyoto University 32, no. 2 (1992): 333–72. http://dx.doi.org/10.1215/kjm/1250519539.

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Більше джерел

Дисертації з теми "Degenerate elliptic operators":

1

Hakulinen, Ville. "Passive advection and the degenerate elliptic operators Mn." Helsinki : University of Helsinki, 2002. http://ethesis.helsinki.fi/julkaisut/mat/matem/vk/hakulinen/.

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2

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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Анотація:

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.

3

Monticelli, D. D. "Maximum principles and applications for a class of degenerate elliptic linear operators." Doctoral thesis, Università degli Studi di Milano, 2006. http://hdl.handle.net/2434/25003.

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In this thesis we deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypotheses on the principal part and on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on classical solutions of the Dirichlet problem for the linear equation. We also prove a Poincaré inequality, which allows us to define the functional setting where we study weak solutions for equations and inequalities involving this class of operators. Then we prove an existence and uniqueness result for weak solutions of the Dirichlet problem on bounded domains of R^n and a Weak Maximum Principle for weak solutions of differential inequalities,involving this class of operators. A good example of such an operator is the Grushin operator on R^(d+k), to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the pioneering result of Gidas-Ni-Nirenberg, and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the celebrated result of Gidas-Spruck and Chen-Li. The method we use to obtain these results is the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

4

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : II. Products in elliptic theory on manifolds with edges." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2633/.

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Exterior tensor products of elliptic operators on smooth manifolds and manifolds with conical singularities are used to obtain examples of elliptic operators on manifolds with edges that do not admit well-posed edge boundary and coboundary conditions.

5

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : I. The index of families of cone-degenerate operators." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2632/.

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We study the index problem for families of elliptic operators on manifolds with conical singularities. The relative index theorem concerning changes of the weight line is obtained. AN index theorem for families whose conormal symbols satisfy some symmetry conditions is derived.

6

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : IV. Obstructions to elliptic problems on manifolds with edges." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2641/.

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The obstruction to the existence of Fredholm problems for elliptic differentail operators on manifolds with edges is a topological invariant of the operator. We give an explicit general formula for this invariant. As an application we compute this obstruction for geometric operators.

7

Giunti, Arianna. "Green\'s function estimates for elliptic and parabolic operators: Applications to quantitative stochastic homogenization and invariance principles for degenerate random environments and interacting particle systems." Doctoral thesis, Universitätsbibliothek Leipzig, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-225533.

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This thesis is divided into two parts: In the first one (Chapters 1 and 2), we deal with problems arising from quantitative homogenization of the random elliptic operator in divergence form $-\\nabla \\cdot a \\nabla$. In Chapter 1 we study existence and stochastic bounds for the Green function $G$ associated to $-\\nabla \\cdot a \\nabla$ in the case of systems. Without assuming any regularity on the coefficient field $a= a(x)$, we prove that for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \\in \\mathbb^d$, there exists a unique Green\'s function centred in $y$ associated to the vectorial operator $-\\nabla \\cdot a\\nabla $ in $\\mathbb{R}^d$, $d> 2$. In addition, we prove that if we introduce a shift-invariant ensemble $\\langle\\cdot \\rangle$ over the set of uniformly elliptic tensor fields, then $\\nabla G$ and its mixed derivatives $\\nabla \\nabla G$ satisfy optimal pointwise $L^1$-bounds in probability. Chapter 2 deals with the homogenization of $-\\nabla \\cdot a \\nabla$ to $-\\nabla \\ah \\nabla$ in the sense that we study the large-scale behaviour of $a$-harmonic functions in exterior domains $\\{ |x| > r \\}$ by comparing them with functions which are $\\ah$-harmonic. More precisely, we make use of the first and second-order correctors to compare an $a$-harmonic function $u$ to the two-scale expansion of suitable $\\ah$-harmonic function $u_h$. We show that there is a direct correspondence between the rate of the sublinear growth of the correctors and the smallness of the relative homogenization error $u- u_h$. The theory of stochastic homogenization of elliptic operators admits an equivalent probabilistic counterpart, which follows from the link between parabolic equations with elliptic operators in divergence form and random walks. This allows to reformulate the problem of homogenization in terms of invariance principle for random walks. The second part of thesis (Chapters 3 and 4) focusses on this interplay between probabilistic and analytic approaches and aims at exploiting it to study invariance principles in the case of degenerate random conductance models and systems of interacting particles. In Chapter 3 we study a random conductance model where we assume that the conductances are independent, stationary and bounded from above but not uniformly away from $0$. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector which imply that the discrete elliptic operator homogenises or, equivalently, that the random conductance model satisfies a quenched invariance principle. In Chapter 4 we turn to a more complicated model, namely the symmetric exclusion process. We show a diffusive upper bound on the transition probability of a tagged particle in this process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

