Geometric Analysis of Fractional Phase Transition Interfaces
We discuss some recent results on phase transition models driven by nonlocal operators, also in relation with their limit (either local or nonlocal) interfaces.
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Abstract We discuss some recent results on phase transition models driven by nonlocal operators, also in relation with their limit (either local or nonlocal) interfaces. Keywords Fractional Laplacian · Nonlocal minimal surfaces · Asymptotics · Rigidity and regularity theory
1 The Fractional Laplacian Operator This note is devoted to report some recent advances concerning the fractional powers of the Laplace operator and some related problems arising in pde’s and geometric measure theory. Namely, the s-Laplacian of a (sufficiently regular) function u can be defined as an integral in the principal value sense by the formula u(x + y) − u(x) s −(−Δ) u(x) := Cn,s P.V. dy n |y|n+2s R u(x + y) − u(x) := Cn,s lim dy, (1) |y|n+2s ε→0+ Rn \Bε where s ∈ (0, 1), and Cn,s = π −2s+n/2
Γ (n/2 + s) Γ (−s)
is a normalization constant (blowing up as s → 1− and s → 0+ , because of the singularities of the Euler Γ -function). Note that the integral here is singular in the G. Franzina (B) Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050 Povo, Trento, Italy e-mail: [email protected] E. Valdinoci Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_8, © Springer-Verlag Italia 2013
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case s ≥ 1/2, but it converges if s < 1/2 as it can be estimated via an elementary argument by splitting the domain of integration. We point out that an equivalent definition may be given by integrating against a singular kernel, which suitably averages a second-order incremental quotient. Indeed, thanks to the symmetry of the kernel under the map y → −y, by performing a standard change of variables, one obtains u(x + y) − u(x) u(x + y) + u(x − y) − 2u(x) 1 dy = dy, n+2s n n 2 R \Bε |y| |y|n+2s R \Bε for all ε > 0; thus (1) becomes Cn,s −(−Δ) u(x) = 2
s
Rn
u(x + y) + u(x − y) − 2u(x) dy. |y|n+2s
(2)
It is often convenient to use the expression in (2), that deals with a convergent Lebesgue integral, rather than the one in (1), that needs a principal value to be wellposed. The fractional Laplacian may be equivalently defined by means of the Fourier symbol |ξ |2s by simply setting, for every s ∈ (0, 1), F (−Δ)s u = |ξ |2s (F u) , (3) for all u ∈ S (Rn ), and this occurs in analogy with the limit case s = 1, when (3) is consistent with the well-known behavior of the (distributional) Fourier transform F on Laplacians. Note that this operator is invariant under the action of the orthogonal transformations in Rn and that the following scaling property holds: (−Δ)s uλ (x) = λ2s (−Δ)s u (λx), for all x ∈ Rn , (4) were we denoted uλ (x) = u(λx). For the basics of the fractional Laplace operators and related functional settings see, for instance, [30] and references therein. After being studied for a long time in potential theory and
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