Hölder continuity of local minimizers (Q1304648)

The authors study the regularity of local minimizers \(u\in W^1_{p,\text{loc}}(\Omega)\), \(p> 1\), \(\Omega\) open in \(\mathbb{R}^n\), of the variational integral \[ J[u]= \int_\Omega F(x,u,\nabla u) dx \] with integrand \(F\) of the form \((\nu>0,\;0\leq \mu\leq 1)\) \[ F(x,u,z)= \nu(\mu+|z|^2)^{{p\over 2}}+ f(x,u,z), \] \(f\) being convex in \(z\) and satisfying the growth estimate \((L> 0)\) \[ 0\leq f(x,u,z)\leq L(1+|z|^2)^{{p\over 2}}. \] Note that no smoothness or ellipticity are imposed on \(f\). The main result states that local minimizers are locally Hölder continuous with any exponent \(0<\alpha<1\). One ingredient of the proof is a variational principle due to Ekeland.