Marcela Sanmartino

An alternative well-posedness property and static spacetimes with naked singularities

In the first part of this paper, we show that the Cauchy problem for wave propagation in some static spacetimes presenting a singular timelike boundary is well-posed, if we only require the waves to have finite energy, although no boundary condition is required. This feature does not come from essential self-adjointness, which is false in these cases, but from a different phenomenon that we call the alternative well-posedness property, whose origin is due to the degeneracy of the metric components near the boundary. Beyond these examples, in the second part, we characterize the type of degeneracy which leads to this phenomenon.

The Calderón Projector for an elliptic operator in divergence form

The Calderón Projector, is one of the most important tools in the study of boundary value problems for elliptic operators. Its construction is well known for elliptic operators with C∞ coefficients on C∞ domains and even for the Laplacian operator on C1 domains. The aim of this article is to extend the results for the Laplacian case to elliptic operators in divergence form with Lipschitz coefficients on C1 domains.

On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n + 2)-dimensional static and spherically symmetric spacetimes. They are related to the properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially self-adjoint, but it does satisfy a weaker property that we call here quasi-essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.