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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.

Random homogenization of p-Laplacian with obstacles on perforated domain and related topics

Tang, Lan, 1980- 09 June 2011 (has links)
Abstract not available. / text

Topics on subelliptic parabolic equations structured on Hörmander vector fields

Frentz, Marie January 2012 (has links)
No description available.

An Obstacle Problem for Mean Curvature Flow

Logaritsch, Philippe 25 October 2016 (has links) (PDF)
We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.

The obstacle problem for second order elliptic operators in nondivergence form

Teka, Kubrom Hisho January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Ivan Blank / We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper ``The regularity of free boundaries in higher dimensions." Finally, we show that blowup limits are in general not unique at free boundary points.

Optimization problems with complementarity constraints in infinite-dimensional spaces

Wachsmuth, Gerd 10 August 2017 (has links) (PDF)
In this thesis we consider optimization problems with complementarity constraints in infinite-dimensional spaces. On the one hand, we deal with the general situation, in which the complementarity constraint is governed by a closed convex cone. We use the local decomposition approach, which is known from finite dimensions, to derive first-order necessary optimality conditions of strongly stationary type. In the non-polyhedric case, stronger conditions are obtained by an additional linearization argument. On the other hand, we consider the optimal control of the obstacle problem. This is a classical example for a problem with complementarity constraints in infinite dimensions. We are concerned with the control-constrained case. Due to the lack of surjectivity, a system of strong stationarity is not necessarily satisfied for all local minimizers. We identify assumptions on the data of the optimal control problem under which strong stationarity of local minimizers can be verified. Moreover, without any additional assumptions on the data, we show that a system of M-stationarity is satisfied provided that some sequence of multipliers converges in capacity. Finally, we also discuss the notion of polyhedric sets. These sets have many applications in infinite-dimensional optimization theory. Since the results concerning polyhedricity are scattered in the literature, we provide a review of the known results. Furthermore, we give some new results concerning polyhedricity of intersections and provide counterexamples which demonstrate that intersections of polyhedric sets may fail to be polyhedric. We also prove a new polyhedricity result for sets in vector-valued Sobolev spaces.

Partial Balayage and Related Concepts in Potential Theory

Roos, Joakim January 2016 (has links)
This thesis consists of three papers, all treating various aspects of the operation partial balayage from potential theory. The first paper concerns the equilibrium measure in the setting of two dimensional weighted potential theory, an important measure arising in various mathematical areas, e.g. random matrix theory and the theory of orthogonal polynomials. In this paper we show that the equilibrium measure satisfies a complementary relation with a partial balayage measure if the weight function is of a certain type. The second paper treats the connection between partial balayage measures and measures arising from scaling limits of a generalisation of the so-called divisible sandpile model on lattices. The standard divisible sandpile can, in a natural way, be considered a discrete version of the partial balayage operation with respect to the Lebesgue measure. The generalisation that is developed in this paper is essentially a discrete version of the partial balayage operation with respect to more general measures than the Lebesgue measure. In the third paper we develop a version of partial balayage on Riemannian manifolds, using the theory of currents. Several known properties of partial balayage measures are shown to have corresponding results in the Riemannian manifold setting, one of which being the main result of the first paper. Moreover, we utilize the developed framework to show that for manifolds of dimension two, harmonic and geodesic balls are locally equivalent if and only if the manifold locally has constant curvature. / Denna avhandling består av tre artiklar som alla behandlar olika aspekter av den potentialteoretiska operationen partiell balayage. Den första artikeln betraktar jämviktsmåttet i tvådimensionell viktad potentialteori, ett viktigt mått inom flertalet matematiska inriktningar såsom slumpmatristeori och teorin om ortogonalpolynom. I denna artikel visas att jämviktsmåttet uppfyller en komplementaritetsrelation med ett partiell balayage-mått om viktfunktionen är av en viss typ. Den andra artikeln behandlar relationen mellan partiell balayage-mått och mått som uppstår från skalningsgränser av en generalisering av den så kallade "delbara sandhögen", en diskret modell för partikelaggregation på gitter. Den vanliga delbara sandhögen kan på ett naturligt sätt betraktas som en diskret version av partiell balayage-operatorn med avseende på Lebesguemåttet. Generaliseringen som utarbetas i denna artikel är väsentligen en diskret version av partiell balayage-operatorn med avseende på mer allmänna mått än Lebesguemåttet. I den tredje artikeln formuleras en version av partiell balayage på riemannska mångfalder utifrån teorin om strömmar. Åtskilliga tidigare kända egenskaper om partiella balayage-mått visas ha motsvarande formuleringar i formuleringen på riemannska mångfalder, bland annat huvudresultatet från den första artikeln. Vidare så utnyttjas det utarbetade ramverket för att visa att tvådimensionella riemannska mångfalder har egenskapen att harmoniska och geodetiska bollar lokalt är ekvivalenta om och endast om mångfalden lokalt har konstant krökning. / <p>QC 20160524</p>

Les équations aux dérivées partielles stochastiques avec obstacle / Stochastic partial differential equations with obstacle

