Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result

Abstract

We deal with symmetry properties for solutions of nonlocal equations of the type ( − Δ ) s v = f ( v ) in R n , where s ∈ ( 0 , 1 ) and the operator ( − Δ ) s is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation { − div ( x α ∇ u ) = 0 on R n × ( 0 , + ∞ ) , − x α u x = f ( u ) on R n × { 0 } , where α ∈ ( − 1 , 1 ) , y ∈ R n , x ∈ ( 0 , + ∞ ) and u = u ( y , x ) . This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γ α : u | ∂ R + n + 1 ↦ − x α u x | ∂ R + n + 1 is ( − Δ ) 1 − α 2 . More generally, we study the so-called boundary reaction equations given by { − div ( μ ( x ) ∇ u ) + g ( x , u ) = 0 on R n × ( 0 , + ∞ ) , − μ ( x ) u x = f ( u ) on R n × { 0 } under some natural assumptions on the diffusion coefficient μ and on the nonlinearities f and g. We prove a geometric formula of Poincaré-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.