Nonstandard growth optimization problems with volume constraint

Abstract

In this article we study some optimal design problems related to nonstandard growth eigenvalues ruled by the $g-$Laplacian operator. More precisely, given $\Omega \subset {\mathbb{R}}^n$ and $\alpha ,c>0$ we consider the optimization problem $\inf \{{\lambda }_{\Omega }(\alpha ,E):E\subset \Omega ,|E|=c\}$, where ${\lambda }_{\Omega }(\alpha ,E)$ is related to the first eigenvalue to $$-\text{div}(g(|\nabla u|)\frac{\nabla u}{|\nabla u|})+g(u)\frac{u}{|u|}+\alpha {\chi }_{E}g(u)\frac{u}{|u|}\text{ in }\Omega$$ subject to Dirichlet, Neumann or Steklov boundary conditions. \\ We analyze existence of an optimal configuration, symmetry properties of them, and the asymptotic behavior as $\alpha$ approaches $+\infty$.

the definition of eigenvalue was clarified