The regularity of solutions to some variational problems, including the p-Laplace equation for 2 ≤ p< 3

Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy.
arrigo.cellina@unimib.it

Received: 1 February 2016
Revised: 6 July 2016
Accepted: 18 September 2016

Abstract

We consider the higher differentiability of solutions to the problem of minimizing ${\mathrm{\int }}_{\mathrm{\Omega }}\mathrm{\left[}\mathit{L}\mathrm{\right(}\mathrm{\nabla }\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{)}\mathrm{+}\mathit{g}\mathrm{(}\mathit{x,v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left)}\mathrm{\right]}\mathrm{d}\mathit{x} \mathrm{on} {\mathit{u}}^{\mathrm{0}}\mathrm{+}{\mathit{W}}_{\mathrm{0}}^{\mathrm{1}\mathit{,p}}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$where L(|ξ|)=1/p|ξ|^p and u⁰ ∈ W^1,p(Ω). We show that, for 2 ≤ p< 3, under suitable regularity assumptions on g, there exists a solution u to the Euler–Lagrange equation associated to the minimization problem, such that $\mathrm{\nabla }\mathit{u}\mathrm{\in }{\mathit{W}}_{\mathrm{loc}}^{\mathrm{1}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathit{.}$ In particular, for g(x,u) = f(x)u with f ∈ W^1,2(Ω) and 2 ≤ p< 3, any W^1,p(Ω) weak solution to the equation $\mathrm{div}\mathrm{\left(}\mathrm{\right|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathit{p}\mathrm{-}\mathrm{2}}\mathrm{\nabla }\mathit{u}\mathrm{)}\mathrm{=}\mathit{f}$is in W^2,2_loc(Ω).