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Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions

El Bahja Hamid

Opuscula Mathematica

2023

Vol. 43, no. 6

759--788

We study the Cauchy–Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard p(x, t), q(x, t)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

Regularity theory for superquadratic energy functionals related to nonlinear Hencky materials in three dimensions

Breit D.

Journal of Applied Analysis

2012

Vol. 18, nr 1

1-31

In this paper we discuss partial regularity results concerning local minimizers u:R3 ⊃ Ω → R3 of variational integrals of the form ∫ Ω {α(|div(w)|)+b(|εD(w)|)}dx, where a and b are N-functions of rather general type. We prove partial regularity results under quite natural conditions between a and b. Furthermore we can extend this to the non-autonomous situation which finally leads to the study of minimizers of the functional ∫ Ω{(1+|div(w)|2)p(x)/2+(1+|εD(w)|2)q(x)/2}dx, where p and q are Lipschitz-functions.