Giuseppe Cardone

We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele–Shaw equation with memory coupled with the convective Cahn–Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model.

We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele-Shaw equation with memory coupled with the convective Cahn-Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model.

We consider a mixed boundary value problem for the Poisson equation in a bounded planar domain. Most of the boundary is endowed with the Neumann condition, while the Dirichlet one is imposed on a periodic family of short segments of length [Formula: see text]. Consequently, the limit problem, corresponding to [Formula: see text], has mixed boundary conditions with two collision points (where the type of boundary condition changes). The asymptotic representation of the solution to the original, singularly perturbed problem includes boundary layers of two types, namely the exponential one near the segments and the power-law one in the vicinity of the collision points. These boundary layers are solutions of certain problems in the half-strip and the half-plane, respectively. The power-law boundary layer causes some surprising phenomena, including the square-root singularities of the main correction asymptotic term at the collision points and their linear depending on the quantity [Formula: see text].

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an example of such a domain perturbation. Let [Formula: see text] be a (not necessarily bounded) domain in [Formula: see text]. We perturb it to [Formula: see text] where [Formula: see text] are closed surfaces with small suitably scaled holes (“windows”) through which the bounded domains enclosed by these surfaces (“resonators”) are connected to the outer domain. When [Formula: see text] goes to zero, the resonators shrink to points. We prove that in the limit [Formula: see text] the spectrum of the Laplacian on [Formula: see text] with the Neumann boundary conditions on [Formula: see text] and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on [Formula: see text] and the numbers [Formula: see text], [Formula: see text], being equal to [Formula: see text] times the limit of the ratio between the capacity of the [Formula: see text]th window and the volume of the [Formula: see text]th resonator. We obtain an estimate on the rate of this convergence with respect to the Hausdorff-type metrics. Also, an application of this result is presented: we construct an unbounded waveguide-like domain with inserted resonators such that the eigenvalues of the Laplacian on this domain lying below the essential spectrum threshold do coincide with the prescribed numbers.