Giampiero Palatucci

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We study existence, uniqueness and other geometric properties of the minimizers of the energy functional ‖u‖Hs(Ω) + ∫ Ω W (u) dx, where ‖u‖Hs(Ω) denotes the total contribution from Ω in the H 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 R. The results(More)
Received (Day Month Year) Communicated by (xxxxxxxxxx) Let Ω be an open bounded set of R 3 and let W and V be two non-negative continuous functions vanishing at α, β and α ′ , β ′ , respectively. We analyze the asymptotic behavior as ε → 0, in terms of Γ-convergence, of the following functional Fε(u) := ε p−2 Z Ω |Du| p dx + 1 ε p−2 p−1 Z Ω W (u)dx + 1 ε Z(More)
We consider an evolution equation arising in the Peierls–Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. these(More)
Let I be an open bounded interval of R and W a non-negative continuous function vanishing only at α, β ∈ R. We investigate the asymptotic behaviour in terms of Γ-convergence of the following functional Gε(u) := ε p−2 ∫∫ I×I ∣∣∣∣u(x)− u(y) x− y ∣∣∣∣pdxdy + 1ε ∫ I W (u) dx (p > 2), as ε→ 0. Mathematics Subject Classification (2000). Primary 82B26, 49J45 ;(More)
We obtain an improved Sobolev inequality in Ḣ spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in Ḣ obtained in(More)
where Ω is an open bounded set of R and W and V are two non-negative continuous functions vanishing at α, β and α, β, respectively. In the previous functional, we fix a = 2 − p and u is a scalar density function, Tu denotes its trace on ∂Ω, d(x, ∂Ω) stands for the distance function to the boundary ∂Ω. We show that the singular limit of the energies Fε leads(More)
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