Boundary Regularity in Variational Problems

Rendiconti Lincei - Matematica e Applicazioni, 2000

Journal de Mathématiques Pures et Appliquées, 2020

This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Hölder regularity under proper geometric conditions. "Unified" means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the p−Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the measure of the complement of the domain near the boundary point concerned could be zero. The key observation in the method is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the Hölder regularity. Moreover, we also give a geometric condition, which guarantees the solvability of the Dirichlet problem for the Laplace equation. The geometric meaning of this condition is more apparent than that of the Wiener criterion. Résumé Dans cet article, nous étudies la relation entre les propriétés géométriques des frontière et la régularité frontière des solutions d'équations elliptiques. Nous prouvons par une nouvelle méthode unifiée la régularité höldérienne ponctuelle dans des conditions géométriques appropriées. Unifié signifie que notre méthode est applicable à l'équa

arXiv: Analysis of PDEs, 2018

We study global regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in rough domains $\Omega$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $\Omega$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.

Nonlinear Analysis, 2018

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in W 1,1 with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.

Communications on Pure and Applied Mathematics, 1991

In this paper we will consider the regularity problem for the following obstacle problem. (1.1) inf Z jrvj p dx among the functions in W(';), where is a bounded C 2 domain in < n (n 2) and ' and are C 2-functions de ned on with ', and 1 < p < 1 such that W(';) = fv 2 W 1;p () : v ? ' 2 W 1;p 0 () and v a.e. in g Remark 1.1. The C 2 assumptions on and ' are purely technical to avoid complications, as the reader may nd out later. Because of the convexity of the integrand, (1.1) has a unique solution u satisfying the variational inequality (see, for example LQ]).

Arabian Journal of Mathematics, 2020

We give an a-priori estimate near the boundary for solutions of a class of higher order degenerate elliptic problems in the general Besov-type spaces $$B^{s,\tau }_{p,q}$$ B p , q s , τ . This paper extends the results found in Hölder spaces $$C^s$$ C s , Sobolev spaces $$H^s$$ H s and Besov spaces $$B^s_{p,q}$$ B p , q s , to the more general framework of Besov-type spaces.

Indiana University Mathematics Journal, 2003

Abstract. This paper is the first part of a program aimed at studying the regularity of sub-elliptic free boundaries. In the setting of Carnot groups we establish the optimal interior regular-ity of the solution to the obstacle problem in terms of the Folland-Stein non-isotropic class Г1, 1 ...

Nonlinear Analysis: Theory, Methods & Applications, 2000

Analysis &amp; PDE, 2012

We continue the development, by reduction to a first-order system for the conormal gradient, of L 2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence-form second-order complex elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere nontangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains. Andreas Rosén was formerly called Andreas Axelsson.

Communications on Pure and …, 2005