A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region

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Mixed Laplace eigenfunctions lack interior critical points in convex domains with connected, small Dirichlet regions.
Two new estimates on the first mixed eigenvalue provide specific conditions for a sufficiently small Dirichlet region.
Analysis focuses on geometric properties of convex domains to understand eigenfunctions and critical points.
Highlights the significance of understanding eigenvalue behavior in relation to Dirichlet region size.

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Analysis shows Laplace eigenfunctions lack critical points in convex domains with small Dirichlet regions, implying eigenvalue estimates matter.

Lawford Hatcher (2025) studied this question.

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Analysis shows Laplace eigenfunctions lack critical points in convex domains with small Dirichlet regions, implying eigenvalue estimates matter.