8

Giunti, Arianna [Verfasser], Felix [Gutachter] Otto, and Antoine [Gutachter] Gloria. "Green\'s function estimates for elliptic and parabolic operators: Applications to quantitative stochastic homogenization and invariance principles for degenerate random environments and interacting particle systems : Green\''s function estimates for elliptic and parabolic operators:Applications to quantitative stochastic homogenization andinvariance principles for degenerate random environments andinteracting particle systems / Arianna Giunti ; Gutachter: Felix Otto, Antoine Gloria." Leipzig : Universitätsbibliothek Leipzig, 2017. http://d-nb.info/1241064598/34.

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9

Perstneva, Polina. "Elliptic measure in domains with boundaries of codimension different from 1." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASM037.

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Cette thèse étudie différentes variantes de la mesure harmonique et leurs relations avec la géométrie de la frontière d'un domaine. Dans la première partie de la thèse, on se concentre sur l'analogue de la mesure harmonique pour les domaines ayant des frontières de dimensions plus petites, définies via la théorie des opérateurs elliptiques dégénérés récemment développée par David et al. Plus précisément, on démontre qu'il n'existe pas de famille à un paramètre non dégénérée de solutions de l'équation LμDμ = 0, ce qui constitue la première étape pour retrouver une forme de l'assertion "si la fonction de distance à la frontière d'un domaine est harmonique, alors la frontière est plate", qui manque à la théorie des opérateurs elliptiques dégénérés. On découvre et explique également pourquoi la stratégie la plus naturelle pour étendre notre résultat à l'absence de solutions individuelles de l'équation LμDμ = 0 ne fonctionne pas. Dans la deuxième partie de la thèse, on s'intéresse aux mesures elliptiques dans le cadre classique. On construit une nouvelle famille d'opérateurs avec des coefficients continus scalaires dont les mesures elliptiques sont absolument continues par rapport aux mesures de Hausdorff sur des flocons de neige symétriques de type Koch. Cette famille enrichit la collection des exemples connus de mesures elliptiques qui se comportent très différemment de la mesure harmonique et des mesures elliptiques d'opérateurs proches, d'une certaine manière, du Laplacien. De plus, nos nouveaux exemples ne sont pas compacts. La construction fournit également une méthode possible pour construire des opérateurs ayant ce type de comportement pour d'autres fractales qui possèdent suffisamment de symétries
This thesis studies different counterparts of the harmonic measure and their relations with the geometry of the boundary of a domain. In the first part of the thesis, we focus on the analogue of harmonic measure for domains with boundaries of smaller dimensions, defined via the theory of degenerate elliptic operators developed recently by David et al. More precisely, we prove that there is no non-degenerate one-parameter family of solutions to the equation LμDμ = 0, which constitutes the first step to recover an analogue of the statement ``if the distance function to the boundary of a domain is harmonic, then the boundary is flat'', missing from the theory of degenerate elliptic operators. We also find out and explain why the most natural strategy to extend our result to the absence of individual solutions to the equation LμDμ = 0 does not work. In the second part of the thesis, we focus on elliptic measures in the classical setting. We construct a new family of operators with scalar continuous coefficients whose elliptic measures are absolutely continuous with respect to the Hausdorff measures on Koch-type symmetric snowflakes. This family enriches the collection of a few known examples of elliptic measures which behave very differently from the harmonic measure and the elliptic measures of operators close in some sense to the Laplacian. Plus, our new examples are non-compact. Our construction also provides a possible method to construct operators with this type of behaviour for other fractals that possess enough symmetries

10

Shen, Lin-hong, and 沈林弘. "Homogenization of some special degenerate second order linear elliptic operators and its numerical computation." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/09316679482569904004.