Zhang, Jing 14 November 2012 (has links)
Cette thèse traite des Équations aux Dérivées Partielles Stochastiques Quasilinéaires. Elle est divisée en deux parties. La première partie concerne le problème d’obstacle pour les équations aux dérivées partielles stochastiques quasilinéaires et la deuxième partie est consacrée à l’étude des équations aux dérivées partielles stochastiques quasilinéaires dirigées par un G-mouvement brownien. Dans la première partie, on montre d’abord l’existence et l’unicité d’un problème d’obstacle pour les équations aux dérivées partielles stochastiques quasilinéaires (en bref OSPDE). Notre méthode est basée sur des techniques analytiques venant de la théorie du potentiel parabolique. La solution est exprimée comme une paire (u,v) où u est un processus prévisible continu qui prend ses valeurs dans un espace de Sobolev et v est une mesure régulière aléatoire satisfaisant la condition de Skohorod. Ensuite, on établit un principe du maximum pour la solution locale des équations aux dérivées partielles stochastiques quasilinéaires avec obstacle. La preuve est basée sur une version de la formule d’Itô et les estimations pour la partie positive d’une solution locale qui est négative sur le bord du domaine considéré. L’objectif de la deuxième partie est d’étudier l’existence et l’unicité de la solution des équations aux dérivées partielles stochastiques dirigées par G-mouvement brownien dans le cadre d’un espace muni d’une espérance sous-linéaire. On établit une formule d’Itô pour la solution et un théorème de comparaison. / This thesis deals with quasilinear Stochastic Partial Differential Equations (in short SPDE). It is divided into two parts, the first part concerns the obstacle problem for quasilinear SPDE and the second part solves quasilinear SPDE driven by G-Brownian motion. In the first part we begin with the existence and uniqueness result for the obstacle problem of quasilinear stochastic partial differential equations (in short OSPDE). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, v) where u is a predictable continuous process which takes values in a proper Sobolev space and v is a random regular measure satisfying minimal Skohorod condition. Then we prove a maximum principle for a local solution of quasilinear stochastic partial differential equations with obstacle. The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is negative on the lateral boundary. The objective of the second part is to study the well-posedness of stochastic partial differential equations driven by G-Brownian motion in the framework of sublinear expectation spaces. One can also establish an Itô formula for the solution and a comparison theorem.

The Double Obstacle Problem on Metric Spaces

Farnana, Zohra January 2008 (has links)
During the last decade, potential theory and p-harmonic functions have been developed in the setting of doubling metric measure spaces supporting a p-Poincar´e inequality. This theory unifies, and has applications in several areas of analysis, such as weighted Sobolev spaces, calculus on Riemannian manifolds and Carnot groups, subelliptic differential operators and potential theory on graphs. In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We show the existence and uniqueness of solutions and their continuity when the obstacles are continuous. Moreover the solution is p-harmonic in the open set where it does not touch the continuous obstacles. The boundary regularity of the solutions is also studied. Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles vary and fix the boundary values and show the convergence of the solutions. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω. / Låt oss börja med att betrakta följande situation: Vi vill förflytta oss från en plats vid ena sidan av en äng till en viss punkt på andra sidan ängen. På båda sidor om ängen finns skogsområden som vi inte får gå in i. Ängen är tyvärr inte homogen utan består av olika sorters mark som vi har noggrant beskrivet på en karta. Vi vill göra förflyttningen på smidigast sätt, men då ängen inte är homogen ska vi förmodligen inte gå rakaste vägen utan ska anpassa vägen optimalt efter terrängen. Detta är ett exempel på ett dubbelhinderproblem där hindren är skogsområdena på sidorna som vi måste hålla oss utanför. Mer abstrakt vill man minimiera energin hos funktioner som tar vissa givna randvärden (de givna start- och slutpunkterna i exemplet ovan) och som håller sig mellan ett undre och ett övre hinder. I denna avhandling studeras detta dubbelhinderproblem i väldigt allmänna situationer. För att kunna lösa hinderproblemet krävs det att vi tillåter ickekontinuerliga lösningar och då visas i avhandlingen att hinderproblemet är entydigt lösbart. Ett huvudresultat i avhandlingen är att om våra hinder är kontinuerliga så blir även lösningen kontinuerlig. Vidare visas diverse konvergenssatser som visar hur lösningarna varierar när hindren eller området i vilket problemet löses varierar. Hinderproblem har utöver eget intresse viktiga tillämpningar i potentialteorin, bland annat för att studera motsvarande energiminimeringsproblem utan hinder. / <p>Incorrect series title in colophon.</p>

An Obstacle Problem for Mean Curvature Flow

Logaritsch, Philippe 19 October 2016 (has links)
We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.

Nonoscillatory second-order procedures for partial differential equations of nonsmooth data

Lee, Philku 07 August 2020 (has links)
Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. This dissertation investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid methods to reduce accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. Parabolic initial-boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This dissertation investigates characteristics of the θ-method and introduces a variable-θ method as a synergistic combination of the Crank-Nicolson (CN) method and the implicit method. It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable-θ method is analyzed for its accuracy and stability. After a theory of morphogenesis in chemical cells was introduced in 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) equations. This dissertation studies a nonoscillatory second-order time-stepping procedure for RD equations incorporating with variable-θ method, as a perturbation of the CN method. We also perform a sensitivity analysis for the numerical solution of RD systems to conclude that it is much more sensitive to the spatial mesh resolution than the temporal one. Moreover, to enhance the spatial approximation of RD equations, this dissertation investigates the averaging scheme, that is, an interpolation of the standard and skewed discrete Laplacian operator and introduce the simple optimizing strategy to minimize the leading truncation error of the scheme.

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