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Let $\Omega\subset \mathbb R^n$ be a bounded open set with Lipschitz boundary, and for each $t\in[0,1]$, let $\phi_t:\Omega\to \mathbb R^n$ be an embedding (i.e. $\phi_t:\Omega\to \phi_t(\Omega)$ is a homeomorphism).

Let $\Omega_t:=\phi_t(\Omega)$. Consider the eigenvalue problem $$ \begin{cases} -\Delta u_t=\lambda_t u_t, &\text{in }\Omega_t \\ u_t=0, &\text{on }\partial\Omega_t \end{cases}$$

where $\lambda_t$ is the first eigenvalue of $-\Delta$ on $\Omega_t$.

How is the regularity of $\lambda_t$ (as a function of $t$) related to the regularity of $\phi_t$ (as a function of $t$ and $x$)?

I would imagine when $\phi_t$ is smooth in both $t$ and $x$, $\lambda_t$ should be smooth in $t$ too. If this is true, what can we say about $\lambda_t$ when $\phi_t$ has lower regularity?

More generally, if we replace $-\Delta u$ by a general divergence elliptic operator $-\nabla(A(x)\nabla u)+b\cdot\nabla u+cu$, how do the regularities of $A,b,c$ relate to that of $\lambda_t$?

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  • $\begingroup$ If you are mainly interested in general elliptic operators, you can just pull back everything to $\Omega$ and turn this into a question about eigenvalues of parameter dependent elliptic operators. At least something like Lipschitz-regularity of those sounds easy enough that it should be in some textbook somewhere. $\endgroup$ Commented Apr 20 at 17:52
  • $\begingroup$ @mlk I did a search on common textbooks and didn't find anything. Do you have any references for that? $\endgroup$ Commented Apr 20 at 17:58

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In the paper by Ahmad El Soufi and Ahmad Ilias, "Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold", Ill. J. Math. 51, No. 2, 645-666 (2007), ArXiv:0705.1263, MR2342681, Zbl 1124.49035, precisely in sections two and three (and the references therein), the authors claim in the slightly more general setting of Riemannian manifolds that if one deforms the domain 'analytically' by acting on $\Omega$ by an analytic one-parameter family of diffeomorphisms $f_\epsilon$ satisfying $f_0 = \mathrm{Id}$ and $f_\epsilon(\partial \Omega) = \partial \Omega_\epsilon$ to produce a family of deformed domains $\Omega_\epsilon = f_\epsilon(\Omega)$, then the eigenvalue functionals $\lambda_{k, \epsilon}$ will always be continuous, and in particular $\lambda_{1 , \epsilon}$ will even be differentiable. Nevertheless, they claim that the general perturbation theory of unbounded self-adjoint operators enables them to show that $\lambda_{k,\epsilon}$ admits a one-sided derivative at zero.

It seems that there are parallels with stability properties of the spectrum of finite-dimensional real-symmetric or Hermitian matrices, where one obtains Lipscitz continuity of the eigenvalues (from an application of the Courant-Fischer minmax principle) and in the case of a simple spectrum, the eigenvalue functionals vary smoothly in the neighbourhood of such a matrix (essentially using the implicit function theorem on the characteristic polynomial).

Would be interested to know how far this connection goes.

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