## 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.

### MSC:

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

### Keywords:

L$${}^{\infty }$$-regularity; Dirichlet problems