Dissertations / Theses: 'Sobolev spaces'

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Clemens, Jason. "Sobolev spaces." Kansas State University, 2014. http://hdl.handle.net/2097/18186.

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Master of Science
Department of Mathematics
Marianne Korten
The goal for this paper is to present material from Gilbarg and Trudinger’s Elliptic Partial Differential Equations of Second Order chapter 7 on Sobolev spaces, in a manner easily accessible to a beginning graduate student. The properties of weak derivatives and there relationship to conventional concepts from calculus are the main focus, that is when do weak and strong derivatives coincide. To enable the progression into the primary focus, the process of mollification is presented and is widely used in estimations. Imbedding theorems and compactness results are briefly covered in the final sections. Finally, we add some exercises at the end to illustrate the use of the ideas presented throughout the paper.

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Gjestland, Fredrik Joachim. "Distributions, Schwartz Space and Fractional Sobolev Spaces." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-23452.

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This thesis derives the theory of distributions, starting with test functions as a basis. Distributions and their derivatives will be analysed and exemplified. Schwartz functions are introduced, and the Fourier transform of Schwartz functions is analysed, creating the basis for Tempered distributions on which we also analyse the Fourier transform. Weak derivatives and Sobolev spaces are defined, and from the Fourier transform we define Sobolev spaces of non-integer order. The theory presented is applied to an initial value problem with a derivative of order one in time and an arbitrary differentiation operator in space, and we take a look at conditions for well-posedness under different differnetiation operators and present some minor results. The Riesz representation theorem and the Lax--Milgram theorem are presented in order to offer a different perspective on the results from the initial value problem.

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Kydyrmina, Nurgul. "Operators in Sobolev Morrey spaces." Doctoral thesis, Università degli studi di Padova, 2013. http://hdl.handle.net/11577/3423455.

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Morrey spaces were introduced by Charles Morrey in 1938. They are a useful tool in the regularity theory of partial differential equations, in real analysis and in mathematical physics. In the nineties of the XX century an active study of general Morrey-type spaces characterized by a functional parameter has started to develop. A number of results on boundedness of classical operators in general Morrey-type spaces were obtained. At the beginning of the XXI century there were new active developments in this area. In the last decade many mathematicians do research on smoothness spaces related to Morrey spaces. Among these spaces the Sobolev-type spaces play an important role. In the thesis Sobolev spaces built on Morrey spaces are studied, which are also referred to as Sobolev Morrey spaces. These are spaces of functions which have derivatives up to certain order in Morrey spaces. We analyze some basic properties of Morrey spaces and of Sobolev Morrey spaces. Then we consider the embedding and multiplication operators in Sobolev Morrey spaces. Finally, the dissertation provides a study of the composition operator in Sobolev Morrey spaces. The results presented in the thesis have been obtained under supervision of Professors V.I. Burenkov and M. Lanza de Cristoforis.
Gli spazi di Morrey sono stati introdotti da Charles Morrey nel 1938. Essi sono uno strumento utile nella teoria della regolarità per equazioni differenziali alle derivate parziali, in analisi reale ed in fisica matematica. Negli anni novanta del XX secolo ha iniziato a svilupparsi un attivo studio degli spazi di Morrey di tipo generalizzato che sono caratterizzati da un parametro funzionale. E' stato ottenuto un cero numero di risultati sulla limitatezza degli operatori classici negli spazi di Morrey di tipo generalizzato. All'inizio del XXI secolo ci sono stati nuovi e attivi sviluppi in questa area. Nell'ultima decade molti matematici hanno svolto ricerche su spazi funzionali relativi agli spazi di Morrey. Tra questi spazi gli spazi di tipo Sobolev giocano un ruolo importante. Nella tesi si studiano Spazi di Sobolev costruiti su spazi di Morrey, anche detti spazi di Sobolev Morrey. Questi sono spazi di funzioni che hanno derivate fino ad un certo ordine negli spazi di Morrey. Si analizzano alcune proprietà di base degli spazi di Morrey e degli spazi di Sobolev-Morrey. Poi si considerano operatori di immersione e di moltiplicazione negli spazi di Sobolev Morrey. La terza parte della tesi presenta uno studio degli operatori di composizione negli spazi di Sobolev Morrey. I risultati presentati nella tesi sono stati ottenuti sotto la supervisione dei Professori V.I. Burenkov and M. Lanza de Cristoforis.

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4

Färm, David. "Upper gradients and Sobolev spaces on metric spaces." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5816.

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The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.

All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.

Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.

This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.

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Park, Young Ja. "Sobolev trace inequality and logarithmic Sobolev trace inequality." Digital version:, 2000. http://wwwlib.umi.com/cr/utexas/fullcit?p9992883.

