Local and global minimizers for a variational energy involving a fractional norm

By Giampiero Palatucci, Ovidiu Savin, Enrico Valdinoci. 2011

193 citations recorded by the source corpus.

AbstractWe study existence, uniqueness, and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int\limits_\Omega W(u)\,{d}x, $$where $${\|u\|_{H^s(\Omega)}}$$ denotes the total contribution from Ω in the Hs norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $${\mathbb{R}^n}$$ . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/d67674a854d93971e9a5eedd2e9bd24360019f98

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In Applied Mathematics

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Citations in the corpus

Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “, the strong three-commutators estimates introduced in [6,7] to deduce the regularity of weak fractional harmonic maps (see also [40]), the strong barriers and density estimates in [37, 39], the pseudo-differential commutator and energy estimates in [35, 36], and many other successful techniques whi”
Local behavior of fractional $p$-minimizers (2015, unknown): “, the sharp 3-commutators estimates introduced in [5] to deduce the regularity of weak fractional harmonic maps, the strong barriers and density estimates in [26, 28, 29], the commutator and energy estimates in [25, 27], and so on.”
Large solutions to elliptic equations involving fractional Laplacian (2013, background): “ linear and nonlinear integral operators, especially, the fractional Laplacian, motivated by great applications and by important advances on the theory of nonlinear partial diﬀerential equations, see [4, 6, 7, 9, 14, 16, 17, 18, 28, 31] for details. In a recent work, Felmer and Quaas [14] considered”
Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations (2014, background): “By the fractional Gagliardo–Nirenberg inequality in [10] and the extension lemma of the fractional Laplacian in [2], we have ‖η0u‖L6/(3−2s)(Br ) s ‖η0u‖Ḣs = (∫”
Overdetermined problems with fractional Laplacian (2013, unknown): “…is a bounded open set with C 2 boundary, η ( x 0 ) stands for the outer normal at x 0 ∈ ∂ Ω and We note that solutions to the first two equations in ( ∗ ) are in C s ( Ω ) (see e.g. [39, Proposition 2.9] or [32, Proposition 1]) and therefore the overdetermined Neuman condition make sense pointwise.”
Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions (2011, background): “In [16], with different techniques than ours it is proved that for potentials G with G(−1) = G(1) = 0 and G > G(−1) = G(1) in (−1, 1), there exists a layer solution to equation (1.”
Γ-convergence for nonlocal phase transitions (2012, unknown): “Since σ is arbitrary the proof is complete (cid:2) We recall the following result about the one-dimensional minimizer which is proved in [22] (see, in particular, Theorems 1.2 and 1.3 there)”
Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian (2012, unknown): “1 in [21] and, also, Lemma 8 in [16]).”
Nonlocal Diffusion and Applications (2015, background): “formula (38) in [75]), KRpvRq `KRpwRq ď KRpuq `KRpuRq.”
Hitchhiker's guide to the fractional Sobolev spaces (2011, unknown): “1007/s00526-013-0656-y [75] G.”