Search

We are concerned with existence, localization and multiplicity of positive radial solutions to Dirichlet problems with Ø-Laplacians in a ball, in both scalar and system cases. Our approach essentially relies on fixed point index computations and a main feature is that it avoids any Harnack type inequality. Applications to some problems involving operators with Uhlenbeck structure are discussed.

e present an existence principle for boundary value problems involving discontinuous ordinary differential equations of the second order using the Krasovskii regularization technique. Especially we obtain sufficient conditions of transversality type for Krasovskii solutions to be also Carathéodory solutions of the original problem. This result is applied on a certain billiard problem, which can be thought as an ordinary differential equation with state-dependent impulses that is equivalent to certain discontinuous differential equation. In particular, we obtain new existence and multiplicity results for Dirichlet problems in billiard spaces with time-varying boundaries

In this thesis solutions to Kolmogorov-Fokker-Planck type equations are studied. It consists of a comprehensive summary and four scientific articles. In the first article a potential theory for certain strongly degenerate parabolic operators in unbounded domains of Lipschitz type is developed. Several fundamental properties such as energy estimates and comparison principles are proven, and in particular the main results are solvability of the continuous Dirichlet problem, a Hölder estimate at the boundary, a Carleson estimate, boundary Harnack inequalities, and a doubling property for the parabolic measure. In the second article the results about the parabolic measure studied in the first article are refined considerably under some further assumptions. The main result states that the parabolic measure is absolutely continuous with respect to the surface measure, and the associated Radon-Nikodym derivative defines a weight in the Muckenhoupt class A∞. In the third article an elliptic, a parabolic, and a Kolmogorov type operator are studied. A structural theorem, which allows results about the parabolic and Kolmogorov type operators to be concluded from the corresponding results about the elliptic operator, is proven. In particular, results about the Lp Dirichlet problem for the Kolmogorov type operator may be derived from the corresponding results for the elliptic operator, using boundary estimates developed in the first article. The established results are then applied on a homogenization problem for the Kolmogorov type operator. In the fourth article the existence and uniqueness, in bounded and unbounded Lipschitz type cylinders, of weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operators of Kolmogorov-Fokker-Planck type is established.

In this thesis solutions to Kolmogorov-Fokker-Planck type equations are studied. It consists of a comprehensive summary and four scientific articles. In the first article a potential theory for certain strongly degenerate parabolic operators in unbounded domains of Lipschitz type is developed. Several fundamental properties such as energy estimates and comparison principles are proven, and in particular the main results are solvability of the continuous Dirichlet problem, a Hölder estimate at the boundary, a Carleson estimate, boundary Harnack inequalities, and a doubling property for the parabolic measure. In the second article the results about the parabolic measure studied in the first article are refined considerably under some further assumptions. The main result states that the parabolic measure is absolutely continuous with respect to the surface measure, and the associated Radon-Nikodym derivative defines a weight in the Muckenhoupt class A∞. In the third article an elliptic, a parabolic, and a Kolmogorov type operator are studied. A structural theorem, which allows results about the parabolic and Kolmogorov type operators to be concluded from the corresponding results about the elliptic operator, is proven. In particular, results about the Lp Dirichlet problem for the Kolmogorov type operator may be derived from the corresponding results for the elliptic operator, using boundary estimates developed in the first article. The established results are then applied on a homogenization problem for the Kolmogorov type operator. In the fourth article the existence and uniqueness, in bounded and unbounded Lipschitz type cylinders, of weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operators of Kolmogorov-Fokker-Planck type is established.

In this thesis solutions to Kolmogorov-Fokker-Planck type equations are studied. It consists of a comprehensive summary and four scientific articles. In the first article a potential theory for certain strongly degenerate parabolic operators in unbounded domains of Lipschitz type is developed. Several fundamental properties such as energy estimates and comparison principles are proven, and in particular the main results are solvability of the continuous Dirichlet problem, a Hölder estimate at the boundary, a Carleson estimate, boundary Harnack inequalities, and a doubling property for the parabolic measure. In the second article the results about the parabolic measure studied in the first article are refined considerably under some further assumptions. The main result states that the parabolic measure is absolutely continuous with respect to the surface measure, and the associated Radon-Nikodym derivative defines a weight in the Muckenhoupt class A∞. In the third article an elliptic, a parabolic, and a Kolmogorov type operator are studied. A structural theorem, which allows results about the parabolic and Kolmogorov type operators to be concluded from the corresponding results about the elliptic operator, is proven. In particular, results about the Lp Dirichlet problem for the Kolmogorov type operator may be derived from the corresponding results for the elliptic operator, using boundary estimates developed in the first article. The established results are then applied on a homogenization problem for the Kolmogorov type operator. In the fourth article the existence and uniqueness, in bounded and unbounded Lipschitz type cylinders, of weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operators of Kolmogorov-Fokker-Planck type is established.

