Let $\Omega$ be an open set of finite measure. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue for the Laplace operator, i.e. $$- \Delta u = \lambda_1(\Omega) u, \ \ \ \ \ \text{in} \ \Omega$$ $$u = 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on} \ \partial \Omega $$ In variational terms $$\lambda_1(\Omega) = \min_{u \in H^1_0(\Omega)} {\frac{\int_{\Omega}{ ||\nabla u || ^2 dx}}{\int_{\Omega}{||u||^2 dx}}}$$

Consider the problem: $$\text{min} \{ \lambda_1(\Omega) : \Omega \subset \mathbb{R}^N, \text{Vol}(\Omega) = m \}$$ i.e. among all measurable open sets of the volume $m$ we find one that minimizes the first eigenvalue.

By Faber-Krahn inequality it is known that the solution to the above problem is the ball of volume $m$.

Let also $$\text{min} \{\lambda_1(\Omega) + C \ \text{Vol}(\Omega) : \Omega \subset \mathbb{R}^n \}$$

be a free boundary problem.

Proposition There exists $C > 0$ such that the problems above are equivalent

(equivalence means the following: if the solution to the first problem exists and unique, then the latter is the unique solution to the second problem and vice versa)

How i can check that?

My progress:

(1) First note that the solution to the first problem exists due to the inequality $$\lambda_1(\Omega) \geq \frac{C}{\text{Vol}(\Omega) ^ {\frac{2}{N}}}$$

(This follows from Sobolev inequality)

(2) The natural idea: given $m := \text{Vol}(\Omega)$ one should find $C > 0$ such that the minimum in the second problem is attained at the argmin of the first and vice versa. The latter means that it would be nice to find a kind of relation of the form: $f(m) = g(C)$, where $f, g$ are sufficiently nice functions.

The questions are:

  • How can i check the equivalence?
  • subquestion: how i can approach the second problem from the scratch, i.e. to find the minimum using some variational arguments? It sound that it's ok to consider a sort of "dilating variation", e.g., for $$J(\Omega) = \lambda_1(\Omega) + |C| \text{Vol}(\Omega)$$ consider $$J_t(\Omega) = \lambda_1(t \Omega) + |C| \text{Vol}(t \Omega) = \frac{\lambda_1(\Omega)}{t^2} + C t^{n} \text{Vol}(\Omega)$$

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