Riesz potential estimates for a class of double phase problems
This paper deals with the equation \[ -\text{div}(\gamma(x) A(x,D)u)= \mu,\tag{1} \] where \(\mu\) is a finite Borel measure defined on the bounded domain \(\Omega\subset\mathbb{R}^n\). The mapping \(A(x,z)\) is \(C^1\) regular in \(z\), \(A_z(\cdot)\) is Carathéodory regular and satisfies several non-standard growth, ellipticity and continuity assumptions. The coefficient \(\gamma\) is Dini-continuous. The authors prove in Theorem 1.1 an optimal pointwise gradient estimate for such a measure data problem via Riesz pontentials of \(\mu\) and log-Dini continuity assumptions imposed on the modulating coeffient \(a(x)\). As a consequence in Theorem 1.2 an optimized \(C^1\) regularity criterion for the weak solution \(u\) of (1) is found. \(u\) is supposed to belong to some appropriate Musiełak-Orlicz spaces.
scientific article; zbMATH DE number 6807498
scientific article