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Besov Regularity of Weak solutions to a Class of Nonlinear Elliptic Equations

Huimin Cheng, Feng Zhou

Abstract

In this article, we study a Besov regularity estimate of weak solutions to a class of nonlinear elliptic equations in divergence form. The main purpose is to establish Calderon-Zygmund type estimate in Besov spaces with more general assumptions on coefficients, non-homogeneous term and integrable index. By involving the Sharp maximal function, we establish an oscillation estimate of weak solutions in Orlicz-Sobolev spaces. By deriving a higher integrability estimate of weak solutions, we obtain the desired regularity estimate which expands the Calderon-Zygmund theory for nonlinear elliptic equations.

Besov Regularity of Weak solutions to a Class of Nonlinear Elliptic Equations

Abstract

In this article, we study a Besov regularity estimate of weak solutions to a class of nonlinear elliptic equations in divergence form. The main purpose is to establish Calderon-Zygmund type estimate in Besov spaces with more general assumptions on coefficients, non-homogeneous term and integrable index. By involving the Sharp maximal function, we establish an oscillation estimate of weak solutions in Orlicz-Sobolev spaces. By deriving a higher integrability estimate of weak solutions, we obtain the desired regularity estimate which expands the Calderon-Zygmund theory for nonlinear elliptic equations.
Paper Structure (11 sections, 7 theorems, 133 equations)

This paper contains 11 sections, 7 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. $N$-function
  4. Function Spaces
  5. Hardy-Littlewood Maximal Function and Sharp Maximal Function
  6. Weak Solution
  7. Iteration Lemma
  8. Oscillation Estimate and Higher Integrability
  9. Oscillation Estimate
  10. Higher Integrability
  11. Proofs of Theorem \ref{['theorem1.3']} and Theorem \ref{['MainTheorem']}

Key Result

Theorem 1.1

Let $0\!<\alpha<\!1$. Assume that $A$ satisfies hypotheses (A1)-(A3) and 1.4 with $0<\mu<1$, and $P(t)$ satisfies (P1) and (P2). If $u\in W^{1,P}_{\rm{loc}}(\Omega)$ is a weak solution to then $V_P(\nabla u)\in B^\alpha_{2,\infty}(\Omega)$ locally.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: 22
  • Definition 2.2
  • Definition 2.3: 24
  • Definition 2.4
  • Lemma 2.5: GM
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 7 more