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6

Spector, Daniel. "Characterization of Sobolev and BV Spaces." Research Showcase @ CMU, 2011. http://repository.cmu.edu/dissertations/78.

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This work presents some new characterizations of Sobolev spaces and the space of functions of Bounded Variation. Additionally it gives new proofs of continuity and lower semicontinuity theorems due to Reshetnyak.

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Neves, Julio Severino. "Fractional Sobolev-type spaces and embeddings." Thesis, University of Sussex, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341514.

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Dines, Nicoleta, Gohar Harutjunjan, and Bert-Wolfgang Schulze. "The Zaremba problem in edge Sobolev spaces." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2661/.

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Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator.

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Davidsson, Johan. "Sobolev Spaces and the Finite Element Method." Thesis, Örebro universitet, Institutionen för naturvetenskap och teknik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:oru:diva-67470.

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In this essay we present the Sobolev spaces and some basic properties of them. The Sobolev spaces serve as a theoretical framework for studying solutions to partial differential equations. The finite element method is presented which is a numerical method for solving partial differential equations.

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Clavero, Nadia F. "Optimal Sobolev Embeddings in Spaces with Mixed Norm." Doctoral thesis, Universitat de Barcelona, 2015. http://hdl.handle.net/10803/292613.

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Este proyecto hace referencia a estimaciones, en espacios funcionales, que relacionan la norma de una función y la de sus derivadas. Concretamente, nuestro principal objetivo es estudiar las estimaciones clásicas de las inclusiones de Sobolev, probadas por Gagliardo y Nirenberg, para derivadas de orden superior y espacios más generales. En particular, estamos interesados en describir el dominio y el rango óptimos para estas inclusiones entre los espacios invariantes por reordenamiento (r.i.) y espacios de normas mixtas.
This thesis project concerns estimates, in function spaces, that relate the norm of a function and that of its derivatives. Speci.cally, our main purpose is to study the classical Sobolev-type inequalities due to Gagliardo and Nirenberg for higher order derivatives and more general spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between rearrangement-invariant spaces (r.i.) and mixed norm spaces.

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11

Flucher, Martin. "Concentration and compactness of functionals on Sobolev spaces /." [S.l.] : [s.n.], 1991. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=9525.

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Mekias, Mohamed. "Restriction to hypersurfaces of non-isotropic Sobolev spaces." Thesis, Massachusetts Institute of Technology, 1993. http://hdl.handle.net/1721.1/100868.

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Airapetyan, Ruben, and Ingo Witt. "Isometric properties of the Hankel Transformation in weighted sobolev spaces." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2500/.

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It is shown that the Hankel transformation Hsub(v) acts in a class of weighted Sobolev spaces. Especially, the isometric mapping property of Hsub(v) which holds on L²(IRsub(+),rdr) is extended to spaces of arbitrary Sobolev order. The novelty in the approach consists in using techniques developed by B.-W. Schulze and others to treat the half-line Rsub(+) as a manifold with a conical singularity at r = 0. This is achieved by pointing out a connection between the Hankel transformation and the Mellin transformation.The procedure proposed leads at the same time to a short proof of the Hankel inversion formula. An application to the existence and higher regularity of solutions, including their asymptotics, to the 1-1-dimensional edge-degenerated wave equation is given.

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Salvato, Maria. "On the solvability of linear PDEs in weighted Sobolev spaces." Doctoral thesis, Universita degli studi di Salerno, 2012. http://hdl.handle.net/10556/335.

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Stefani, Giorgio. "A distributional approach to fractional Sobolev spaces and fractional variation." Doctoral thesis, Scuola Normale Superiore, 2020. http://hdl.handle.net/11384/85724.

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In this thesis, we present the distributional approach to fractional Sobolev spaces and fractional variation developed in [20, 22, 23]. The new space BVᵅ(ℝⁿ) of functions with bounded fractional variation in ℝⁿ of order α ∈ (0, 1) is distributionally defined by exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical BV theory, we give a new notion of set E of (locally) finite fractional Caccioppoli α-perimeter and we define its fractional reduced boundary FᵅE. We are able to show that Wᵅ,¹(ℝⁿ) ⊂ BV ᵅ(ℝⁿ) continuously and, similarly, that sets with (locally) finite standard fractional α-perimeter have (locally) finite fractional Caccioppoli α-perimeter, so that our theory provides a natural extension of the known fractional framework. We first extend De Giorgi’s Blow-up Theorem to sets of locally finite fractional Caccioppoli α-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets. We then prove that the fractional α-variation converges to the standard De Giorgi’s variation both pointwise and in the Γ-limit sense as α → 1- and, similarly, that the fractional β-variation converges to the fractional α-variation both pointwise and in the Γ-limit sense as β → α- for any given α ∈ (0, 1). Finally, by exploiting some new interpolation inequalities on the fractional operators involved, we prove that the fractional α-gradient converges to the Riesz transform as α → 0⁺ in Lp for p ∈ (1,+∞) and in the Hardy space and that the α-rescaled fractional α-variation converges to the integral mean of the function as α → 0⁺.

