Gamma convergence of an energy functional related to the fractional Laplacian

By María del Mar González. 2008

65 citations recorded by the source corpus.

We prove a Γ-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (−Δ)s for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (−Δ)1/2 was considered by Alberti–Bouchitté–Seppecher in relation to a model in capillarity with line tension effect. However, the proof in our setting requires some new ingredients such as the Caffarelli–Silvestre extension for the fractional Laplacian and new trace inequalities for weighted Sobolev spaces.

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

View Paper: https://www.semanticscholar.org/paper/04c2d213e95ad6bff42b09734c020ec79f7e9302

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Sharp energy estimates for nonlinear fractional diffusion equations (2012, background): “D symmetry result in R2 of [6] and (since it applies to all nonlinearities f) the necessary conditions on f proved in [5] for a monotone solution in R to exist.”
On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres (2013, background): “[13, 14, 15, 16, 17, 37, 58, 72], specially in the case s = 1/2 which is related to models for dislocations in crystals [35, 38] (see also [25, 27] for s 6= 1/2), and to thin-film micromagnetics [46, 47].”
Fractional conformal Laplacians and fractional Yamabe problems (2010, background): “It was observed by Chang and González in [11] that the generalized eigenvalue problem (2.3) on a non-compact manifold (Xn+1, g+) is equivalent to a linear degenerate elliptic problem on the compact manifold (X̄n+1, ḡ), for ḡ = ρ2g+.”
Γ-convergence for nonlocal phase transitions (2012, unknown): “We also refer to [5, 16, 18], where other types of nonlocal models have been considered (in particular, a threedimensional fluid with boundary and weight inhomogeneity of distance type, whose energy bounds the Gagliardo norm, see Theorem 19 in [18]).”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, background): “Other related issues may be found in [15,25,27,30,40,50,52,54,61,68,73,86,87].”
Local and global minimizers for a variational energy involving a fractional norm (2011, unknown): “The 1D minimizer We are ready to deal with 1-D minimizers (for related observations when s ∈ (1/2, 1) see [10] and [12]).”