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.


49N60 Regularity of solutions in optimal control