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This book presents in a detailed and self-contained way a new and
important density result in the analysis of fractional partial
differential equations, while also covering several fundamental
facts about space- and time-fractional equations.
Working in the fractional Laplace framework, this book provides
models and theorems related to nonlocal diffusion phenomena. In
addition to a simple probabilistic interpretation, some
applications to water waves, crystal dislocations, nonlocal phase
transitions, nonlocal minimal surfaces and Schroedinger equations
are given. Furthermore, an example of an s-harmonic function, its
harmonic extension and some insight into a fractional version of a
classical conjecture due to De Giorgi are presented. Although the
aim is primarily to gather some introductory material concerning
applications of the fractional Laplacian, some of the proofs and
results are new. The work is entirely self-contained, and readers
who wish to pursue related subjects of interest are invited to
consult the rich bibliography for guidance.
These lecture notes are devoted to the analysis of a nonlocal
equation in the whole of Euclidean space. In studying this
equation, all the necessary material is introduced in the most
self-contained way possible, giving precise references to the
literature when necessary. The results presented are original, but
no particular prerequisite or knowledge of the previous literature
is needed to read this text. The work is accessible to a wide
audience and can also serve as introductory research material on
the topic of nonlocal nonlinear equations.
This book collects together lectures by some of the leaders in the
field of partial differential equations and geometric measure
theory. It features a wide variety of research topics in which a
crucial role is played by the interaction of fine analytic
techniques and deep geometric observations, combining the intuitive
and geometric aspects of mathematics with analytical ideas and
variational methods. The problems addressed are challenging and
complex, and often require the use of several refined techniques to
overcome the major difficulties encountered. The lectures, given
during the course "Partial Differential Equations and Geometric
Measure Theory'' in Cetraro, June 2-7, 2014, should help to
encourage further research in the area. The enthusiasm of the
speakers and the participants of this CIME course is reflected in
the text.
This book is the outcome of a conference held at the Centro De
Giorgi of the Scuola Normale of Pisa in September 2012. The aim of
the conference was to discuss recent results on nonlinear partial
differential equations, and more specifically geometric evolutions
and reaction-diffusion equations. Particular attention was paid to
self-similar solutions, such as solitons and travelling waves,
asymptotic behaviour, formation of singularities and qualitative
properties of solutions. These problems arise in many models from
Physics, Biology, Image Processing and Applied Mathematics in
general, and have attracted a lot of attention in recent years.
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