Regularity of minimizers of autonomous convex variational integrals
We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,Ω) = \int_Ω /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon Ω\subset {\mathbb R}^n \to {\mathbb R}^N$ satisfying a Dirichlet boundary condition. The integrands $F$ are assumed to be autonomous, convex and of $(p,q)$ growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range $1<p \leq q < p^{\ast}$, where $p^{\ast}$ denotes the Sobolev conjugate exponent of $p$.
math.AP↗