This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space., Målet med detta arbete är att vara en introduktion på kandidatnivå till ämnesfältet potentialteori, genom att studera ett viktigt matematiskt problem inom potentialteori, Dirichletproblemet. Genom att undersöka den historiska utvecklingen av olika lösningsmetoder för problemet i mer och mer allmänna sammanhang, i kombination med de matematiska koncepten som utvecklades för att användas i dessa lösningsmetoder, ges en översikt av olika grundläggande tekniker inom potentialteori, samt en sammanfattaning av de olika matematiska resultaten som har att göra med Dirichletproblemet i det Euklidiska rummet.

We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile of an initial hypersurface. We show that our solution agrees with a classical mean curvature flow under the Dirichlet condition. We moreover prove that our solution agrees with a level-set flow under the Dirichlet condition constructed by P. Sternberg and W. P. Ziemer (1994), where the initial hypersurface is contained in a strictly mean-convex domain and the prescribed boundary is on the boundary of the domain.

We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile of an initial hypersurface. We show that our solution agrees with a classical mean curvature flow under the Dirichlet condition. We moreover prove that our solution agrees with a level-set flow under the Dirichlet condition constructed by P. Sternberg and W. P. Ziemer (1994), where the initial hypersurface is contained in a strictly mean-convex domain and the prescribed boundary is on the boundary of the domain.

This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space., Målet med detta arbete är att vara en introduktion på kandidatnivå till ämnesfältet potentialteori, genom att studera ett viktigt matematiskt problem inom potentialteori, Dirichletproblemet. Genom att undersöka den historiska utvecklingen av olika lösningsmetoder för problemet i mer och mer allmänna sammanhang, i kombination med de matematiska koncepten som utvecklades för att användas i dessa lösningsmetoder, ges en översikt av olika grundläggande tekniker inom potentialteori, samt en sammanfattaning av de olika matematiska resultaten som har att göra med Dirichletproblemet i det Euklidiska rummet.

In this thesis solutions to Kolmogorov-Fokker-Planck type equations are studied. It consists of a comprehensive summary and four scientific articles. In the first article a potential theory for certain strongly degenerate parabolic operators in unbounded domains of Lipschitz type is developed. Several fundamental properties such as energy estimates and comparison principles are proven, and in particular the main results are solvability of the continuous Dirichlet problem, a Hölder estimate at the boundary, a Carleson estimate, boundary Harnack inequalities, and a doubling property for the parabolic measure. In the second article the results about the parabolic measure studied in the first article are refined considerably under some further assumptions. The main result states that the parabolic measure is absolutely continuous with respect to the surface measure, and the associated Radon-Nikodym derivative defines a weight in the Muckenhoupt class A∞. In the third article an elliptic, a parabolic, and a Kolmogorov type operator are studied. A structural theorem, which allows results about the parabolic and Kolmogorov type operators to be concluded from the corresponding results about the elliptic operator, is proven. In particular, results about the Lp Dirichlet problem for the Kolmogorov type operator may be derived from the corresponding results for the elliptic operator, using boundary estimates developed in the first article. The established results are then applied on a homogenization problem for the Kolmogorov type operator. In the fourth article the existence and uniqueness, in bounded and unbounded Lipschitz type cylinders, of weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operators of Kolmogorov-Fokker-Planck type is established.

