On very weak solutions of certain elliptic systems with double phase growth
Yoshiki Kaiho
TL;DR
This work proves a higher integrability (self-improving) property for very weak solutions of higher-order elliptic systems with double phase growth. The authors develop a higher-order Lipschitz truncation adapted to the double-phase structure using a weighted mean value polynomial, prove a Sobolev–Poincaré inequality tailored to the operator, and derive a reverse Hölder inequality for the principal part. The main result shows there exists a $\delta\in(0,1)$ such that $\int_{\Omega_0} (|D^m u|^p + a(x)|D^m u|^q)^{1/\delta} dx < \infty$ for all $\Omega_0\Subset \Omega$, implying a very weak solution is in fact a weak solution under the stated conditions. The approach combines Lipschitz truncation, a Sobolev–Poincaré framework for a double-phase operator, reverse Hölder inequalities, and Gehring's lemma to achieve a sharp higher-integrability conclusion with explicit data-dependence.
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_Ω \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}(Ω), \end{equation} where $n,m \in \mathbb{N},\ n\ge 2,\,1 < p \le q < \infty,\,Ω\subset \mathbb{R}^n$ is an open set and $a:Ω\rightarrow [0,\infty)$ 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 Hölder inequality and an application of Gehring's lemma. Our contributions include estimates for weighted mean value polynomials and sharp Sobolev--Poincaré-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).