Zhikov, Vasilij V. On some variational problems. (English) Zbl 0917.49006 Russ. J. Math. Phys. 5, No. 1, 105-116 (1997). 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) Cited in 1 ReviewCited in 180 Documents MSC: 49J20 Existence theories for optimal control problems involving partial differential equations 49N60 Regularity of solutions in optimal control 35J60 Nonlinear elliptic equations Keywords:nonlinear elliptic system; existence of solutions; Meyers type estimate; variational problems; thermistor problem; regularity; Lavrentiev phenomenon PDF BibTeX XML Cite \textit{V. V. Zhikov}, Russ. J. Math. Phys. 5, No. 1, 105--116 (1997; Zbl 0917.49006)