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Inahama, Yuzuru. "Logarithmic Sobolev Inequality on Free Loop Groups for Heat Ker-nel Measures Associated with the General Sobolev Spaces." 京都大学 (Kyoto University), 2001. http://hdl.handle.net/2433/150808.

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17

Döhring, Nicolas [Verfasser]. "Regularity of Random Fields in Scales of Besov Spaces and Generalized Sobolev Spaces / Nicolas Döhring." München : Verlag Dr. Hut, 2016. http://d-nb.info/1084386887/34.

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Blair, Matthew D. "Strichartz estimates for wave equations with coefficients of Sobolev regularity /." Thesis, Connect to this title online; UW restricted, 2005. http://hdl.handle.net/1773/5745.

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Georgoulis, Emmanuil H. "Discontinuous Galerkin methods on shape-regular and anisotropic meshes." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.270366.

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Prats, Soler Martí. "Singular integral operators on sobolev spaces on domains and quasiconformal mappings." Doctoral thesis, Universitat Autònoma de Barcelona, 2015. http://hdl.handle.net/10803/314193.

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En aquesta tesi s’obtenen nous resultats sobre l’acotació d’operadors de Calderón-Zygmund en espais de Sobolev en dominis de Rd. En primer lloc es demostra un teorema de tipus T(P) vàlid per a Wn,p(U), a on U és un domini uniforme acotat de Rd, n és un nombre natural arbitrari, i p>d. Essencialment, el resultat obtingut afirma que un operador de Calderón-Zygmund de convolució és acotat en aquest espai si i solament si per a tot polinomi P de grau menor que n restringit al domini, T(P) pertany a Wn,p(U). Per a índexs p menors o iguals que d, es demostra una condició suficient per a l'acotació en termes de mesures de Carleson. En el cas n=1 i p<=d, es comprova que aquesta caracterització en termes de mesures de Carleson és també una condició necessària. El cas en què n és no enter i 02. La darrera aportació de la tesi és l'aplicació dels resultats anteriorment descrits a l'estudi de la regularitat de l'equació de Beltrami que satisfan les aplicacions quasiconformes. Essencialment, es demostra que si el coeficient de Beltrami pertany a l'espai Wn,p(U), essent U un domini Lipschitz del pla complex amb parametritzacions de la frontera en un cert espai de Besov i p>2, llavors l'aplicació quasiconforme associada està en l'espai Wn,p(U).
In this dissertation some new results on the boundedness of Calderón-Zygmund operators on Sobolev spaces on domains in Rd. First a T(P)-theorem is obtained which is valid for Wn,p (U), where U is a bounded uniform domain of Rd, n is a given natural number and p>d. Essentially, the result obtained states that a convolution Calderón-Zygmund operator is bounded on this function space if and only if T(P) belongs to Wn,p (U) for every polynomial P of degree smaller than n restricted to the domain. For indices p less or equal than d, a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of n=1 and p<=d, this Carleson condition is shown to be necessary in fact. The case where n is not integer and 02. Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity Wn,p(U) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain U with boundary parameterizations in a certain Besov space and p>2, implies that the mapping itself is in Wn+1,p(U).

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21

Schrohe, Elmar, and Jörg Seiler. "Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces." Universität Potsdam, 1999. http://opus.kobv.de/ubp/volltexte/2008/2562/.

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Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces, it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on Lp(B).

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Malý, Lukáš. "Sobolev-Type Spaces : Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces." Doctoral thesis, Linköpings universitet, Matematik och tillämpad matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-105616.

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This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike $\mathcal{C}^1$ -smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.

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23

Buffa, Vito. "Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amslaurea.unibo.it/6893/.

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The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

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Silvestre, Albero María Pilar. "Capacitary function spaces and applications." Doctoral thesis, Universitat de Barcelona, 2012. http://hdl.handle.net/10803/77717.

