Shape Optimization of a Weighted Two-Phase Dirichlet Eigenvalue

This article is concerned with a spectral optimization problem: in smooth bounded domain $${\Omega }$$ , for function m and nonnegative parameter $$\alpha $$ consider the first eigenvalue $$\lambda _\alpha (m)$$ of operator $${\mathcal {L}}_m$$ given by {L}}_m(u)= -{\text {div}} \left( 1+\alpha m)\nabla u\right) -mu$$ . Assuming uniform pointwise integral bounds on m, we investigate issue minimizing respect to m. Such problem related so-called “two phase extremal problem” arises naturally, instance population dynamics where it survival ability species domain. We prove that unless ball, this has no “regular” solution. then provide careful analysis case ball by: (1) characterizing solution among all radially symmetric resources distributions, help new method involving homogenized version problem; (2) proving more general setting stability result centered distribution monotonicity principle second order shape derivatives which significantly simplifies analysis.