Consider the solution $$u(t,x) = \sum_{i=1}^\infty e^{-\lambda_i t} \hat{g}_i \phi_i(x)$$ of the Heat equation with initial data $g$ and homogeneous Dirichlet or Neumann boundary data (where we denote by $\lambda_i$ and $\phi_i$ eigenvalues and eigenfunctions of the Laplacian with the Dirichlet or Neumann boundary condition).

How can one estimate $$\Vert D^2 u \Vert_\infty?$$

Clearly if we are in only $1$ space dimension $D^2 = \partial_{xx}^2$, so we can use the heat equation to perform the estimate by computing one derivative in time. What happens in general if we are working on a bounded domain $\Omega \subset \mathbb R^N$ with $N >1$?


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