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The first part of the thesis is devoted to the analysis on a capacity space, with capacities as substitutes of measures in the study of function spaces. The goal is to extend to the associated function lattices some aspects of the theory of Banach function spaces, to show how the general theory can be applied to classical function spaces such as Lorentz spaces, and to complete the real interpolation theory for these spaces included in [CeClM] and [Ce]. In the second part of the thesis, we present an integral inequality connecting a function space norm of the gradient of a function to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the estimates obtained by V. Maz’ya and S. Costea, and sharp capacitary inequalities due to V. Maz’ya in the case of the Sobolev norm. The inequality, obtained under appropriate convexity conditions on the function space, gives a characterization of Sobolev type inequalities involving two measures, necessary and sufficient conditions for Sobolev isocapacitary type inequalities, and self-improvements for integrability of Lipschitz functions.
La primera part està dedicada a l’anàlisi d’un espai de capacitat, amb capacitats com a substituts de les mesures en l’estudi d’espais de funcions. L’objectiu és estendre als recicles de funcions associats alguns aspectes de la la teoria d’espais de funcions de Banach, mostrar com la teoria general pot ser aplicada a espais funcionals clàssics com els espais de Lorentz, i completar la teoria d’interpolació real d’aquests espais inclosos en [CeClM] i [Ce]. A la segona part de la tesi es presenta una desigualtat integral que connecta la norma del gradient d’una funció en un espai de funcions amb la integral de la corresponent capacitat del conductor entre dues superfícies de nivell de la funció, que estén les estimacions obtingudes per V. Maz’ya i S. Costea, i desigualtats capacitàries fortes de V. Maz’ya en el cas de la norma de Sobolev. La desigualtat, obtinguda sota condicions de convexitat pel espai funcional, permet una caracterització de les desigualtats de tipus Sobolev per dues mesures, condicions necessàries i suficients per desigualtats isocapacitàries de tipus Sobolev, i la millora de l’autointegrabilitat de les funcions de Lipschitz.

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Khanfar, Abeer. "Multiple Solutions on a Ball for a Generalized Lane Emden Equation." ScholarWorks@UNO, 2008. http://scholarworks.uno.edu/td/901.

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In this work we study the Generalized Lane-Emden equation and the interplay between the exponents involved and their consequences on the existence and non existence of radial solutions on a unit ball in n dimensions. We extend the analysis to the phase plane for a clear understanding of the behavior of solutions and the relationship between their existence and the growth of nonlinear terms, where we investigate the critical exponent p and a sub-critical exponent, which we refer to as ^p. We discover a structural change of solutions due the existence of this sub-critical exponent which we relate to the same change in behavior of the Lane- Emden equation solutions, for ; = 0; andp = 2, due to the same sub-critical exponent. We hypothesize that this sub-critical exponent may be related to a weighted trace embedding.

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Ortiz, Vargas Walter Andrés. "Sobolev inequalities: isoperimetry and symmetrization." Doctoral thesis, Universitat Autònoma de Barcelona, 2019. http://hdl.handle.net/10803/669904.

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MENEGATTI, GIORGIO. "Sobolev classes and bounded variation functions on domains of Wiener spaces, and applications." Doctoral thesis, Università degli studi di Ferrara, 2018. http://hdl.handle.net/11392/2488305.

