Alfred Wagner

Optimal Shape Problems for Eigenvalues

ABSTRACT We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.

On some rescaled shape optimization problems

Abstract We consider Cheeger-like shape optimization problems of the form Min{|Ω| α J(Ω) : Ω ⊂ D} where D is a given bounded domain and α is above the natural scaling. We show the existence of a solution and analyze as J(Ω) the particular cases of the compliance functional C(Ω) and of the first eigenvalue λ1(Ω) of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.

Sobolev Constants in Disconnected Domains

A Sobolev constant is studied which generalizes the torsional rigidity. Some qualitative properties are derived. It is then used to estimate the L∞‐ norm of quasilinear boundary value problems with variable coefficients. The techniques used are direct methods from the calculus of variations and level line methods. An optimization problem is discussed which is crucial for avoiding the coarea formula for the estimates.A Sobolev constant is studied which generalizes the torsional rigidity. Some qualitative properties are derived. It is then used to estimate the L∞‐ norm of quasilinear boundary value problems with variable coefficients. The techniques used are direct methods from the calculus of variations and level line methods. An optimization problem is discussed which is crucial for avoiding the coarea formula for the estimates.

Lateral overdetermination of the FitzHugh-Nagumo system

The FitzHugh–Nagumo system, composed of a semilinear parabolic equation coupled to a linear ordinary equation, captures the qualitative dynamics of an excitable fibre. The semilinear term encodes the fibres nonlinear membrane conductance that underlies its excitability. We here establish conditions under which a semilinear term exists that corresponds to specified Neumann data at each end and Dirichlet data at one end.

Second Domain Variation for Problems with Robin Boundary Conditions

In this paper, the first and second domain variations for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions are computed. Extremal properties of the ball among nearly spherical domains of given volume are derived. The discussion leads to a Steklov eigenvalue problem. As a by-product, a general characterization of the optimal shapes is obtained.

Domain Derivatives for Energy Functionals with Boundary Integrals

This paper deals with domain derivatives of energy functionals related to elliptic boundary value problems. Emphasis is put on boundary conditions of mixed type which give rise to a boundary integral in the energy. A formal computation for rather general functionals is given. It turns out that in the radial case the first derivative vanishes provided the perturbations are volume preserving. In the simplest case of a torsion problem with Robin boundary conditions, the sign of the first variation shows that the energy is monotone with respect to domain inclusion for nearly circular domains. In this case also the second variation is derived.

Shape optimization for an elliptic operator with infinitely many positive and negative eigenvalues

Abstract This paper deals with an eigenvalue problem possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domains.