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search.filters.applied.f.access: openAccess × Author: Valdinoci, Enrico × End date: 2019 × Start date: 2010 × DDC: 510 × Type: Text × Subject: uniqueness ×

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    Ground states and concentration phenomena for the fractional Schrödinger equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fall, Mouhamed Moustapha; Mahmoudi, Fethi; Valdinoci, Enrico
    We consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique.
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    A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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.