In this thesis solutions to Kolmogorov-Fokker-Planck type equations are studied. It consists of a comprehensive summary and four scientific articles. In the first article a potential theory for certain strongly degenerate parabolic operators in unbounded domains of Lipschitz type is developed. Several fundamental properties such as energy estimates and comparison principles are proven, and in particular the main results are solvability of the continuous Dirichlet problem, a Hölder estimate at the boundary, a Carleson estimate, boundary Harnack inequalities, and a doubling property for the parabolic measure. In the second article the results about the parabolic measure studied in the first article are refined considerably under some further assumptions. The main result states that the parabolic measure is absolutely continuous with respect to the surface measure, and the associated Radon-Nikodym derivative defines a weight in the Muckenhoupt class A∞. In the third article an elliptic, a parabolic, and a Kolmogorov type operator are studied. A structural theorem, which allows results about the parabolic and Kolmogorov type operators to be concluded from the corresponding results about the elliptic operator, is proven. In particular, results about the Lp Dirichlet problem for the Kolmogorov type operator may be derived from the corresponding results for the elliptic operator, using boundary estimates developed in the first article. The established results are then applied on a homogenization problem for the Kolmogorov type operator. In the fourth article the existence and uniqueness, in bounded and unbounded Lipschitz type cylinders, of weak solutions to Cauchy-Dirichlet problems for strongly degenerate parabolic operators of Kolmogorov-Fokker-Planck type is established.

We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

We study stable solutions to the equation , posed in a bounded domain of . For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions . This result, which was known only for , follows from a new interior Hölder estimate that is completely independent of the nonlinearity f. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabré, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the Hölder bound for , a universal estimate in all dimensions. Our bound is expected to hold for , but this has been settled only in the radial case or when . For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when , even in the radial case., Peer Reviewed, Postprint (published version)

We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving -Laplacian operators in a ball. In particular, -Laplacian and Minkowski-curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack-type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions

We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

This thesis presents a class of narrow-stencil finite difference methods for approximating the viscosity solution of second-order linear elliptic Dirichlet boundary value problems. The methods are simple to motivate and implement. This thesis proves admissibility and stability results for the simple narrow-stencil finite difference methods as well as optimal convergence rates when the underlying solution to the partial differential equation (PDE) is sufficiently smooth. The results in this thesis extend the analytic techniques first developed by Feng and Lewis when approximating viscosity solutions of fully nonlinear elliptic PDEs using the Lax-Friedrich’s-like method. Numerical tests are presented to gauge the performance of the methods and to validate the convergence results of the thesis.

In this Bachelor's thesis, a solution to the Dirichlet problem using Brownian motion is given. Brownian motion is constructed using Kolmogorov's existence and continuity theorems. Blumenthal's zero-one law and the strong Markov property in various formulations are proven. Using these results, a solution to the Dirichlet problem is given using Brownian motion. The cone condition which gives conditions on the domain guaranteeing existence of solution is proven., This work deals with two mathematical concepts from seemingly disparate worlds: the Dirichlet problem and Brownian motion. The Dirichlet problem deals with very smooth functions, whereas Brownian motion is prototypically the random movement of a particle suspended in a liquid. The intuition for the Dirichlet problem comes from physics. Imagine some object with a given temperature distribution on its surface. The problem is to find a function which would tell us the temperature at any point inside the object. This work culminates in formulating this function in terms of average properties of randomly moving particles.

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderon-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Holder continuous with the same Holder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions., QC 20211110

In this Bachelor's thesis, a solution to the Dirichlet problem using Brownian motion is given. Brownian motion is constructed using Kolmogorov's existence and continuity theorems. Blumenthal's zero-one law and the strong Markov property in various formulations are proven. Using these results, a solution to the Dirichlet problem is given using Brownian motion. The cone condition which gives conditions on the domain guaranteeing existence of solution is proven., This work deals with two mathematical concepts from seemingly disparate worlds: the Dirichlet problem and Brownian motion. The Dirichlet problem deals with very smooth functions, whereas Brownian motion is prototypically the random movement of a particle suspended in a liquid. The intuition for the Dirichlet problem comes from physics. Imagine some object with a given temperature distribution on its surface. The problem is to find a function which would tell us the temperature at any point inside the object. This work culminates in formulating this function in terms of average properties of randomly moving particles.

