Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

Abstract

We prove boundedness, Hölder continuity, Harnack inequality results for local quasiminima to elliptic double phase problems of p-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach and they are based on the De Giorgi method, a careful phase analysis and estimates in the intrinsic geometries.

Higher integrability and stability of $(p,q)$-quasiminimizers

“…Next, we report the De Giorgi Lemma, which has a key role in our paper and whose proof can be found in [40]. In order to do so, we start introducing some notations.…”

“…Quasiminimizers have been an active research topic for several years in the setting of doubling metric measure spaces supporting a Poincaré inequality. For instance, the boundary continuity for (p, q)-quasiminimizers on a bounded set Ω with fixed boundary data has been examined in [40], furthermore, it was proven that (p, q)quasiminimizers are (locally) Hölder continuous. In [24], Kinnunen, Marola, and Martio proved that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided that the limit function is finite at some point.…”

“…This paper is motivated by the work of Maasalo and Zatorska-Goldstein [32] and is a continuation of [40]. The novelty is that, as already anticipated, we include both p-Laplace and q-Laplace operators, involving also some measurable coefficient functions a and b, assuming only they are bounded away from zero and infinity.…”

“…In particular, in Section 2.3, we present Newtonian spaces, results related to the Poincaré inequality and also some useful properties concerning the space with zero boundary values. Section 3 is devoted to introduce the concept of (p, q)-quasiminimizers and report the De Giorgi type inequality, as proven in [40]. Section 4 deals with the higher integrability problem for quasiminimizers and Section 5 contains the proof of the stability result.…”