2016, Sprekels, Jürgen, Valdinoci, Enrico

In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the sth power of a positive definite operator having a discrete spectrum in R+. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter s. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter s serves as the control parameter that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new class of identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter s also the domain of definition, and thus the underlying function space, of the fractional operator changes.

2014, Castro, Agnese di, Novaga, Matteo, Ruffini, Berardo, Valdinoci, Enrico

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of fractional perimeters. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers.