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A variational method for functionals depending on eigenvalues
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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.
Forward citations
Cited by 2 Pith papers
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Maximizing higher eigenvalues in dimensions three and above
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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Extremising eigenvalues of the GJMS operators in a fixed conformal class
Conformal eigenvalue extremals for GJMS operators of any order s and any index k exist under a gap condition for positive eigenvalues and unconditionally for negative eigenvalues, assuming a unique continuation property.
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