Ricerc@Sapienza

We prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, the initial data being a finite superposition of Dirac masses and the flux being Lipschitz continuous, bounded and sufficiently smooth. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by
$$
-\Delta u= h(u){f} \ \ \text{in}\,\ \Omega,
$$
where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.

We compute the Morse index of 1-spike solutions of the semilinear elliptic problem
()
where is a smooth bounded domain and is sufficiently large.

When Ω is convex, our result, combined with the characterization in [21], a result in [40] and with recent uniform estimates in [37], gives the uniqueness of the solution to (), for p large. This proves, in dimension two and for p large, a longstanding conjecture.

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The main idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form.
The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with the diffusion equation.