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form ut(x; t) = ∫ J(x - y)u(y; t) dy - h∑(x)u(x; t)+f(x; u(x; t)) with x in a perturbed domain Ω∑ C Ω which is thought as a fixed set Ω from where we remove a subset A∑ called the holes. We choose an appropriated families of functions h∑ € L∞ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside Ω. Moreover, we take J as a non-singular kernel and f as a nonlocal nonlinearity. Under the assumption that the characteristic functions of Ω€ have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds M× ϱR. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to - ∞ provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast., SCOPUS: ar.j, info:eu-repo/semantics/published

Дисертацiя присвячена побудовi i дослiдженню крайових задач в областi на нескiнченновимiрних многовидах i просторах. Запропонована умова рiвномiрностi атласу рiманового многовиду, виконання якої дозволяє довести метричну повноту многовиду за внутрiшньою метрикою. Доведено, що при виконаннi певних додаткових умов, якi, зокрема, виконуються при умовi рiвномiрностi атласу, внутрiшня метрика є узгодженою з вихiдною топологiєю многовиду. Показано, що при виконаннi певних умов межа областi та поверхня сумiсного рiвня функцiй в гiльбертовому просторi є рiмановими многовидами з рiвномiрними атласами. Запропоновано L2-версiю лапласiана за мiрою на (нескiнченновимiрному) рiмановому многовидi. Наведено модельний приклад рiвномiрного рiманового многовиду, для якого реалiзуються всi умови, використанi при побудовi уведеного лапласiана i при доведеннi коректностi задач Дiрiхле певного класу. Дослiджено дифеоморфне вiдображення мiж нескiнченновимiрними рiмановими многовидами з рiвномiрними атласами як спосiб розширення класу коректних крайових задач. Наведено два приклади використання методу дифеоморфiзмiв.

We establish that if the distribution function of a measurable function v given on a bounded domain Ω of Rn (n > 2) satisfies, for sufficiently large k, the estimate meas{|v| > k} 6 k−αϕ(k)/ψ(k), where α > 0, ϕ: [1, +∞) → R is a nonnegative nonincreasing measurable function such that the integral of the function s → ϕ(s)/s over [1, +∞) is finite, and ψ: [0, +∞) → R is a positive continuous function with some additional properties, then |v|αψ(|v|) ∈ L1(Ω). In so doing, the function ψ can be bounded or unbounded. We give corollaries of the corresponding theorems for some specific ratios of the functions ϕ and ψ. In particular, we consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas{|v| > k} 6 Ck−α(ln k)−β with C, α > 0 and β > 0. In this case, we strengthen our previous result for β > 1 and, on the whole, we show how the integrability properties of the function v differ depending on which of the intervals [0, 1] or (1, +∞) contains β. We also consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas{|v| > k} 6 Ck−α(ln ln k)−β with C, α > 0 and β > 0. We give examples showing the accuracy of the obtained results in the corresponding scales of classes close to Lα(Ω). Finally, we give applications of these results to entropy and weak solutions of the Dirichlet problem for nonlinear elliptic second-order equations with right-hand side in some classes close to L1(Ω) and defined by the logarithmic function or its double composition. © 2019 Krasovskii Institute of Mathematics and Mechanics. All Rights Reserved.

