TITLE: An inequality for eigenvalues of the Dirichlet problem
QUESTION [5 upvotes]: Is either of these inequalities true?
$$\lambda(tA + (1-t)B)\geq t\lambda(A) + (1-t)\lambda(B)$$
or
$$\lambda(tA + (1-t)B)\leq t\lambda(A) + (1-t)\lambda(B),$$
where $0\leq t \leq 1$, $A,B$ are bounded domains in $\mathbb{R}^n$ and $\lambda(\Omega)$ is an eigenvalue of the problem
$$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, \partial\Omega.$$

REPLY [3 votes]: The inequalities you state cannot hold. It is known that $\lambda(tA) = t^{-2}\lambda(A)$ so choosing $A = sB$ you would get
$$ \frac{1}{(ts+1-t)^2}\lambda(B) \leq \geq (t/s^2+1-t)\lambda(B)$$
Neither of the inequalities can hold for any $s$. Take $s=0.5$ and $s\to 0$.
However, a Brunn-Minkowski inequality holds for the $1$-homogeneous function $\lambda^{-1/2}$. In the paper https://core.ac.uk/download/pdf/81213094.pdf and the references therein you can see that the following inequality holds
$$ 
\lambda(tA+(1-t)B)^{-1/2} \geq t \lambda(A)^{-1/2}+(1-t)\lambda(B)^{-1/2}
$$
Therefore, an inequality of the type that you want holds for $\lambda^{-1/2}$. Equality cases are also described in the paper above, for the case when $A$ and $B$ are convex.