Sobolev and Lipschitz regularity for local minimizers of widely degenerate anisotropic functionals (original) (raw)

Partial regularity for anisotropic functionals of higher order

ESAIM: Control, Optimisation and Calculus of Variations, 2007

We prove a C k,α partial regularity result for local minimizers of variational integrals of the type I(u) = Ω f (D k u(x))dx, assuming that the integrand f satisfies (p, q) growth conditions. Mathematics Subject Classification. 35G99, 49N60, 49N99.

Regularity under sharp anisotropic general growth conditions

Discrete and Continuous Dynamical Systems - Series B, 2008

We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type (1) below, under sharp assumptions on the exponents p i in terms of p * : the Sobolev conjugate exponent of p; i.e., p * = np n−p ,

Pointwise bounds for minimizers of some anisotropic functionals

Nonlinear Analysis, 2018

We consider variational integral functionals ∫ Ω g(x, u(x), Du(x)) dx, where Ω is a bounded open subset in R n and the integrand g(x, s, ξ) = f (x, ξ)+b(x)s is not subjected to any growth condition from above neither satisfies any regularity assumption, growing linearly with respect to the s-variable. Their interest relies on the fact that they are strictly connected with optimal transport problems with congestion effects. The aim of this paper is to find sufficient conditions on the boundary datum u * in order to obtain global and explicit estimates for the solutions of the following minimization problem min {∫ Ω g(x, u(x), Du(x)) dx; u ∈ u * + W 1,1 0 (Ω) } .