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Interior pointwise $C^α$ regularity for elliptic and parabolic equations with divergence-free drifts

Yuanyuan Lian

Abstract

We investigate the interior pointwise $C^α$ regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior $C^α$ regularity for some $0<α<1$ has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise $C^α$ regularity for any $0<α<1$ provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.

Interior pointwise $C^α$ regularity for elliptic and parabolic equations with divergence-free drifts

Abstract

We investigate the interior pointwise regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior regularity for some has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise regularity for any provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.
Paper Structure (3 sections, 10 theorems, 132 equations)

This paper contains 3 sections, 10 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. $C^{\alpha}$ regularity for elliptic equations
  3. $C^{\alpha}$ regularity for parabolic equations

Key Result

Theorem 1.20

Let $0<\alpha<1$ and $u\in H^1(B_1)$ be a weak solution of (E) in e.equation. Suppose that where $\delta$ is small and depends only on $n, p,\alpha$ (or $n, \alpha$ or $n, q,\alpha$) (see e1.5). Then $u\in C^{\alpha}(0)$, i.e., there exists a constant $P$ such that and where $C$ depends only on $n, p,\alpha$ (or $n, \alpha$ or $n, q,\alpha$).

Theorems & Definitions (36)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 26 more