This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.

In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].

A class of parametric optimal control problems for semilinear parabolic equations is considered. Using recent regularity results for solutions of such equations, sufficient conditions are derived under which the solutions to optimal control problems are locally Lipschitz continuous functions of the parameter in the L1-norm. It is shown that these conditions are also necessary, provided that the dependence of data on the parameter is sufficiently strong.

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.

Orientador: Marcelo da Silva Montenegro / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-16T14:07:09Z (GMT). No. of bitstreams: 1 Melo_AlisonMarceloVanDerLaan_M.pdf: 595460 bytes, checksum: f3496cc25c882ea841e02b15bffe5256 (MD5) Previous issue date: 2010 / Resumo: Observação: O resumo, na íntegra poderá ser visualizado no texto completo da tese digital / Abstract: Note: The complete abstract is available with the full electronic digital thesis or dissertations / Mestrado / Matematica / Mestre em Matemática

Dans la première partie, on démontre des inégalités de Carleman pour des systèmes paraboliques. Au chapitre 1, on démontre des inégalités de stabilité pour un système parabolique 2 x 2 en utilisant des inégalités de Carleman avec une seule observation. Il s'agit d'un problème inverse pour l'identification des coefficients et les conditions initiales du système. Le chapitre2 est consacré aux inégalités de Carleman pour des systèmes paraboliques dont les coefficients de diffusion sont de classe C1 par morceaux ou à variations bornées. A la fin, on donne quelques applications à la contrôlabilité à zéro. La seconde partie est consacrée à l'étude d'un problème de diffraction d'ondes acoustiques dans un demi-plan homogène. Il s'agit d'un problème aux limites associé à l'équation de Helmholtz dans le demi-plan supérieur avec une donnée de Neumann non homogène au bord. On apporte des éléments de réponse sur la question d'unicité et d'existence des solutions pour certaines classes de la donnée au bord. / In the first part, we prove Carleman estimates for parabolic systems. In chapter1, we prove stability inequalities for 2 x 2 parabolic system using Carleman estimates with one observation. It is concerns to the identification of the coefficients and initial conditions of the system. The chapter2 is devoted to th Carleman estimates of parabolic systems for which the diffusion coefficients are assumed to be ofclass piecewise C1 or with bounded variations. In the end, we give some applications to the null controllability. The second part is devoted to the study of the scattering problem of acoustics waves in a homogeneous half-plane. It is about a boundary value problem associated to the Helmholtz equation in theupper half-plane with a nonhomogeneous Neumann boundary data. We provide some answers to the question of uniqueness and existence of solutions for some classes of the boundary data.

Le thème de cette thèse est l'analyse mathématique de modèles décrivant l'intrusion saline dans les aquifères côtiers. On a choisi d'adopter la simplicité de l'approche avec interface nette : il n'y a pas de transfert de masse entre l'eau douce et l'eau salée (resp. entre la zone saturée et la zone sèche). On compense la difficulté mathématique liée à l'analyse des interfaces libres par un processus de moyennisation verticale nous permettant de réduire le problème initialement 3D à un système d'edps définies sur un domaine, Ω, 2D. Un second modèle est obtenu en combinant l'approche 'interface nette' à celle avec interface diffuse ; cette approche est déduite de la théorie introduite par Allen-Cahn, utilisant des fonctions de phase pour décrire les phénomènes de transition entre les milieux d'eau douce et d'eau salée (respectivement les milieux saturé et insaturé). Le problème d'origine 3D est alors réduit à un système fortement couplé d'edps quasi-linéaires de type parabolique dans le cas des aquifères libres décrivant l'évolution des profondeurs des 2 surfaces libres et de type elliptique-parabolique dans le cas des aquifères confinés, les inconnues étant alors la profondeur de l'interface eau salée par rapport à eau douce et la charge hydraulique de l'eau douce. Dans la première partie de la thèse, des résultats d'existence globale en temps sont démontrés montrant que l'approche couplée interface nette-interface diffuse est plus pertinente puisqu'elle permet d'établir un principe du maximum plus physique (plus précisèment une hiérarchie entre les 2 surfaces libres). En revanche, dans le cas de l'aquifère confiné, nous montrons que les deux approches conduisent à des résultats similaires. Dans la seconde partie de la thèse, nous prouvons l'unicité de la solution dans le cas non dégénéré, la preuve reposant sur un résultat de régularité du gradient de la solution dans l'espace Lr (ΩT), r > 2, (ΩT = (0,T) x Ω). Puis nous nous intéressons à un problème d'identification des conductivités hydrauliques dans le cas instationnaire. Ce problème est formulé par un problème d'optimisation dont la fonction coût mesure l'écart quadratique entre les charges hydrauliques expérimentales et celles données par le modèle. / The theme of this thesis is the analysis of mathematical models describing saltwater intrusion in coastal aquifers. The simplicity of sharp interface approach is chosen : there is no mass transfer between fresh water and salt water (respectively between the saturated zone and the area dry). We compensate the mathematical difficulty of the analysis of free interfaces by a vertical averaging process allowing us to reduce the 3D problem to system of pde's defined on a 2D domain Ω. A second model is obtained by combining the approach of 'sharp interface' in that with 'diffuse interface' ; this approach is derived from the theory introduced by Allen-Cahn, using phase functions to describe the phenomena of transition between fresh water and salt water (respectively the saturated and unsaturated areas). The 3D problem is then reduced to a strongly coupled system of quasi-linear parabolic equations in the unconfined case describing the evolution of the DEPTHS of two free surfaces and elliptical-parabolic equations in the case of confined aquifer, the unknowns being the depth of salt water/fresh water interface and the fresh water hydraulic head. In the first part of the thesis, the results of global in time existence are demonstrated showing that the sharp-diffuse interface approach is more relevant since it allows to establish a mor physical maximum principle (more precisely a hierarchy between the two free surfaces). In contrast, in the case of confined aquifer, we show that both approach leads to similar results. In the second part of the thesis, we prove the uniqueness of the solution in the non-degenerate case. The proof is based on a regularity result of the gradient of the solution in the space Lr (ΩT), r > 2, (ΩT = (0,T) x Ω). Then we are interest in a problem of identification of hydraulic conductivities in the unsteady case. This problem is formulated by an optimization problem whose cost function measures the squared difference between experimental hydraulic heads and those given by the model.

Neste trabalho estudamos a dinâmica assintótica de uma classe de equações diferenciais de reação-difusão definidas em abertos de \'R POT. 3\' arbitrários, limitados ou não, com condições de fronteira de Dirichlet. Utilizando a técnica de estimativas de truncamento, como nos artigos de Prizzi e Rybakowski, mostramos a existência de atratores globais / In this work we study the asymptotic behavior of a class of semilinear reaction-diffusion equations defined on an arbitrary open set of R3, bounded or not, with Dirichlet boundary conditions. Using the tail-estimates technic based on papers of Prizzi and Rybakowski, we prove existence of global attractors

In this paper, optimal control problems for semilinear parabolic equations with distributed and boundary controls are considered. Pointwise constraints on the control and on the state are given. Main emphasis is laid on the discussion of second order sufficient optimality conditions. Sufficiency for local optimality is verified under different assumptions imposed on the dimension of the domain and on the smoothness of the given data.

On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary. / text