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

#### 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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Bucur, D.; Buttazzo, Giuseppe; Velichkov, B.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/466883