The optimal shape of a dendrite sealed at both ends
On a Bernoulli problem with geometric constraints
A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by ≥ 0 and its boundary to contain a segment of the hyperplane { = 0} where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution...
Erratum of “The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent”
On a Bernoulli problem with geometric constraints
A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by ≥ 0 and its boundary to contain a segment of the hyperplane { = 0} where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution...
The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of...
On the best observation of wave and Schrödinger equations in quantum ergodic billiards
This paper is a proceedings version of the ongoing work [
Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
We consider the wave and Schrödinger equations on a bounded open connected subset of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset of during a time interval with . It is well known that, if the pair satisfies the Geometric Control Condition ( being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be...
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