Hessian Estimates for Laplacian Eigenfunctions

We study Laplacian eigenfunctions $-\Delta \phi_k = \lambda_k \phi_k$ with a Dirichlet condition on bounded domains $\Omega \subset \mathbb{R}^n$ with smooth boundary. Classical estimates for how large such an eigenfunction and its derivative can be are $$ \left\| \phi_k \right\|_{L^{\infty}(\Omega)} \lesssim \lambda_k^{\frac{n-1}{4}} \qquad \mbox{and} \qquad \left\| \nabla \phi_k \right\|_{L^{\infty}(\Omega)} \lesssim \lambda_k^{\frac{n+1}{4}}$$ and these are sharp. We prove a sharp estimate for the second derivatives $$ \| D^2 \phi_k(x)\|_{L^{\infty}(\Omega)} \lesssim \lambda_k^{\frac{n+3}{4}}.$$ The main ingredient is a quantitative version of the following principle for solutions of the heat equation that may be of interest in itself: if $u(t,\cdot)$ has second derivatives much larger than $u(0,\cdot)$, then there is a time $0 < s < t$ such that $u(s,\cdot)$ has a large second derivative on the boundary.

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