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Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties
González Nogueras, María del Mar; Saéz, Mariel; Sire, Yannick
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
We investigate the equation; (-Delta(Hn))(gamma) w = f(w) in H-n,; where (-Delta(Hn))(gamma) corresponds to the fractional Laplacian on hyperbolic space for gamma is an element of(0, 1) and f is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to +/- 1 at any point of the two hemispheres S-+/- subset of partial derivative H-infinity(n) and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane Pi. We prove that under additional conditions on the nonlinearity uniqueness holds up to isometry. Then we provide several symmetry results and qualitative properties of the layer solutions. Finally, we consider the multilayer case, at least when gamma is close to one.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística
Differential equations
Fractional Laplacian
Hyperbolic space
Layer solution
Equacions diferencials parcials
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Classificació AMS::58 Global analysis, analysis on manifolds::58J Partial differential equations on manifolds; differential operators
Attribution-NonCommercial-NoDerivs 3.0 Spain

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