Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

Abstract

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for A-superharmonic functions with nonlinear operator A : Ω × R n → R n having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. Finally, we give a variant of Hedberg-Wolff theorem on characterization of the dual of the Orlicz space.

Measure data systems with Orlicz growth

“…It has locally integrable gradient if and only if p > 2 − 1/n. For the Orlicz counterpart of this fact see [19,Corollary 2.4].…”

“…by [3,15,18,23]. For pioneering work on the equations with Orlicz growth we refer to [37,49,56], while for regularity to their measure data counterparts see [5,10,14,16,19]. The cornerstone of studies on quasilinear elliptic systems in divergence form has been laid in [57,58].…”

Wolff potentials and measure data vectorial problems with Orlicz growth

“…Orlicz one [21,69], as well as a counterpart proven for systems involving p-Laplace operator [26,63]. We study the nonstandard growth version of pointwise estimates involving suitably generalized potential of the Wolff type and infer their regularity consequences.…”

“…In the scalar case one can infer continuity or Hölder continuity of the solution to −divA(x, Du) = µ when the dependence of the operator on the spacial variable is merely bounded and measurable and the growth of A with respect to the second variable is governed by an arbitrary doubling Young function (cf. [21]). The same is not possible for systems even with far less complicated growth and null datum, cf.…”

“…see [21,69]. To our best knowledge, the only reference one can find on the related results for systems are [26,63] that involves problems with the p-Laplace operator.…”