Existence and Lipschitz Regularity for Minima (original) (raw)

A boundary regularity result for minimizers of variational integrals with nonstandard growth

Nonlinear Analysis, 2018

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in W 1,1 with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.

Regularity results for minimizers of irregular integrals with (p,q) growth

Forum Mathematicum, 2000

We consider variational integrals f Du dx where u X 3 R N and the convex function f has pY q growth jzj p f z Ljzj q 1, p`q. We prove local Lipschitz continuity of minimizers in the scalar case and in some vectorial cases. We further prove higher integrability and di¨erentiability for local minimizers. The results cover the case in which f is degenerate convex. A main feature of the paper is that we do not assume that f is di¨erentiable everywhere.

Regularity results for vectorial minimizers of a class of degenerate convex integrals

Journal of Differential Equations

We establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with (p, q)-growth conditions. We assume that the nonhomogeneous densities are uniformly convex and have a radial structure, with respect to the gradient variable, only at infinity. The results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand.

Lipschitz minimizers for a class of integral functionals under the bounded slope condition

arXiv (Cornell University), 2020

We consider the functional Ω g(∇u + X *) dL 2n where g is convex and X * (x, y) = 2(−y, x) and we study the minimizers in BV(Ω) of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on g, there exists a unique minimizer which is also Lipschitz continuous.