Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result

By Yannick Sire, Enrico Valdinoci. 2008

304 citations recorded by the source corpus.

We deal with symmetry properties for solutions of nonlocal equations of the type (−Δ)sv=f(v)in Rn, where s∈(0,1) and the operator (−Δ)s is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation {−div(xα∇u)=0on Rn×(0,+∞),−xαux=f(u)on Rn×{0}, where α∈(−1,1), y∈Rn, x∈(0,+∞) and u=u(y,x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γα:u|∂R+n+1↦−xαux|∂R+n+1 is (−Δ)1−α2. More generally, we study the so-called boundary reaction equations given by {−div(μ(x)∇u)+g(x,u)=0on Rn×(0,+∞),−μ(x)ux=f(u)on Rn×{0} under some natural assumptions on the diffusion coefficient μ and on the nonlinearities f and g. We prove a geometric formula of Poincare-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.

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

View Paper: https://www.semanticscholar.org/paper/8a06b8e9d480b1546dfcd3bb8c9ac9fa69780f81

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Citations in the corpus

Local and global minimizers for a variational energy involving a fractional norm (2011, unknown): “(5) We refer to [9,25,26,16,8,10,11] for several results in this direction.”
Bernstein and De Giorgi type problems: new results via a geometric approach (2008, background): “Results and techniques developed here are also the keystone of the forthcoming papers [15] and [25], in which higher dimensional problems and fractional operators will be considered.”
Nonexistence results for a class of fractional elliptic boundary value problems (2012, unknown): “[8, 9, 12,14,26].”
Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions (2011, methodology): “We also refer the reader to [21, 22] where some rigidity properties of boundary reactions have been established through a more geometric approach.”
Harnack inequality for fractional sub-Laplacians in Carnot groups (2012, background): “We mention, for instance, the thin obstacle problem, phase transition problems, and the study of a general class of conformally covariant operators in conformal geometry: see, for instance, [6], [32] and [9].”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “onicity and rigidity of solutions to semilinear elliptic equations of fractional order in the whole space RN or in smooth bounded sets has attracted considerable attention in the last years, see e.g. [3,8,9,10,11,15,19,20,28,29,43]. In striking contrast, if is unbounded, but dierent from the whole s”
Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations (2013, unknown): “[2, 3, 4, 7, 9, 10, 24, 25] and references therein.”
ENERGY ESTIMATES AND 1-D SYMMETRY FOR NONLINEAR EQUATIONS INVOLVING THE HALF-LAPLACIAN (2010, methodology): “, for the equation (−∆)u = f(u) in R(2), with 0 < s < 1, has been established by the first author and Sire [4, 5] and by Sire and Valdinoci [16].”
Semilinear elliptic equations for the fractional Laplacian with Hardy potential (2011, background): “We refer the reader to papers [10], [39], [40], [24], [4] (and the references there in) for a nice expository.”
Nonlocal Diffusion and Applications (2015, methodology): “In the case n “ 2, for any s P p0, 1q, the result is proved using the harmonic extension of the fractional Laplacian in [15] and [88].”
Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations (2013, background): “[32], finance [19], phase transition [2, 11, 65, 33, 40], stratified materials [61], anomalous diffusion [56], crystal dislocation [67, 41], soft thin films [47], semipermeable membranes and flame propagation [12], conservation laws [9], ultra-relativistic limits of quantum mechanics [34], quasi-geo”
The Dirichlet Problem for the fractional p-Laplacian evolution equation (2015, unknown): “Among the first, we mention the work on nonlocal obstacle models, [18, 19], on semilinear elliptic problems of nonlocal type [14, 48, 49], on phase transitions [51], and the work on nonlocal perimeters [17, 21].”
Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, background): “Existence and related results were studied by Cabré and Tan [4], Dipierro, Palatucci and Valdinoci [12], Felmer, Quaas and Tan [13], and Servadei and Valdinoci [24].”
A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations (2009, unknown): “Several applications to phase transition problems driven by either the fractional Laplacian or non-local interactions have recently appeared in the literature: see, for instance, [1,7,18,11,38,77] and references therein.”