L\({}^{\infty}\)-regularity for variational problems with sharp non- standard growth conditions. (English) Zbl 0711.49058

Summary: It is proved that the solutions of Dirichlet problems related to a class of differential equations which includes the following \[ \sum^{n}_{i=1}\frac{\partial}{\partial x_ i}(| u_{x_ i}|^{q_ i-2}u_{x_ i})=\sum^{n}_{i=1}\frac{\partial}{\partial x_ i}(f_ i)\text{ in } \Omega,\quad u=u_ 0\text{ on } \partial \Omega \] where \(\Omega\) is a bounded open subset of \({\mathbb{R}}^ n\), \(\mu\) is a scalar function, \(f_ i\in L^{\infty}(\Omega)\) and \(q_ i>1\) for \(i=1,2,...,n\), are bounded in \({\bar \Omega}\) (if \(u_ 0\) given on the boundary is bounded), under the assumption that the exponents \(q_ i\) satisfy the inequality \(\bar q^*>q\), where \[ q=\max_{i}\{q_ i\},\quad \frac{1}{\bar q}=\frac{1}{n}\sum^{n}_{i=1}\frac{1}{q_ i},\quad \bar q^*=\frac{n\bar q}{n-\bar q}\quad (\bar q<n). \] An analogous result is also given for integrals of variational calculus.


49N60 Regularity of solutions in optimal control
35B65 Smoothness and regularity of solutions to PDEs
35J20 Variational methods for second-order elliptic equations