Lipschitz Bounds and Nonuniform Ellipticity

Abstract

We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 Wiley Periodicals, Inc.

Double phase problems with variable growth and convection for the Baouendi–Grushin operator

“…We refer to the works of Adimurthi et al [2], Baroni et al [8], Colasuonno and Pucci [12], Colombo and Mingione [13] for relevant applications of the Caffarelli-Kohn-Nirenberg inequality. For recent contributions to the study of double-phase problems we refer to Beck and Mingione [9], Papageorgiou et al [24,25], and Zhang and Rȃdulescu [31].…”

Anisotropic double-phase problems with indefinite potential: multiplicity of solutions

“…The functional P p,q defined in (3) appears as an upgraded version of V. Again, in this case, the modulating coefficient a(x) dictates the geometry of the composite made by two differential materials, with hardening exponents p and q, respectively. The study of non-autonomous functionals characterized by the fact that the energy density changes its ellipticity and growth properties according to the point has been continued in a series of remarkable papers by Mingione et al [6,7,9]. This work continues the recent paper by Papageorgiou, Rȃdulescu & Repovš [26], where the authors consider parametric equations driven by the p(z)-Laplacian plus an indefinite potential term.…”

Higher integrability for variational integrals with non-standard growth

“…Moreover, in two dimensions we cannot improve the previous results on higher differentiability and partial regularity of, e.g., [7,18], see [8] for a full regularity result under Assumption 3 with n = 2 and q p < 2. Finally, we mention the recent paper [3] where optimal Lipschitz-estimates with respect to a right-hand side are proven for functionals with ( p, q)-growth.…”