Giampiero Palatucci - Recent Developments in Nonlocal Theory

Giampiero Palatucci, Tuomo Kuusi (Eds.)

Recent developments in Nonlocal Theory

ISBN 978-3-11-057155-4

e-ISBN (PDF) 978-3-11-057156-1

e-ISBN (EPUB) 978-3-11-057203-2

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Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

2018 Giampiero Palatucci, Tuomo Kuusi and Chapters Contributors, published by De Gruyter Poland Ltd, Walter de Gruyter GmbH, Berlin/Boston
Cover image: Mauro Palatucci
Printed on acid-free paper
Printed in Germany

www.degruyter.com

This book is devoted to recent advances in the theory and applications of Partial Differential Equations and energy functionals related to the fractional Laplacian operator () s and to more general integro-differential operators with singular kernel of fractional differentiability order 0 < s < 1.

After being investigated firstly in Potential Theory and Harmonic Analysis, fractional operators defined via singular integral are currently attracting great attention in different research fields related to Partial Differential Equations with nonlocal terms, since they naturally arise in many different contexts. The literature is really too wide to attempt any reasonable account here, and the progress achieved in the last few years has been very important.

For this, we proposed the leading experts in the field to present their community recent results together with strategy, methods, sketches of the proofs, and related open problems.

The contributions to this book are the following,

Chapter 1: Heat kernel for nonsymmetric nonlocal operators

Z.-Q. CHEN and X. ZHANG present a survey on the recent progress in the study of heat kernels for a wide class of nonsymmetric nonlocal operators, by focusing on the existence and some sharp estimates of the heat kernels and their corresponding connection to jump diffusions.

Chapter 2: Fractional harmonic maps

F. DA LIO gives an overview of the recent results on the regularity and the compactness of fractional harmonic maps, by mainly focusing on the so-called horizontal 1/2- harmonic maps, which arise from several geometric problems such as for instance in the study of free boundary manifolds. The author describes the techniques that have been introduced in a series of very recent important papers in order to investigate the regularity of these maps. Some natural applications to geometric problems are also mentioned.

Chapter 3: Obstacle problems involving the fractional Laplacian

D. DANIELLI and S. SALSA investigate fractional obstacle problems, by firstly presenting the very important results concerning the analysis of the solution and the free boundary of the obstacle for the fractional Laplacian, mainly based on the extension method. Then, the authors consider the two time-dependent models which can be seen as the parabolic counterparts of the stationary fractional obstacle problem as well as the Signorini problem in the cylinder, by discussing some regularity properties of the solutions and the free boundary.

Chapter 4: Nonlocal minimal surfaces: interior regularity, quantitative estimates and boundary stickiness

S. DIPIERRO and E. VALDINOCI present various important results related to the surfaces which minimize the nonlocal perimeter functional. The authors discuss the interior regularity and some rigidity properties (in both a quantitative and a qualitative way) of these nonlocal minimal surfaces, together with their quite surprising boundary behavior.

Chapter 5: Eigenvalue bounds for the fractional Laplacian: a review

R. L. FRANK reviews some recent developments concerning the eigenvalues of the fractional Laplacian and fractional Schrdinger operators. In particular, the author focuses his attention on LiebThirring inequalities and their generalizations, as well as semi-classical asymptotics.

Chapter 6: A survey on the conformal fractional Laplacian and some geometric applications

M. D. M. GONZALEZ reports on recent developments on the conformal fractional Laplacian, both from the analytic and geometric points of view, with a special sight towards the Partial Differential Equations community. Among other investigations, the author explains the construction of the conformal fractional Laplacian from a purely analytic point of view, by relating its original definition coming from Scattering Theory to a Dirichlet-to-Neumann operator for a related elliptic extension problem, thus allowing for an analytic treatment of Yamabe-type problems in the nonlocal framework. Several examples and related opens problems are presented.

Chapter 7: Jump processes, nonlocal operators and regularity

M. KASSMANN reviews some basic concepts of Probability Theory, by focusing on the jump processes and their connection to nonlocal operators. Then, the author explains how to use jump processes for proving regularity results for a very general class of integro-differential equations.

Chapter 8: Regularity issues involving the fractional p -Laplacian

T. KUUSI, G. MINGIONE, and Y. SIRE deal with a general class of nonlinear integro- differential equations involving measure data, mainly focusing on zero order potential estimates. The nonlocal elliptic operators considered are possibly degenerate or singular and cover the case of the fractional p-Laplacian operator with measurable coefficients. The authors report recent related existence and regularity results by providing different, more streamlined proofs.

Chapter 9: Boundary regularity, Pohozaev identities, and nonexistence results

X. ROS-OTON surveys some recent results on nonlocal Dirichlet problems driven by a class of integro-differential operators, whose model case is the fractional Laplacian. The author discusses in detail the fine boundary regularity of the solutions, by sketching the main proofs and the involved blow-up techniques. Related Pohozaev identities strongly based on the aforementioned boundary regularity results are also presented, by showing how they can be used in order to deduce nonexistence and unique continuation properties.

Chapter 10: Variational and topological methods for nonlocal fractional periodic equations

G. MOLICA BISCI reports on recent existence and multiplicity results for nonlocal fractional problems under periodic boundary conditions. The abstract approach is based on variational and topological methods. More precisely, for subcritical equations, mountain pass and linking-type nontrivial solutions are obtained, as well as solutions for parametric problems, followed by equations at resonance and the obtention of multiple solutions using pseudo-index theory. Finally, in order to overcome the difficulties related to the lack of compactness in the critical case, the author performs truncation arguments and the Moser iteration scheme in the fractional Sobolev framework. Some related open problems are briefly presented.

Chapter 11: Change of scales for crystal dislocation dynamics

S. PATRIZI presents various results for a class of evolutionary equations driven by fractional operators, naturally arising in Crystallography, whose corresponding solution has the physical meaning of the atom dislocation inside a crystal structure. Since different scales come into play in such a description, different models have been adopted in order to deal with phenomena at atomic, microscopic, mesoscopic and macroscopic scale. By looking at the asymptotic states of the solutions of equations modeling the dynamics of dislocations at a given scale, it is shown in particular that one can deduce the model for the motion of dislocations at a larger scale.