Revisiting multi-phase variational problems: A Muckenhoupt weight approach

Abstract

In this paper, we investigate the regularity theory of local minimizers of multi-phase energy functionals. As a key feature of our work, instead of the classical H\"older continuity assumptions on the modulating coefficients and interaction between the growth exponents, we assume that these coefficients belong to a suitable class of Muckenhoupt weights. The presence of multiple growth phases with the degenerate or singular nature of Muckenhoupt weights poses substantial analytical difficulties that prevent a direct application of classical theory. Our approach requires a refinement of localized energy estimates and an adapted iteration scheme that exploits the reverse H\"older properties of the Muckenhoupt weights. As our main results, we establish the higher integrability, local boundedness, and H\"older continuity of local minimizers. Most notably, we prove the Harnack inequality for non-negative local minimizers, which, to the best of our knowledge, stands as the first result of its kind in the multi-phase setting involving Muckenhoupt modulating coefficients. This paper is a contribution toward a better understanding of the qualitative behavior of minimizers in non-uniformly elliptic variational problems, and offers a new framework that complements the existing literature beyond the classical H\"older continuity of modulating coefficients.