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Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes

Tadele Mengesha, Enrique Otarola, Abner J. Salgado

Abstract

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$ with $p \in (1,\infty$). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.

Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes

Abstract

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class with ). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.
Paper Structure (11 sections, 8 theorems, 88 equations)

This paper contains 11 sections, 8 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Notations, preliminaries, and structural assumptions
  3. Notations
  4. $A_p$ weights
  5. Assumptions on the nonlinearity
  6. Linear elliptic problems with continuous coefficients
  7. Quasilinear problems
  8. Discretization
  9. The Ritz projection
  10. The linear problem
  11. The quasilinear problem

Key Result

proposition 1

Let $D \subset \mathbb{R}^d$ be bounded and convex. Let $p \in (1,\infty)$ and $\omega \in A_p$. For every ${\mathcal{F}} \in W^{-1,p}(\omega,D)$, there is a unique $U \in W^{1,p}_0(\omega,D)$ such that $-\Delta U = {\mathcal{F}}$ in $\mathscr{D}'(D)$. Moreover, where $C_\Delta$ does not depend on $D$ and depends on $\omega$ only through $[\omega]_{A_p}$.

Theorems & Definitions (18)

  • remark 1: structure conditions
  • proposition 1: weighted stability
  • theorem 1: well posedness
  • lemma 1: Gårding--like inequality
  • proof
  • corollary 1: a priori estimate
  • proof
  • proof : Proof of Theorem \ref{['thm:LinearWellPosed']}
  • theorem 2: existence and uniqueness
  • proof
  • ...and 8 more