Harnack's inequality for general solutions with nonstandard growth

Abstract

Abstract. We prove Harnack's inequality for general solutions of elliptic equationswhere A and B satisfy natural structural conditions with respect to a variable growth exponent p(x). The proof is based on a modification of the Caccioppoli inequality, which enables us to use existing versions of the Moser iteration.

The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth

“…In most cases the assumptions in these results coincides with optimal assumptions in wellknown special cases. The important special case are the variable exponent case F (x, t) := |t| p(x) [1,25,28,32,45], the Orlicz case F (x, t) := ϕ(|t|) [2,36,37], and the double phase case F (x, t) := |t| p + a(x)|t| q [3,14,15]. The survey [11] includes more references of variational problems and partial differential equations of generalized Orlicz growth, while the recent monographs [21,34] present the theory of the underlying function spaces.…”

The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate

“…There are some results in the literature proved with the use of Caccioppoli-type inequality for weak subsolutions or supersolutions to PDEs. We would like to mention Liouville theorems [1,17], Harnack inequalities [2,4,25,27,39] and their further consequences-maximum and comparison principles [20,22]. The already mentioned applications are important in the regularity theory as some versions of the Harnack inequality imply that solutions are locally Hölder continuous [3].…”

Parabolic Harnack estimates for anisotropic slow diffusion

“…The most up-to-date result for (1.6) is in [6], where the Lipschitz regularity of its bounded solutions is proved for any choice of p i > 1. The Harnack inequality for non-standard elliptic problems has been the object of various works: [2,5,19,20,23,31,34] focus on isotropic equations with non-standard growth of p(x)-type, while [21,27] deal with energies with Uhlenbeck structure and non-standard growth. However, none of the frameworks considered therein cover the anisotropic equation (1.6): indeed, its Euler-Lagrange equation is degenerate/singular on the union of the coordinate axes, while non-standard functionals of p(x)-or Uhlenbeck-type exhibit this problem only at the origin.…”