Reference about the affect of the change of boundary to the eigenvalue of Laplacian

Question

In my view , it is hard to know how the principl eigenvalue exchange when boundary change. For example , when we deform the boundary under some geometric flow, how to change the eigenvalue of Laplacian ? I fail to find relevant paper.

Answer 1

The one thing I can say is that it changes continuously when the domain is smoothly deformed.

Recall that the smallest eigenvalue of $-\Delta$ with zero boundary conditions can be described as the reciprocal of the optimal constant in Poincare's inequality, i.e. $$ \lambda(\Omega) = \inf \frac{\int_\Omega |\nabla u|^2}{\int_\Omega |u|^2}, $$ where the infimum is taken over all nonzero functions $u \in W_0^{1,2}(\Omega)$ or equivalently $u \in C_c^\infty(\Omega)$.

Say there is a diffeomorphism $X \ni x \mapsto y \in Y$ with bi-Lipschitz constant $L \geqslant 1$. If two functions $u,v$ are related by $v(x) = u(y(x))$, then \begin{align} \int_X |\nabla v(x)|^2 dx & = \int_X \left|\nabla u(y(x)) \cdot \frac{dy}{dx}\right|^2 dx \\ & \leqslant L^2 \int_Y |\nabla u(y)|^2 \left|\det \frac{dx}{dy}\right| dy \\ & \leqslant L^{n+2} \int_Y |\nabla u(y)|^2 dy, \end{align} similarly $$ \int_X |v(x)|^2 dx \geqslant L^{-n} \int_Y |u(y)|^2 dy. $$

This shows that $\lambda(X) \leqslant L^{2n+2} \lambda(Y)$; for the same reason $\lambda(X) \geqslant L^{-2n-2} \lambda(Y)$. If you deform your domain $\Omega$ by a group of diffeomorphisms $\Phi_t$, the bi-Lipschitz constant of $\Phi_t$ is close to $1$ for small $t$, so $\lambda(\Phi_t(\Omega))$ is continuous in $t$.