Second order regularity of solutions of elliptic equations in divergence form with Sobolev coefficients

TL;DR

This paper establishes $L^p$ estimates for second derivatives of weak solutions to elliptic divergence form equations with Sobolev coefficients, advancing understanding of regularity under minimal coefficient regularity assumptions.

Contribution

It provides new $L^p$ regularity estimates for second derivatives of solutions with Sobolev coefficients, extending classical results to less regular coefficient settings.

Findings

Derived $L^2$ estimates for second derivatives involving Sobolev coefficients.

Established conditions on $f$ and coefficients for regularity estimates.

Provided explicit bounds depending on the Sobolev norm of coefficients.

Abstract

We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $\Omega\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(\Omega) \bigcap L^s(\Omega)$ $$\|\Delta u\|_{2} \leq \begin{cases} c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_q^2\|f\|_s, & \text{if } 1 < s < d/2, \frac{1}{2}=\frac{2}{q}+ \frac{1}{s} - \frac{2}{d}\\ c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_4^2\|f\|_s, & \text{if } s > d/2 \end{cases}.$$