Nonstandard growth optimization problems with volume constraint

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

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