In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on $R^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in $R^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.
Spectral optimization problems for potentials and measures
BUTTAZZO, GIUSEPPE;Velichkov B.
2014-01-01
Abstract
In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on $R^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\int V^{-p}dx \le m$ with $0
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