Nonlocal Diffusion and Applications

By C. Bucur, Enrico Valdinoci. 2015

631 citations recorded by the source corpus.

Introduction.- 1 A probabilistic motivation.-1.1 The random walk with arbitrarily long jumps.- 1.2 A payoff model.-2 An introduction to the fractional Laplacian.-2.1 Preliminary notions.- 2.2 Fractional Sobolev Inequality and Generalized Coarea Formula.- 2.3 Maximum Principle and Harnack Inequality.- 2.4 An s-harmonic function.- 2.5 All functions are locally s-harmonic up to a small error.- 2.6 A function with constant fractional Laplacian on the ball.- 3 Extension problems.- 3.1 Water wave model.- 3.2 Crystal dislocation.- 3.3 An approach to the extension problem via the Fourier transform.- 4 Nonlocal phase transitions.- 4.1 The fractional Allen-Cahn equation.- 4.2 A nonlocal version of a conjecture by De Giorgi.- 5 Nonlocal minimal surfaces.- 5.1 Graphs and s-minimal surfaces.- 5.2 Non-existence of singular cones in dimension 2 5.3 Boundary regularity.- 6 A nonlocal nonlinear stationary Schrodinger type equation.- 6.1 From the nonlocal Uncertainty Principle to a fractional weighted inequality.- Alternative proofs of some results.- A.1 Another proof of Theorem A.2 Another proof of Lemma 2.3.- References.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/140fde9442af4918047ff351f838fcd4eeb59136

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Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “The recent monograph [58] by Bucur and Valdinoci (2016) introduces fractional operators and more generally nonlocal diffusion, and then goes on to study a number of stationary problems.”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, methodology): “Fractional Laplacians have a close connection to Levý processes and have been used in many areas of mathematics and physics, for example to model anomalous and nonlocal diffusion, and also in the formulation of fractional quantum mechanics where the fractional Schrödinger equation arises naturally a”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “See for instance [14] for the equivalence of (1.”
Asymptotic method of moving planes for fractional parabolic equations (2020, background): “29), we derive that for each fixed t ∈ [0, 2] and for any x ∈ Bδ(x̄) (−∆)s(χD∪Dλ(x)w̃(x, t)) = CN,sP.”
On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent (2017, methodology): “We also refer to the recent monographs [15, 5, 10] for a thorough variational approach of nonlocal problems.”
What is the fractional Laplacian? A comparative review with new results (2018, background): “5 0 2 4 6 8 [0, 2 ] 10 10 10 10 10”