Partial and full boundary regularity for minimizers of functionals with nonquadratic growth.(English)Zbl 1066.49022

The authors discuss variational integrals of the form $F(u,\Omega) =\int_\Omega(A(x,u)Du\cdot Du)^{p/2}\,dx$ with exponent $$1 <p< \infty$$ and coefficients $$A = (A^{\alpha\beta}_{ij})_{1\leq \alpha,\beta\leq n,1\leq i,j\leq N}$$ being continuous, elliptic and bounded on $$\Omega\times \mathbb{R}^N$$. Here $$\Omega$$ is a bounded Lipschitz domain in $$\mathbb{R}^n$$, $$n\geq 2$$, and $$F(\cdot,\Omega)$$ is defined on the Sobolev space $$W^1_p(\Omega,\mathbb{R}^N)$$. Given a function $$h$$ of class $$W^1_s(\Omega,\mathbb{R}^N)$$ for some exponent $$s > n$$, the authors adress the question of boundary regularity for $$F(\cdot,Q)$$-minimizers subject to the condition that $$u = h$$ on $$\partial\Omega$$. As a first result it is shown that $$u$$ is partially Hölder continuous up to the boundary with Hölder exponent $$1-n/s$$, in a second part full regularity under more restrictive hypothesis on the coefficients $$A(x,u)$$ is established.

MSC:

 49N60 Regularity of solutions in optimal control