If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem $$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, \partial\Omega.$$

Does there exists some result to the limit $$\lim_{t\to 0^+}\frac{\lambda_1(B+t(A-B)) - \lambda_1(B)}{t}$$ in one dimension case ($n=1$), even for special sets $A,B$? And for $\lambda_k, \,\, k>1$ when ($n=1$)? And in higher dimensions?

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    $\begingroup$ So you're asking if $\lambda_1(\Omega)$ is "differentiable" w.r.t. $\Omega$... is that a natural question to ask? For example: Is it at least known that this function is "continuous" w.r.t. $\Omega$ (what ever that means) ? $\endgroup$ – Johannes Hahn Oct 16 '15 at 17:56

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