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 Ω ⊂ Rn and α, c > 0, we consider the optimization problem inf{λΩ(α, E): E ⊂ Ω, |E| = c}, where λΩ(α, E) is related to the first eigenvalue to − div(g(|∇u|) |∇∇uu|) + g(u) |uu| + αχEg(u) |uu| 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 +∞.