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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 study existence, uniqueness and other geometric properties of the minimizers of the energy functional u 2 H s (Ω) + Ω W (u) dx, where u H s (Ω) denotes the total contribution from Ω in the H s 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 n. The… (More)
We obtain an improved Sobolev inequality in ˙ H s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimiz-ers 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 ˙ H s… (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 p dxdy + 1 ε I W (u) dx (p > 2), as ε → 0.
We use variational methods to study the asymptotic behavior of solutions of p-Laplacian problems with nearly subcritical nonlinearity in general, possibly non-smooth, bounded domains.
We study the Γ-convergence of the following functional (p > 2) F ε (u) := ε p−2 Ω |Du| p d(x, ∂Ω) a dx+ 1 ε p−2 p−1 Ω W (u)d(x, ∂Ω) − a p−1 dx+ 1 √ ε ∂Ω V (T u)dH 2 , where Ω is an open bounded set of R 3 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… (More)