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Bounded weak solutions with Orlicz space data: an overview

David Cruz-Uribe

Abstract

It is well known that non-negative solutions to the Dirichlet problem $Δu =f$ in a bounded domain $Ω$, where $f\in L^q(Ω)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(Ω)} \leq C\|f\|_{L^q(Ω)}$. We generalize this result by replacing the Laplacian with a degenerate elliptic operator, and we show that we can take the data $f$ in an Orlicz space $L^A(Ω)$ that, in the classical case, lies strictly between $L^{\frac{n}{2}}(Ω)$ and $L^q(Ω)$, $q>\frac{n}2$.

Bounded weak solutions with Orlicz space data: an overview

Abstract

It is well known that non-negative solutions to the Dirichlet problem in a bounded domain , where , , satisfy . We generalize this result by replacing the Laplacian with a degenerate elliptic operator, and we show that we can take the data in an Orlicz space that, in the classical case, lies strictly between and , .

Paper Structure

This paper contains 3 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction: uniformly elliptic operators
  2. Degenerate elliptic operators
  3. Further remarks

Key Result

Theorem 1.1

Let $f\in L^q(\Omega)$, $q>\frac{n}{2}$, $Q$ uniformly elliptic, and $v=1$. If $u$ is a non-negative weak solution of dp, then

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1