Anomalous diffusion equations: regularity and geometric properties of solutions and free boundaries
Progetto The purpose of this project is the analytical study of the regularity, qualitative, symmetry, and variational
properties of solutions to partial differential equations and systems ruled by anomalous diffusions. The
main subject of study are some second order equations whose coefficients are degenerate or singular on
hypersurfaces, which are however deeply related to nonlocal operators (fractional Laplacians). Important
achievement of the proposal is the extension to the parabolic framework of the Schauder regularity theory
for degenerate equations developed by myself and collaborators in the past years. The regularization
approach we used for the elliptic case should prove to be efficient also in the present case. Thus, boundary
regularity under mild assumptions on the regularity of domains would be crucial for unique continuation
properties from boundary points for solutions to fractional equations. Then, given an equation, the ratio of
two solutions which share the same zero set solves a degenerate equation whose coefficients vanish with a
certain rate on the common nodal set: boundary Harnack type principles on nodal sets can then be proved
passing through the Schauder theory for the associated degenerate equation. Finally, my interest is to
consider such anomalous diffusions when another phenomenon appears: the strong repulsive interaction
between densities or populations. Such competition feature gives rise to spatial segregation and formation
of free boundaries. Target of the proposal is understanding how anomalous diffusions and competition
features influence each other. Major tool for achieving the targets of the project is the validity of
monotonicity features of suitably rescaled energies which allow blow-up analysis: the classification of the
homogeneity degrees of the limiting entire blow-up configurations better points out the regularity and
geometric properties of solutions near nodal sets, and consequently the properties of free interfaces.