On very weak solutions of certain elliptic systems with double phase growth

TL;DR

This paper establishes higher integrability for very weak solutions of higher-order elliptic systems with double phase operators, extending previous results to more general settings involving derivatives, coefficients, and growth conditions.

Contribution

It introduces new estimates and inequalities for double phase operators, generalizing recent work to broader classes of elliptic systems with complex growth behaviors.

Findings

Proved higher integrability of very weak solutions.

Developed estimates for weighted mean value polynomials.

Established sharp Sobolev--Poincaré-type inequalities.

Abstract

In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_{\Omega} \left( |D^m u|^{p-2}D^m u + a(x)|D^m u|^{q-2}D^m u \right) \cdot D^m \varphi = 0 \quad \text{for any } \varphi \in C_c^{\infty}(\Omega), \end{equation} where $n,m \in \mathbb{N},\ n\ge 2,\,1 < p \le q < \infty,\,\Omega \subset \mathbb{R}^n$ is an open set and $a:\Omega \rightarrow [0,\infty)$ is a measurable function. The proof is based on a construction of an appropriate test function by the Lipschitz truncation technique, a deduction of a reverse H\"older inequality and an application of Gehring's lemma. Our contributions include estimates for weighted mean value polynomials and sharp Sobolev--Poincar\'e-type inequalities for the double phase operator. Our result can be viewed as a generalization with respect to the derivative order, the coefficient function and the growth conditions of the recent paper by Baasandorj, Byun and Kim (Trans. Amer. Math. Soc. 376:8733-8768,2023).