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The main thread of this work is the bounded variation (BV) functions in abstract Wiener spaces (a topic in infinite-dimensional analysis). In the first Part of this work, we present some known results, and we introduce the concepts of Wiener space, of Sobolev space in Wiener spaces, of BV functions (and finite perimeter sets) in Wiener spaces, and of BV functions in convex sets of Wiener spaces (by following the definition in V. I. Bogachev, A. Y. Pilipenko, A. V. Shaposhnikov, “Sobolev Functions on Infinite-dimensional domains”, J. Math. Anal. Appl., 2014); moreover, we introduce the trace theory on subsets of a Wiener space (by following P. Celada, A. Lunardi, “Traces of Sobolev functions on regular surfaces in infinite dimensions”, J. Funct. Anal., 2014), and the concept of Mosco convergence. In the second Part we present some new results. In Chapter 6, we consider a subset O of a Wiener space which satisfies a regularity condition, and we prove that a function in W^{1,2}(O) has null trace if and only if it is the limit of a sequence of functions with support contained in O. The main chapter is Chapter 7, which is devoted to the extension in the Wiener spaces setting of a result given in the section 8 of (V. Barbu, M. Röckner, “Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise”, Arch. Ration. Mech. Anal., 2013): if O is a convex bounded set with regular boundary in R^{d} and L is the Laplace operator in O with null Dirichlet boundary condition, then the normalized resolvent of L is contractive in sense L^1 respect to the gradient. We extend this result to the case of L Ornstein-Uhlenbeck operator in O with null Dirichlet boundary condition, with Gaussian measure (by using the results of Chapter 6): in this case O must satisfy a condition (which we call Gaussian convexity) which takes the place of the convexity in the Gaussian setting. Moreover, we extend the result also to the case of: L Laplace operator in an open convex O with null Neumann boundary condition, with Lebesgue measure; L Ornstein-Uhlenbeck operator in an open convex O with null Neumann boundary condition, with Gaussian measure. In the last part of Chapter 7, we use the preceding results to give an alternative definition of BV function (in the case L^2(O)). In Chapter 8, let X the set of continuous functions on [0,1] with starting point 0, provided with the measure induced by the Brownian motion with starting point 0; it is a Wiener space. For every A subset of X, we define Ξ_A, set of functions in X with image in A. In (M. Hino, H. Uchida, “Reflecting Ornstein–Uhlenbeck processes on pinned path spaces”, Res. Inst. Math. Sci. (RIMS), 2008) it is proved that, if d ≥ 2 and A is an open subset of R^d which satisfies an uniform outer ball condition then Ξ_A has finite perimeter in the sense of Gaussian measure. We present a weaker condition on A (in dimension sufficiently great) such that Ξ_A has finite perimeter: in particular, A can be the complement of a convex unbounded symmetric cone.
L’argomento principale di questo lavoro sono le funzioni a variazione limitata (BV) in spazi di Wiener astratti (un argomento di analisi infinito-dimensionale). Nella prima parte di questo lavoro, presentiamo alcuni risultati noti, e introduciamo i concetti di spazi di Wiener, di classi di Sobolev su spazi di Wiener, di funzioni BV (e insiemi di perimetro finito) in spazi di Wiener, e di funzioni BV in sottoinsiemi convessi di Spazi di Wiener (seguendo la definizione in V. I. Bogachev, A. Y. Pilipenko, A. V. Shaposhnikov, “Sobolev Functions on Infinite-dimensional domains”, J. Math. Anal. Appl., 2014); inoltre, introduciamo la teoria delle tracce su sottoinsiemi di uno spazio di Wiener( seguendo P. Celada, A. Lunardi, “Traces of Sobolev functions on regular surfaces in infinite dimensions”, J. Funct. Anal., 2014), e il concetto di convergenza di Mosco. Nella seconda parte presentiamo alcuni risultati originali. Nel capitolo 6, consideriamo un sottoinsieme O di uno spazio di Wiener che soddisfa a una condizione di regolarità, e proviamo che una funzione in W^{1,2} (O) ha traccia nulla se e solo se è il limite di una sequenza di funzioni con supporto contenuto in O. Il capitolo principale è il 7, che è dedicato all'estensione all'ambito degli spazi di Wiener di un risultato dato nella sezione 8 di (V. Barbu, M. Röckner, “Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise”, Arch. Ration. Mech. Anal., 2013): se O è un insieme convesso limitato con frontiera regolare in R^{d} e L è l'operatore di Laplace in O con condizione al bordo di Dirichlet nulla, allora il risolvente normalizzato di L è contrattivo nel senso L^1 rispetto al gradiente. Estendiamo questo risultato al caso di L operatore di Ornstein-Uhlenbeck in O con condizione al bordo di Dirichlet nulla, con misura gaussiana (usando i risultati del Capitolo 6): in questo caso O deve soddisfare una condizione (che chiamiamo convessità Gaussiana) che nel caso gaussiano prende il posto della convessità. Inoltre, estendiamo il risultato anche al caso di: L operatore di Laplace in un insieme aperto e convesso O con condizione al bordo di Neumann nulla, con misura di Lebesgue; L operatore in un insieme aperto e convesso O con condizione al bordo di Neumann nulla, con misura gaussiana. Nell'ultima parte del Capitolo 7, usiamo i precedenti risultati per dare una definizione alternativa di funzione BV in O (nel caso L^2(O) ). Nel Capitolo 8, sia X l'insieme delle funzioni continue in R^d su [ 0,1 ] con punti di partenza nell’origine fornito della misura indotta dal moto browniano con punto di partenza nell’origine; è uno spazio di Wiener. Per ogni A sottoinsieme di X, definiamo Ξ_A, insieme delle funzioni in X con immagine in A. In (M. Hino, H. Uchida, “Reflecting Ornstein–Uhlenbeck processes on pinned path spaces”, Res. Inst. Math. Sci. (RIMS), 2008) viene dimostrato che, se d ≥ 2 e A è un insieme aperto in R^d che soddisfa una condizione di uniforme palla esterna, allora Ξ_A ha perimetro finito nel senso della misura gaussiana. Presentiamo una condizione più debole su A (in dimensione sufficientemente grande) tale che Ξ_A ha perimetro finito: in particolare, A può essere il complementare di un cono convesso illimitato simmetrico.

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Arden, Greg. "Approximation properties of subdivision surfaces /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/5765.

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Santiago, Landerson Bezerra. "O nÃcleo do calor em uma variedade riemanniana." Universidade Federal do CearÃ, 2011. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5674.