Дисертацiя присвячена побудовi i дослiдженню крайових задач в областi на нескiнченновимiрних многовидах i просторах. Запропонована умова рiвномiрностi атласу рiманового многовиду, виконання якої дозволяє довести метричну повноту многовиду за внутрiшньою метрикою. Доведено, що при виконаннi певних додаткових умов, якi, зокрема, виконуються при умовi рiвномiрностi атласу, внутрiшня метрика є узгодженою з вихiдною топологiєю многовиду. Показано, що при виконаннi певних умов межа областi та поверхня сумiсного рiвня функцiй в гiльбертовому просторi є рiмановими многовидами з рiвномiрними атласами. Запропоновано L2-версiю лапласiана за мiрою на (нескiнченновимiрному) рiмановому многовидi. Наведено модельний приклад рiвномiрного рiманового многовиду, для якого реалiзуються всi умови, використанi при побудовi уведеного лапласiана i при доведеннi коректностi задач Дiрiхле певного класу. Дослiджено дифеоморфне вiдображення мiж нескiнченновимiрними рiмановими многовидами з рiвномiрними атласами як спосiб розширення класу коректних крайових задач. Наведено два приклади використання методу дифеоморфiзмiв., The thesis deals with constructing and studying boundary value problems in domains in infinite-dimensional spaces and manifolds. Research of boundary value problems with infinite-dimensional argument is one of the most important tasks of functional analysis. One of the research subjects of functional analysis are infinitedimensional topological vector spaces, their mappings and relevant objects. Historically functional analysis emerged as a mean to research Fourier transformation, differential and integral equations. Starting from the second half of XX-th century functional analysis expanded to include a range of new sections via generalizing classical finite-dimensional theory results to infinite-dimensional case. The main part of the thesis consists of an introduction, four sections, divided into subsections, conclusions, list of references and an appendix with the list of the author’s publications concerning the topic of the thesis and the scientific seminars and conferences, at which the obtained results were reported. The introduction grounds the relevance of the research topic, gives short historical review of its state, formulates the purpose and tasks of the research, indicates the scientific novelty and also points out where the results of the dissertation have been discussed and published. Section 1 provides review of works, which are relevant to the topic of the dissertation research. A review of classical results of Riemannian geometry, that relate to the subject of research, is given, i.e., definition of a Riemannian manifold, metric tensor existence, Riemannian connection, Levi–Civita connection and completeness of a Riemannian manifold. A number of modern papers on Riemannian geometry, that consider elliptical equations on Riemannian manifolds, is reviewed. Fomin and Skorokhod directional differentiabilities of measures, differentiabilty along vector fields, are reviewed, connection between them is established. Section 2 considers infinite-dimensional, Диссертация посвящена построению и исследованию краевых задач в области на бесконечномерных многообразиях и пространствах. Предложено условие равномерности атласа риманова многообразия, выполнение которого позволяет доказать метрическую полноту многообразия по внутренней метрике. Доказано, что при выполнении некоторых дополнительных условий, которые, в том числе, выполняются в случае равномерности атласа, внутренняя метрика согласована с исходной топологией многообразия. Показано, что при выполнении некоторых условий граница области и поверхность совместного уровня в гильбертовом пространстве есть римановы многообразия с равномерными атласами. Предложена L2-версия лапласиана по мере на (бесконечномерном) римановом многообразии. Приведен модельный пример равномерного риманова многообразия, для которого реализуются все условия, использованные при построении введеного лапласиана и при доказательстве корректности задач Дирихле определённого класса. Исследовано диффеоморфное отображение между бесконечномерными римановыми многообразиями с равномерными атласами как способ расширения класса корректных краевых задач. Приведено два примера использования метода диффеоморфизмов.

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings

Let be a bounded domain, and let f be a real-valued function defined on the whole topological boundary . The aim of this paper is to find a characterization of the functions f which can be extended to the inside to a m-subharmonic function under suitable assumptions on . We shall do so using a function algebraic approach with focus on m-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of m-subharmonic functions.

In this bachelor's thesis we will solve the Dirichlet problem with an Lp(T) boundary function. First, we will focus on the holomorphic version of the Dirichlet problem and introduce Hardy space theory, from which will follow a sufficient condition on the Fourier coefficients of the boundary function. Then we will prove the Marcinkiewicz interpolation theorem. After that we introduce the conjugate function "tilde f", which equals the Hilbert transform of f, and use functional analysis to prove an important duality argument of the Hilbert transform. Finally, we will give several different proofs for the boundedness of the map f ↦ tilde f using the Marcinkiewicz interpolation theorem and the duality argument: the last proof will be done rigorously from scratch, i.e. without relying on (unproved) arguments from other literature., Applied Mathematics

Postprint (author's final draft)

In this bachelor's thesis we will solve the Dirichlet problem with an Lp(T) boundary function. First, we will focus on the holomorphic version of the Dirichlet problem and introduce Hardy space theory, from which will follow a sufficient condition on the Fourier coefficients of the boundary function. Then we will prove the Marcinkiewicz interpolation theorem. After that we introduce the conjugate function "tilde f", which equals the Hilbert transform of f, and use functional analysis to prove an important duality argument of the Hilbert transform. Finally, we will give several different proofs for the boundedness of the map f ↦ tilde f using the Marcinkiewicz interpolation theorem and the duality argument: the last proof will be done rigorously from scratch, i.e. without relying on (unproved) arguments from other literature., Applied Mathematics

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan–Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound K(P)≤-ϕ(ϕ-1)/r(x)2and a pointwise pinching condition, SCOPUS: ar.j, info:eu-repo/semantics/published

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound K(P) and a pointwise pinching condition |K(P)| for some constants phi > 1 and C-K >= 1, where P and P ' a re any 2-dimensional subspaces of TxM containing the (radial) vector del(x) and r (x) = d(o,x) is the distance to a fixed point o. M. We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions n = dim M > 4/phi+ 1.

