Nonlocal Diffusion and Applications
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 Schrödinger 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.
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ˆ ˆ ˇ ˇ ˇ Allen-Cahn equation arXiv preprint atoms ball boundary Br.x Bucur Caffarelli catenoids change of variable classical collision compute concludes the proof conjecture consider constant contact points contradiction define dimension domain estimates Euler-Lagrange equation exists formula Fourier transform fractional Laplacian fractional perimeter fractional Sobolev geometric Giorgi graph Harnack Inequality Hence hyperplanes infinity integral jtj1C2s jx yjnC2s jyjnC2s dy kinetic energy Laplace operator Lemma level sets Lévy Processes Math mathematical Maximum Principle nonlinear Nonlocal Diffusion nonlocal mean curvature nonlocal minimal surfaces Notice obtain operator perturbation phase transitions Plancherel Theorem problem proof of Theorem prove random walk recall result RnC1C s-harmonic function s-minimal cone s-minimal set Schrödinger equation smooth Sobolev Inequality Sobolev spaces space su.x supconvolution u;B R Valdinoci vanishes