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Em uma variedade riemanniana conexa e compacta introduziremos o conceito de espectro do operador laplaciano. Utilizando a existÃncia e a unicidade do nÃcleo do calor em uma variedade riemanniana,provaremos o teorema de decomposiÃÃo de Hodge. Este teorema afirma que o espaÃo de Hilbert L2(M, g) se decompÃe em uma soma direta de subespaÃos de dimensÃo finita, onde cada subespaÃo Ã o auto-espaÃo associado a um autovalor do laplaciano. AlÃm disso, os autovalores formam uma sequÃncia nÃo-negativa que acumula somente no infinito. Em seguida iniciaremos a construÃÃo do nÃcleo do calor e, por fim, mostraremos que se duas variedades riemannianas sÃo isospectrais entÃo elas possuem o mesmo volume.
In a connected and compact Riemannian Manifold we will introduce the concept of spectre of Laplace operator. Using the existence and unicity of the heat kernel in Riemannian manifold we proof the Hodge composition theorem. This theorem states that the Hilbert space L2(M, g) decompose in direct sum of subspaces with finite dimesion, where each subspace is the eigen-space relative of a eigenvalue of the laplacian. Furthermore, the eigenvalues form a nonnegative sequence the accumulate only in the infinity. After that we begin the construction of the heat kernel and, finally, we show that two isospetral Riemannian manifolds have the same volume.

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30

Mahavier, William Ted. "A Numerical Method for Solving Singular Differential Equations Utilizing Steepest Descent in Weighted Sobolev Spaces." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc278653/.

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We develop a numerical method for solving singular differential equations and demonstrate the method on a variety of singular problems including first order ordinary differential equations, second order ordinary differential equations which have variational principles, and one partial differential equation.

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Tami, Abdelkader. "Etude d'un problème pour le bilaplacien dans une famille d'ouverts du plan." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4362/document.

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L’objet de cette thèse est l’étude du problème Δ 2uω = fω avec les conditions aux limites Uω = Δ uω = 0, le second membre étant supposé dépendre continûment de ω dans L2(ω), où ω = {(r, θ); 0 < r < 1, 0 < θ < ω} , 0 < ω ≤ π, est une famille de secteurs tronqués du plan. Si ω < π on sait d’après Blum et Rannacher (1980) que la solution de ce problème uω se décompose au voisinage de l’origine en uω = u1,ω + u2,ω + u3,ω, (1) où u1,ω, u2,ω sont les parties singulières de uω et u3,ω la partie régulière. En effet, au voisinage de l’origine u1,ω (resp. u2,ω, u3,ω) est de régularité H1+πω−ǫ (resp. H2+πω−ǫ, H4) pour tout Q > 0, tandis que la solution uπ appartient, au moins au voisinage de l’origine, à l’espace H4(π), où π est le demi-disque supérieur de centre 0 et de rayon r = 1. On voit clairement une résolution de la singularité près de l’angle π dont la description est l’objectif principal de ce travail. Le résultat obtenu est que la décomposition (1) de uω est uniforme par rapport à ω, lorsque ω → π, pour les meilleures topologies possibles pour chacun des termes, et converge terme à terme vers le développement limité de uπ au voisinage de 0
In this work, we study the family of problems Δ 2uω = fω with boundary conditionuω = Δ uω = 0. There, the second member is assumed to depend smoothly on ω in L2(ω), where ω = {(r, θ); 0 < r < 1, 0 < θ < ω} , 0 < ω ≤ π, is a family of truncated sectors of the plane. If ω < π it is known from Blum et Rannacher (1980) that the solution uω decomposes as uω = u1,ω + u2,ω + u3,ω, (1) where u1,ω, u2,ω are singular and u3,ω is regular. Indeed, near the origin, u1,ω(resp. u2,ω, u3,ω) is of regularity H1+πω−ǫ (resp. H2+πω−ǫ, H4) for every Q > 0, while the solution uπ is, in the neighborhood of the origin again, of regularity H4. One clearly sees a resolution of the singularity near the angle π whose descriptionis the main objective of this work. The obtained result is that there exists a decomposition (1) of uω which is uniform with respect to ω, when ω → π, with the best possible topologies for each term, and which term by term convergestowards the Taylor expansion of uπ near 0

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Oliveira, Andrielber da Silva. "Metodos de interpolação real e espaços de Sobolev e Besov sobre a esfera Sd." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306560.

