A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian

作者：Dipierroa Serena^*; Pinamonti Andrea

来源：Journal of Differential Equations, 2013, 255(1): 85-119.

DOI：10.1016/j.jde.2013.04.001

• Elliptic systems
• Fractional Laplacian
• Monotone solutions
• Stable solutions
• Phase separation
• Poincare-type inequality

We study the symmetry properties for solutions of elliptic systems of the type %26lt;br%26gt;{(-Delta)(s1) u = F-1(u, v), (-Delta)(s2) v = F2(u, v), %26lt;br%26gt;where F is an element of C-loc(1, 1)(R-2). s(1), s(2) is an element of (0, 1) and the operator (-Delta)(s) is wthe so-called fractional Laplacian. We obtain some Poincare-type formulas for the a-harmonic extension in the half-space, that we use to prove a symmetry result both for stable and for monotone solutions.

• 出版日期2013-7-1