We study the infinitesimal generator of the Poisson semigroup in L associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L-based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains.

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold M of dimension n ≥ 2 for a large class of operators containing, in particular, the p-Laplacian and the minimal graph operator. We extend several existence results obtained for the p-Laplacian to our class of operators. As an application of our main result, we prove the solvability of the asymptotic Dirichlet problem for the minimal graph equation for any continuous boundary data on a (possibly non rotationally symmetric) manifold whose sectional curvatures are allowed to decay to 0 quadratically., SCOPUS: ar.j, info:eu-repo/semantics/published

On this Bachelor's Thesis we apply the method of layer potentials on two different contexts. On the first part of this work we will prove some important properties of the single and double layer potentials for the Laplacian on smooth domains and we will use them to prove uniqueness and existence of solution for the Dirichlet and Neumann problems (both exterior and interior). On the second part we use the method of the layer potentials to look for an approximate solution for the flow of air around a very thin wing and we use the solution found to compute the approximate force exerted over the wing.

We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by Mingarelli (1988)., Validerad; 2016; Nivå 2; 2016-10-19 (andbra)

We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by Mingarelli (1988)., Validerad; 2016; Nivå 2; 2016-10-19 (andbra)

We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence-type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence-type elliptic equations set in halfspaces and with in finitely smooth data may converge to their corresponding boundary layer tails as slowly as one wishes depending on the position of the hyperplane. Second, we construct a Dirichlet problem for divergence-type elliptic operators set in a bounded domain, and with all data being C-infinity-smooth, for which the boundary value homogenization holds with arbitrarily slow speed., QC 20170119

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form -y''+q(x)y=λw(x)y, on I = [a,b], where q(x) and w(x) are real-valued

We show that the boundedness of the Hardy-Littlewood maximal operator on a Kothe function space X and on its Kothe dual X is equivalent to the well-posedness of the X-Dirichlet and X-Dirichlet problems in Rn + in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space H1, and the Beurling-Hardy space HAp for p € (1,∞). Based on the well-posedness of the Lp-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.

We review the classical Cauchy-Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy-Kovalevskaya and Holmgren theorems supply a complete answer to the existence and uniqueness issues. Here, we consider a global uniqueness problem of Holmgren's type. Perhaps surprisingly, we obtain a connection with the theory of quadrature identities, which demonstrates that rather subtle algebraic properties of the curve come into play. For instance, if is the interior domain of an ellipse, and I is a proper arc of the ellipse , then there exists a nontrivial biharmonic function u in which is three-flat on I (i.e., all partial derivatives of u of order vanish on I) if and only if the ellipse is a circle. Another instance of the same phenomenon is that if is bounded and simply connected with -smooth Jordan curve boundary, and if the arc is nowhere real-analytic, then we have local uniqueness already with sub-Cauchy data: if a function is biharmonic in for some planar neighborhood of I, and is three-flat on I, then it vanishes identically on , provided that is connected. Finally, we consider a three-dimensional setting, and analyze it partially using analogues of the square of the standard Cauchy-Riemann operator. In a special case when the domain is of periodized cylindrical type, we find a connection with the massive Laplacian [the Helmholz operator with imaginary wave number] and the theory of generalized analytic (or pseudoanalytic) functions of Bers and Vekua., QC 20150915

The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p,p)-Poincaré inequality, 1

In this article we address the question whether the complex Monge-Ampere equation is solvable for measures with large singular part. We prove that under some conditions there is no solution when the right-hand side is carried by a smooth subvariety in C-n of dimension k < n.

In this paper we solve the relative homotopy Dirichlet problem for (Formula presented.)-harmonic maps from compact manifolds with boundary to compact manifolds of non-positive sectional curvature. The proof, which is based on the direct calculus of variations, uses some ideas of B. White to define the relative d-homotopy type of Sobolev maps. One of the main points of the proof consists in showing that the regularity theory by Hardt and Lin can be applied. A comprehensive uniqueness result for general complete targets with non-positive curvature is also given.

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian (-Delta)(s) with s > 1. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case s is an element of (0,1).; As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions (-Delta)(s)phi = lambda phi in Omega, phi equivalent to 0 in R-n\Omega., Peer Reviewed, Postprint (published version)

The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p,p)-Poincaré inequality, 1