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Orientador: Sergio Antonio Tozoni
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-06T13:07:08Z (GMT). No. of bitstreams: 1 Oliveira_AndrielberdaSilva_M.pdf: 1284065 bytes, checksum: 0117263cf98921db674e49f5f57d460d (MD5) Previous issue date: 2006
Resumo: O objetivo da dissertação é realizar um estudo dos espaços de Besov sobre a esfera unitária d-dimensional real Sd. No primeiro capítulo são estudados espaços de interpolação utilizando dois métodos de interpolação real. Em particular são estudados os Teoremas de Equivalência e de Reiteração para os J-método e K-método. No segundo capítulo é realizado um estudo rápido sobre análise harmônica na esfera Sd, incluindo um estudo sobre harmônicos esféricos, harmônicos zonais, somas de Cesàro e sobre um teorema de multiplicadores. O terceiro e último capítulo é o mais importante e nele são aplicados os resultados dos capítulos anteriores. São introduzidos os espaços de Besov, decompondo uma função suave definida sobre a esfera d-dimensional, em uma série de harmônicos esféricos e usando uma seqüência de polinômios zonais que podem ser vistos como uma generalização natural dos polinômios de Vallée Poussin definidos sobre o círculo unitário. O principal resultado estudado diz que todo espaço de Besov pode ser obtido como espaço de interpolação de dois espaços de Sobolev
Abstract: The purpose of this work is to make a study about Besov¿s spaces on the unit d-dimensional real sphere Sd. In the first chapter are studied spaces of interpolation using two real interpolation methods. In particular, are studied The Equivalence Theorem and The Reiteration Theorem for the J-method and the K-method. In the second chapter it is made a short study about harmonic analysis on the sphere Sd, including a study about spherics harmonics, zonal harmonics, Cesàro sums and about a multiplier theorem. The third and last chapter is the most important of this work. In this chapter are applied the results of the others chapters. Are introduced the Besov spaces, decomposing a smooth function defined on the d-dimensional sphere, in a series of harmonics spherics and using a sequence o zonal polynomials which can be seen as a natural generalization of the Vallée Poussin polynomials defined on the unit circle. The main result studied says that every Besov¿s space can be got as a interpolation space of two Sobolev¿s spaces
Mestrado
Mestre em Matemática

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Bruder, Andrea S. Littlejohn Lance L. "Applied left-definite theory the Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators /." Waco, Tex. : Baylor University, 2009. http://hdl.handle.net/2104/5327.

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Thesis (Ph.D.)--Baylor University, 2009.
Subscript in abstract: n and n=0 in {Pn([alpha],[beta])(x)} [infinity] n=0, [mu] in (f,g)[mu], and R in [integral]Rfgd[mu]. Superscript in abstract: ([alpha],[beta]) and [infinity] in {Pn([alpha],[beta])(x)} [infinity] n=0. Includes bibliographical references (p. 115-119).

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Brewster, Kevin. "Surface to surface changes of variables and applications." Diss., Columbia, Mo. : University of Missouri-Columbia, 2008. http://hdl.handle.net/10355/5759.

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Thesis (Masters of Science for Teachers)--University of Missouri-Columbia, 2008.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on August 25, 2008) Vita. Includes bibliographical references.

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Bento, Antonio Jorge Gomes. "Interpolation, measures of non-compactness, entropy numbers and s-numbers." Thesis, University of Sussex, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.344067.

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36

Bihun, Oksana Chicone Carmen Charles. "Approximate isometries and distortion energy functionals." Diss., Columbia, Mo. : University of Missouri--Columbia, 2009. http://hdl.handle.net/10355/6129.

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Title from PDF of title page (University of Missouri--Columbia, viewed on Feb. 11, 2010). The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Dissertation advisor: Professor Carmen Chicone. Vita. Includes bibliographical references.

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37

Algervik, Robert. "Embedding Theorems for Mixed Norm Spaces and Applications." Licentiate thesis, Karlstad : Faculty of Technology and Science, Mathematics, Karlstads universitet, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-2874.

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38

Sloane, Craig Andrew. "Hardy-Sobolev-Maz'ya inequalities for fractional integrals on halfspaces and convex domains." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41125.

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This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation. After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace. Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.

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Bandyopadhyay, Jogia. "Optimal concentration for SU(1,1) coherent state transforms and an analogue of the Lieb-Wehrl conjecture for SU(1,1)." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24801.

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Thesis (Ph.D.)--Physics, Georgia Institute of Technology, 2008.
Committee Chair: Eric A. Carlen; Committee Member: Jean Bellissard; Committee Member: Michael Loss; Committee Member: Predrag Cvitanovic.

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40

McCabe, Terence W. (Terence William). "Minimization of a Nonlinear Elasticity Functional Using Steepest Descent." Thesis, University of North Texas, 1988. https://digital.library.unt.edu/ark:/67531/metadc331296/.

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Dima, Ute [Verfasser], and Arnd [Akademischer Betreuer] Rösch. "Regularization in fractional order Sobolev spaces for a parameter identification problem / Ute Dima ; Betreuer: Arnd Rösch." Duisburg, 2017. http://d-nb.info/1147681287/34.

