Academic literature on the topic 'Regularity of weak solutions'

We prove that a smooth generic embedded CR submanifold of C^n obeys the maximum principle for continuous CR functions if and only if it is weakly 1-concave. The proof of the maximum principle in the original manuscript has later been generalized to embedded weakly q-concave CR submanifolds of certain complex manifolds. We give a generalization of a known result regarding automatic smoothness of solutions to the homogeneous problem for the tangential CR vector fields given local holomorphic extension. This generalization ensures that a given locally integrable structure is hypocomplex at the origin if and only if it does not allow solutions near the origin which cannot be represented by a smooth function near the origin. We give a sufficient condition under which it holds true that if a smooth CR function f on a smooth generic embedded CR submanifold, M, of C^n, vanishes to infinite order along a C^infty-smooth curve  \gamma in M, then f vanishes on an M-neighborhood of \gamma. We prove a local maximum principle for certain locally integrable structures.<br><p>Funding  by FMB, based at Uppsala University.</p>

Cette thèse est consacrée à l’étude du système de Boussinesq stationnaire:-νΔu+(u⋅∇)u+∇π=θg, div u=0,dans Ω(1a)-κΔθ+u⋅∇θ=h,dans Ω (1b)où Ω⊂R^3 est un ouvert, borné et connexe; les inconnues du système sont u,π et θ: la vitesse, la pression et la température du fluide, respectivement; ν&gt;0 est la viscosité cinématique du fluide, κ&gt;0 est la diffusivité thermique du fluide, g est l’accélération de la pesanteur et h est une source de chaleur appliquée au fluide.L’objectif de cette thèse est l’étude de la théorie L^p pour le système de Boussinesq en considérant deux différents types de conditions aux limites du champ de vitesse. En effet, dans une première partie, nous considérons une condition de Dirichlet non homogèneu=u_b, sur Γ (2)où Γ désigne la frontière du domaine. Dans une deuxième partie, nous considérons une condition de Navier non homogèneu⋅n=0,2[D(u)n]_τ+αu_τ=a,sur Γ(3)où D(u)=1/2 (∇u+(∇u)^T ) est le tenseur de déformation associé au champ de vitesse u, n est le vecteur normal unitaire extérieur, τ est le correspondant vecteur tangent unitaire, α et a sont une fonction scalaire de friction et un champ de vecteur tangentiel donnés sur la frontière, respectivement. De plus, la condition aux limites pour la température sera, dans les deux premières parties, une condition aux limites de Dirichlet non homogèneθ=θ_b, sur Γ. (4)Alors, premièrement, nous étudions l’existence et l’unicité d’une solution faible pour le problème (1), (2) et (4) dans le cas hilbertien. Également, l’existence de solutions généralisées pour p≥3/2 et des solutions fortes pour 12 et des solutions fortes pour p≥6/5 pour le problème (1), (3) et (4). Notez que l’hypothèse d’une frontière non-connexe, mentionnée précédemment, ne figurait pas dans cette partie du travail en raison de la restriction d’imperméabilité de la frontière.Enfin, dans la dernière partie de cette thèse, nous étudions la théorie L^p pour les équations de Stokes avec la condition de Navier (3). Plus précisément, nous examinons la régularité W^(1,p) pour p≥2 et la régularité W^(2,p) pour p≥6/5.Mots clés: système de Boussinesq; régularité L^p; solutions faibles; solutions fortes; solutions très faibles<br>This thesis is dedicated to the study of the stationary Boussinesq system:-νΔu+(u⋅∇)u+∇π=θg, div u=0,in Ω(1a)-κΔθ+u⋅∇θ=h,in Ω (1b)where Ω⊂R^3 is an open bounded connected set; u,π and θ are the velocity field, pressure and temperature of the fluid, respectively, and stand for the unknowns of the system; ν&gt;0 is the kinematic viscosity of the fluid, κ&gt;0 is the thermal diffusivity of the fluid, g is the gravitational acceleration and h is a heat source applied to the fluid.The aim of this thesis is the study of the L^p-theory for the stationary Boussinesq system in the context of two different types of boundary conditions for the velocity field. Indeed, in the first part of the thesis, we will consider a non-homogeneous Dirichlet boundary conditionu=u_b, on Γ (2)where Γ denotes the boundary of the domain; meanwhile in the second part, the velocity field will be prescribed through a non-homogeneous Navier boundary conditionu⋅n=0,2[D(u)n]_τ+αu_τ=a,on Γ(3)where D(u)=1/2 (∇u+(∇u)^T ) is the strain tensor associated with the velocity field u, n is the unit outward normal vector, τ is the corresponding unit tangent vector, α and a are a friction scalar function and a tangential vector field defined both on the boundary, respectively. Further, the boundary condition for the temperature will be, in the first two parts of the thesis, a non-homogeneous Dirichlet boundary conditionθ=θ_b, on Γ. (4)Then, firstly, we study the existence and uniqueness of the weak solution for the problem (1), (2) and (4) in the hilbertian case. Also, the existence of generalized solutions for p≥3/2 and strong solutions for 12 and strong solutions for p≥6/5 for the problem (1), (3) and (4). Note that the assumption of a non-connected boundary, which was mentioned before, will not appear here due to the impermeability restriction on the boundary.Finally, in the last part of this thesis, we study the L^p-theory for the Stokes equations with Navier boundary condition (3). Specifically, we deal with the W^(1,p)-regularity for p≥2 and the W^(2,p)-regularity for p≥6/5.Keywords: Boussinesq system; L^p-regularity; weak solutions; strong solutions; very weak solutions

