In this paper we study the behavior of solutions of the boundary value problem for the Poisson equation in a partially perforated domain with arbitrary density of cavities and mixed type conditions on their boundary. The corresponding spectral problem is also considered. A short communication of similar results can be found in [1].
We present a simple proof of the fact that if is a bounded domain in , , which is convex and symmetric with respect to orthogonal directions, , then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues must intersect the boundary. This result was proved by Payne in the case for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.
We consider a size structured cell population model where a mother cell gives birth to two daughter cells. We know that the asymptotic behavior of the density of cells is given by the solution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue gives the exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends on the division rule for the mother cell, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We use a...