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42

Medeiros, Luiz Adauto, and Juan Límaco. "On the Kirchhoff equation in noncylindrical domains of R." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97270.

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43

Chappa, Eduardo. "The x-ray transform of tensor fields /." Thesis, Connect to this title online; UW restricted, 2002. http://hdl.handle.net/1773/5750.

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44

Brauer, Uwe, and Lavi Karp. "Local existence of classical solutions for the Einstein-Euler system using weighted Sobolev spaces of fractional order." Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2009/3017/.

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We prove the existence of a class of local in time solutions, including static solutions, of the Einstein-Euler system. This result is the relativistic generalisation of a similar result for the Euler-Poisson system obtained by Gamblin [8]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article.

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45

Gonçalves, Breno Lucas da Costa. "An introduction to distribution theory, Fourier transform, Sobolev spaces and its applications to the Black-Scholes Equation." Master's thesis, Instituto Superior de Economia e Gestão, 2019. http://hdl.handle.net/10400.5/19247.

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Mestrado em Mathematical Finance
A maior parte da teoria citada em cursos introdutórios de análise matemática e cálculo foi elaborada ainda antes do século XVIII. No entanto, em meados de 1950, Laurent Schwartz desenvolveu um novo conceito que mudou e redefiniu a noção de função. A definição de derivada e transformada de Fourier são dois dos conceitos mais importantes em análise e essenciais para o estudo de equações diferenciais parciais. No entanto, note-se que nem todas as funções são diferenciáveis ou possuem uma transformada de Fourier. A teoria de distribuições permite esta correção ao incorporar as funções clássicas numa classe maior de objetos matemáticos. Esta dissertação tem como objetivo completar o artigo publicado por D. da Silva, K. Igibayeva, A. Khoroshevskay e Z.Sakayevz. De modo a alcançar o nosso objectivo apresentam-se inicialmente as ferramentas necessárias para uma introdução à teoria de distribuições, transformadas de Fourier e espaços de Sobolev que serão usadas para o cálculo explícito da solução da equação do calor.
Most of the mathematical theory study in standard courses of calculus were developed even before the eighteen century. However, around the year of 1950, Laurent Schwartz came up with a new concept that would change and redefine the concept of function. Differentiability and the Fourier transform are two of the most important notions in analysis and genuinely essential when working with partial differential equations. It is well-known that not all functions are differentiable or have a Fourier transform. The theory of distributions allows us to correct this issue by embedding classical functions into a larger class of mathematical objects. This dissertation aims to complete the article published by D. Da Silva, K. Igibayeva, A. Khoroshevskay and Z.Sakayeva (2018). To accomplish our goal, we provide and develop the necessary tools for an introductory course in distribution theory, Fourier transforms, Sobolev spaces and use them to solve the heat diffusion equation.
info:eu-repo/semantics/publishedVersion

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Benetti, Francesco. "The Cauchy problem for the wave equation." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18369/.

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In questo lavoro viene esposta la teoria del problema di Cauchy per l'equazione delle onde in un mezzo omogeneo e isotropo in dimensione qualunque. I primi due capitoli sono incentrati sull'approcio classico alla soluzione del problema. In particolare, nel primo capitolo si studia il problema in tutto lo spazio, mentre nel secondo in un dominio limitato, con condizioni al contorno. Nel terzo capitolo viene esposta la teoria degli Spazi di Sobolev, che verrà poi applicata nel capitolo successivo, nella cosiddetta formulazione debole del problema. L'ultimo capitolo è dedicato alle applicazioni fisiche: vengono studiate le onde elettromagnetiche e le onde gravitazionali, la cui recente scoperta ha aperto nuovi orizzonti nello studio del cosmo.

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Roth, John Charles. "Perturbations of Kähler-Einstein metrics /." Thesis, Connect to this title online; UW restricted, 1999. http://hdl.handle.net/1773/5737.

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48

Algervik, Robert. "Embedding Theorems for Mixed Norm Spaces and Applications." Doctoral thesis, Karlstads universitet, Avdelningen för matematik, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-5646.

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This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005). Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale. We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev spaces. The methods used are based on non-increasing rearrangements, and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces by Kolyada. We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces, and in the description of Sobolev type embeddings. In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces follow.

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Montealegre, Scott Juan. "Initial value problem for a coupled system of Kadomtsev-Petviashvili II equations in Sobolev spaces of negative indices." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95255.

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Mantegazza, Carlo. "Smooth geometric evolutions of hypersurfaces and singular approximation of mean curvature flow." Doctoral thesis, Scuola Normale Superiore, 2014. http://hdl.handle.net/11384/85686.

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Dissertations / Theses on the topic 'Sobolev spaces'

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