Removable sets in non-uniformly elliptic problems

Abstract

We analyze fine properties of solutions to quasilinear elliptic equations with double phase structure and characterize, in the terms of intrinsic Hausdorff measures, the removable sets for Hölder continuous solutions.2010 Mathematics Subject Classification. 35J60, 35J70.

Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy

“…The degeneracy term appearing in (1.1) is modelled upon the Double-Phase energy, which first appeared in [24][25][26] in the study of the Lavrentiev phenomenon and Homogeneization theory. It received lots of attention also from the viewpoint of regularity theory, look at [1,2,11,13] for a rather comprehensive account on the regularity of local minimizers of the variational integral W 1,p (Ω) ∋ w → min Ω |Dw| p + a(x)|Dw| q dx, a ∈ C 0,α (Ω), see also [10] for the obstacle problem and some potential theoretic considerations, [19] for the manifold constrained case, [12,18] for nonlinear Calderón-Zygmund-type results and [20] for the regularity features of viscosity solutions of the fractional Double-Phase operator |w(x) − w(y)| p−2 (w(x) − w(y)) |x − y| n+sp + a(x, y) |w(x) − w(y)| q−2 (w(x) − w(y)) |x − y| n+tq dy.…”

“…Linkages with interpolation methods [19], and Calderón-Zygmund estimates [16,22], have also been established, while, on a more applied sides, applications to image restorations problems have been recently given [27]. See also [14] for the obstacle problem and some potential theoretic considerations, [23] for the manifold constrained case and [24] for the regularity features of viscosity solutions of equations related to the fractional double phase integral This last paper is particularly important in our setting as it provides another instance of the basic regularity assumptions we are going to consider here; see comments after theorem 2.…”

Removable sets in elliptic equations with Musielak–Orlicz growth

“…Double phase growth operators. Within the framework developed in [15], in [12] removable sets are characterized for solutions to…”

Generalized Superharmonic Functions with Strongly Nonlinear Operator

“…Inspired by the significant attention paid lately to problems with strongly nonstandard and non-uniformly elliptic growth e.g. [11,15,21,23,30,47,50] we aim at developing basics of potential theory for problems with essentially broader class of operators embracing in one theory as special cases Orlicz, variable exponent and double-phase generalizations of p-Laplacian. To cover whole the mentioned range of general growth problems we employ the framework described in the monograph [26].…”