On some variational problems. (English) Zbl 0917.49006

The author establishes a Meyers type estimate for solutions of variational problems with the integrand \[ f(x,\nabla u)=|\nabla u+ g(x)|^{p(x)}/p(x) \] provided that \[ \begin{aligned} p(x) & \geq \alpha\geq 1,\\ | p(x) & - p(y)|\leq C/|\ln| x-y| |,\quad | x-y|\leq 1/2.\end{aligned}\tag{1} \] This estimate enables the author to prove the existence of a solution for the thermistor problem \[ \text{div}(|\nabla u|^{p(x)-2}\nabla u)= 0\quad\text{in }\Omega,\quad\Delta p+\lambda|\nabla u|^{p(x)}= 0\quad\text{in }\Omega, \]
\[ u= h\quad\text{on }\partial\Omega,\quad h\in W^{1,\infty}(\Omega);\quad p= \alpha\quad\text{on }\partial\Omega,\quad \alpha\in R,\quad \alpha>1, \] with \(\lambda>0\) small enough.
There are given various examples which illustrate the role of the condition (1) for the regularity of solutions of variational problems and the connections with the Lavrentiev phenomenon.
Reviewer: U.Raitums (Riga)


49J20 Existence theories for optimal control problems involving partial differential equations
49N60 Regularity of solutions in optimal control
35J60 Nonlinear elliptic equations