On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds

We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.

Sublinear elliptic problems with a Hardy potential

In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.

Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign

We consider the energy of the torsion problem with Robin boundary conditions in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the case of positive elasticity constants, the ball does not provide a minimum. For nearly spherical domains and elasticity constants close to zero the energy is largest for the ball. This result is true for general domains in the plane under an additional condition on the first non-trivial Steklov eigenvalue. For more general elasticity constants the situation is more involved and it is strongly related to the particular domain perturbation. The methods used in this paper are the series representation of the solution in terms of Steklov eigenfunctions, the first and second shape derivatives and an isoperimetric inequality of Payne and Weinberger cite{PaWe61} for the torsional rigidity.

Isoperimetric inequalities for the principal eigenvalue of a membrane and the energy of problems with Robin boundary conditions

An inequality for the reverse Bossel-Daners inequality is derived by means of the harmonic transplantation and the first shape derivative. This method is then applied to elliptic boundary value problems with inhomogeneous Neumann conditions. The first variation is computed and an isoperimetric inequality is derived for the minimal energy.

Second variation of domain functionals and applications to problems with Robin boundary conditions

In this paper the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions is computed. Minimality and maximality properties of the ball among nearly circular domains of given volume are derived. The discussion leads to the investigation of the eigenvalues of a Steklov eigenvalue problem. As a byproduct a general characterization of the optimal shapes is obtained.