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A variational method for functionals depending on eigenvalues

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arxiv 2211.15632 v2 pith:W73TT3EK submitted 2022-11-28 math.AP math.DGmath.FA

Romain Petrides This is my paper · ORCID
classification math.APmath.DGmath.FA
keywords eigenvaluesfunctionalscombinationsdependingmethodsequencesvariationalclassical
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abstract

We perform a systematic variational method for functionals depending on eigenvalues of Riemannian manifolds. It is based on a new concept of Palais Smale sequences that can be constructed thanks to a generalization of classical min-max methods on $C^1$ functionals to locally-Lipschitz functionals. We prove convergence results on these Palais-Smale sequences emerging from combinations of Laplace eigenvalues or combinations of Steklov eigenvalues in dimension 2.

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  1. Maximizing higher eigenvalues in dimensions three and above

    math.SP 2025-06 conditional novelty 8.0 of 10

    For every closed manifold of dimension at least 3 and every k, the maximal k-th eigenvalue functional is attained by a measure induced by a locally stable harmonic map into a sphere.

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