Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients

TL;DR

This paper proves an extrapolation result for the solvability of the Dirichlet problem for elliptic equations with complex coefficients, extending solvability from one Lebesgue space to a broader range under certain ellipticity conditions.

Contribution

It establishes a new extrapolation theorem for the $L^p$ Dirichlet problem for elliptic operators with complex coefficients satisfying $p$-ellipticity, broadening the known solvability range.

Findings

Solvability extends from $L^q$ to $L^p$ for a range of $p$ under $p$-ellipticity.

If coefficients are real or in two dimensions, solvability holds for all $p$ in $[q, \infty)$.

Provides conditions under which the Dirichlet problem is solvable for complex elliptic operators.

Abstract

In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as $p$-{\it ellipticity}. Specifically, let $\Omega$ be a chord-arc domain in $\mathbb R^n$ and the operator $\mathcal L = \partial_{i}\left(A_{ij}(x)\partial_{j}\right) +B_{i}(x)\partial_{i} $ be elliptic, with $|B_i(x)| \le K\delta(x)^{-1}$ for a small $K$. Let $p_0 = \sup\{p>1: A \,\,\text{is}\,\, \text{$p$-elliptic}\}$. We establish that if the $L^q$ Dirichlet problem is solvable for $\mathcal L$ for some $1<q< \frac{p_0(n-1)}{(n-2)}$, then the $L^p$ Dirichlet problem is solvable for all $p$ in the range $[q, \frac{p_0(n-1)}{(n-2)})$. In particular, if the matrix $A$ is real, or $n=2$, the $L^p$ Dirichlet problem is solvable for $p$ in the range $[q, \infty)$.