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We show that the nonnegative weak subsolutions of \begin{equation*} (u^{q})_t-\nabla\cdot(|\nabla u|^{p-2}\nabla u)=0,\ 1 < p < d_\mu,\ q>0 \end{equation*} are locally bounded in measure spaces equipped with a doubling non-trivial Borel measure $\mu$, of dimension $d_\mu$, supporting a weak Poincaré inequality.

We extend some properties to the solutions of free boundary problems of obstacle type with two phases for a class of heterogeneous quasilinear elliptic operators. Under a natural non degeneracy assumption on the interface, corresponding to the zero level set, we prove a continuous dependence result for the characteristic functions of each phase and we establish sharp estimates on the variation of its Lebesgue measure with respect to the $L^1$-variation of the data, in a rather general framework. For elliptic quasilinear equations that have solutions with integrable second order derivatives, we show that the characteristic functions of both phases are of bounded variations for non degenerating heterogeneous forces. This extends recent results for the obstacle problem and is a first result on the regularity of the free boundary of the heterogeneous two phases problem.

We prove existence of weak and strong solution for equations with the operator $\nabla\times(\alpha\nabla\times\cdot)$ in the space of divergence free functions belonging to $\boldsymbol L^p(\Omega)$, with curl in $\boldsymbol L^p(\Omega)$ and tangential or normal boundary conditions. We use these results to prove existence of a coupled stationary heating-electromagnetic problem, extending results, in the stationary case, existing in the literature for the evolutive hilbertian case.

Joint work with Chérif Amrouche.

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting some interactions between heterogeneities and geometric parameters like curvature and torsion.

In this talk we discuss the numerical discretization of the time-dependent Maxwell's equations. We use the nodal discontinuous Galerkin method for the integration in space due to the fact that it is a high-order accurate method that can easily handle complex geometries. Moreover, local refinement strategies can be incorporated due to the ability of the method to deal with irregular meshes with hanging nodes and local spaces of different orders. We present some numerical examples in the context of modeling scattered electromagnetic wave's propagation through human eye's structures.