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You could find a proof in Variation et optimisation des formes by Antoine Henrot and Michel Pierre, Section 2.2.3. The second property mentioned there is the following: An increasing sequence $( …

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \in [ …

I am interested in equations of the form $|\nabla d|= F(x)$, where $F(x)$ is piecewise constant and $d(x) = 0$ on $\Gamma_D$, a subset of the boundary. In particular, like in the figure, one can consi …

In Variation et optimisation des formes by A. Henrot and M. Pierre, in Chapter 2, Exercises section there is the following statement: Exercise 2.12 Let $(\Omega_n)$ be a sequence of open sets, hav …

I am interested in finding references regarding estimates of the form $$ \| D^2 u\|_{L^2(\Omega)} \leq C(\|f\|_{L^2(\Omega)}+\|g\|_{S} )$$ where $\|D^2 u\|_{L^2(\Omega)}^2 = \sum\limits_{i,j \in \{1,2 …

When minimizing numerically the eigenvalues of the Laplacian under area constraint in 2D it is observed that optimal shapes tend to have multiple eigenvalues. Take for example the simulations shown he …

This question is solved in the PhD thesis of Nicolas Landais: Problèmes de régularité en optimisation de forme. You can read the thesis here. The result is presented in Chapter 6. The conclusion is th …

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar domai …