A first-variation formula for eigenvalue functionals at the essential-spectrum edge characterizes all maximizing weights for Laplace and Steklov problems.
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For n at least 7 on positive manifolds with non-negative constant boundary mean curvature and a nonumbilic point, least-energy nodal solutions to the Yamabe boundary-value problem exist.
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Eigenvalue optimization via a first-variation formula
A first-variation formula for eigenvalue functionals at the essential-spectrum edge characterizes all maximizing weights for Laplace and Steklov problems.
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Sign-changing solutions to the Yamabe problem on manifolds with boundary
For n at least 7 on positive manifolds with non-negative constant boundary mean curvature and a nonumbilic point, least-energy nodal solutions to the Yamabe boundary-value problem exist.