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碩士
國立清華大學
數學系
103
Abstract Homogenization of some special degenerate second order linear elliptic operators and its numerical computation Lin-Hong Shen, Avisor:Assistant Professor Chia-Chieh Chu Department of Mathematics National Tsing Hua University, Hsin-Chu City,Taiwan In many area, homogenization is an alternative way to find out the asymptotic behaviour of partial differential equation. This arti- cle is about homogenization process of degenerate second order linear elliptic operators. In this article, we give both theoretical and com- putational analysis to the asymptotic behaviour of the solution of the equation. −div(a( x )Duh) = f on Ω , uh |∂Ω= 0 on ∂Ω , when Eh tends to zero, where aij (x) is Y -periodic, nonnegative defi- nite for almost every x in domain Ω and vanishes at some points in Ω. We find out that the homogenization process of degenerate ellip- tic equation in rectangle domain is still available for some particular coefficient functions with its inverse is integrable Key words: homogenization, degenerate elliptic equation, asymp- totic behaviour, numerical analysis

Книги з теми "Degenerate elliptic operators":

1

Popivanov, Peter R. The degenerate oblique derivative problem for elliptic and parabolic equations. Berlin: Akademie Verlag, 1997.

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2

Cannarsa, Piermarco. Global Carleman estimates for degenerate parabolic operators with applications. Providence, Rhode Island: American Mathematical Society, 2016.

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3

Workshop in Nonlinear Elliptic Partial Differential Equations (2009 Université libre de Bruxelles). Nonlinear elliptic partial differential equations: Workshop in celebration of Jean-Pierre Gossez's 65th birthday, September 2-4, 2009, Université libre de Bruxelles, Belgium. Edited by Gossez J. P. 1943- and Bonheure Denis. Providence, R.I: American Mathematical Society, 2011.

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4

Epstein, Charles L., and Rafe Mazzeo. Degenerate Diffusion Operators Arising in Population Biology (AM-185). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.001.0001.

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This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

5

Epstein, Charles L. Degenerate diffusion operators arising in population biology. 2013.

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6

Krylov, Nikolai. Probabilistic methods of investigating interior smoothness of harmonic functions associated with degenerate elliptic operators. Edizioni della Normale, 2007.

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7

Epstein, Charles L., and Rafe Mazzeo. Introduction. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0001.

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This book proves the existence, uniqueness and regularity results for a class of degenerate elliptic operators known as generalized Kimura diffusions, which act on functions defined on manifolds with corners. It presents a generalization of the Hopf boundary point maximum principle that demonstrates, in the general case, how regularity implies uniqueness. The book is divided in three parts. Part I deals with Wright–Fisher geometry and the maximum principle; Part II is devoted to an analysis of model problems, and includes degenerate Hölder spaces; and Part III discusses generalized Kimura diffusions. This introductory chapter provides an overview of generalized Kimura diffusions and their applications in probability theory, model problems, perturbation theory, main results, and alternate approaches to the study of similar degenerate elliptic and parabolic equations.

Частини книг з теми "Degenerate elliptic operators":

1

Levendorskii, Serge. "General Calculus of Pseudodifferential Operators." In Degenerate Elliptic Equations, 9–73. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_2.

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2

Levendorskii, Serge. "Spectral Asymptotics of Degenerate Elliptic Operators." In Degenerate Elliptic Equations, 301–34. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_11.

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3

Levendorskii, Serge. "Model Classes of Degenerate Elliptic Differential Operators." In Degenerate Elliptic Equations, 75–127. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_3.

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4

Levendorskii, Serge. "General Classes of Degenerate Elliptic Differential Operators." In Degenerate Elliptic Equations, 129–62. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_4.

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5

Levendorskii, Serge. "L p — Theory for Degenerate Elliptic Operators." In Degenerate Elliptic Equations, 171–85. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_6.

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6

Levendorskii, Serge. "Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics." In Degenerate Elliptic Equations, 335–87. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_12.

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7

Levendorskii, Serge. "Degenerate Elliptic Operators in Non — Power — Like Degeneration Case." In Degenerate Elliptic Equations, 163–70. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_5.

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8

Levendorskii, Serge. "Some Classes of Hypoelliptic Pseudodifferential Operators on Closed Manifold." In Degenerate Elliptic Equations, 203–44. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_8.

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9

Kohn, J. J. "Lectures on Degenerate Elliptic Problems." In Pseudodifferential Operators with Applications, 89–151. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11092-4_5.

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10

Levendorskii, Serge. "Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary." In Degenerate Elliptic Equations, 245–78. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-1215-6_9.

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