Dans cette thèse, nous nous intéressons à l'étude de trois équations aux dérivées partielles et d'évolution non-locales en espace et en temps. Les solutions de ces trois solutions peuvent exploser en temps fini. Dans une première partie de cette thèse, nous considérons l'équation de la chaleur nonlinéaire avec une puissance fractionnaire du laplacien, et obtenons notamment que, dans le cas d'exposant sur-critique, le comportement asymptotique de la solution lorsque $t\rightarrow+\infty$ est déterminé par le terme de diffusion anormale. D'autre part, dans le cas d'exposant sous-critique, l'effet du terme non-linéaire domine. Dans une deuxième partie, nous étudions une équation parabolique avec le laplacien fractionnaire et un terme non-linéaire et non-local en temps. On montre que la solution est globale dans le cas sur-critique pour toute donnée initiale ayant une mesure assez petite, tandis que dans le cas sous-critique, on montre que la solution explose en temps fini $T_{\max}>0$ pour toute condition initiale positive et non-triviale. Dans ce dernier cas, on cherche le comportement de la norme $L^1$ de la solution en précisant le taux d'explosion lorsque $t$ s'approche du temps d'explosion $T_{\max}.$ Nous cherchons encore les conditions nécessaires à l'existence locale et globale de la solution. Une toisième partie est consacré à une généralisation de la deuxième partie au cas de systèmes $2\times 2$ avec le laplacien ordinaire. On étudie l'existence locale de la solution ainsi qu'un résultat sur l'explosion de la solution avec les mêmes propriétés étudiées dans le troisième chapitre. Dans la dernière partie, nous étudions une équation hyperbolique dans $\mathbb{R}^N,$ pour tout $N\geq2,$ avec un terme non-linéaire non-local en temps. Nous obtenons un résultat d'existence locale de la solution sous des conditions restrictives sur les données initiales, la dimension de l'espace et les exposants du terme non-linéaire. De plus on obtient, sous certaines conditions sur les exposants, que la solution explose en temps fini, pour toute condition initiale ayant de moyenne strictement positive.

We investigate solutions of the coupled Diral Klein-Gordon Equations in one and three space dimensions. Through analysis of the Fourier representations of the solutions to these equations, we introduce the ‘Null Structure’ as developed by Klainerman and Machedon. This structure allows us to prove the necessary estimates, both fixed time and bilinear space-time, that allow us to show existence of solutions of these equations with initial data of lower regularity than previously required. We also study global existence for a two dimensional wave equation with a critical non-linearity.

In der vorliegenden Arbeit untersuchen wir schwache Lösungen, die zu einem geeigneten Sobolevraum gehören, q-elliptischer und parabolischer Systeme partieller Differentialgleichungen auf deren Regularität für den Fall 1<br>In the present work we study the regularity of weak solution to q-elliptic and parabolic systems partial differential equations in appropriate Sobolev spaces in case 1

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003.<br>Includes bibliographical references (p. 57-58).<br>We examine the regularity properties of solutions to an elliptic free boundary problem, near a Neumann fixed boundary. Consider a nonnegative function u which minimizes the functional ... on a bounded, convex domain ... This function u is harmonic in its positive phase and satisfies ... along the free boundary ... , in a weak sense. We prove various basic properties of such a minimizer near the portion of the boundary ... on which ... weakly. These results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains. The main result is that the minimizer u is Lipschitz continuous. The proof in dimension 2 is by means of conformal mapping as well as a simplified monotonicity formula. In higher dimensions, the proof is via a maximum principle estimate for ...<br>by Sarah Groff Raynor.<br>Ph.D.

<p>In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α < 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.</p>

We study the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. In order to deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guarantee ellipticity for the local part of the Fock operator on the open stretched cone R+ × S². This proves existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed-point approach based on Cances and Le Bris analysis of the level-shifting algorithm, we show via another bootstrap argument, that the corresponding self-consistent-field solutions of the Hartree-Fock equation have the same type of asymptotic regularity.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.<br>Includes bibliographical references (p. 71-72).<br>In the first part of this dissertation, we provide the first example of a singular energy minimizing free boundary. This singular solution occurs in dimension 7 and higher, and in fact it is conjectured that there are no singular minimizers in dimension lower than 7. Our example is the analogue of the 8-dimensional Simons cone in the theory of minimal surfaces. The minimality of the Simons cone is closely related to the existence of a complete minimal graph in dimension 9, which is not a hyperplane. The first step toward solving the analogous problem in the free boundary context, consists in developing a local existence and regularity theory for monotone solutions to a free boundary problem. This is the objective of the second part of our thesis. We also provide a partial result in the global context..<br>by Daniela De Silva.<br>Ph.D.