Symmetry results for fractional elliptic systems and related problems

Abstract

We study elliptic gradient systems with fractional laplacian operators on the whole space $$(-\Delta {)}^{\mathbf{s}}\mathbf{u}=\nabla H(\mathbf{u})\text{in}{\mathbf{R}}^n,$$ where $\mathbf{u}:{\mathbf{R}}^n\to {\mathbf{R}}^m$, $H\in C^{2,\gamma }({\mathbf{R}}^m)$ for $\gamma >\max (0,1-2\min \left\{s_i\right\})$, $\mathbf{s}=(s_1,\cdots ,s_m)$ for $0<s_i<1$ and $\nabla H(\mathbf{u})=(H_{u_i}(u_1,u_2,\cdots ,u_m){)}_i$. We prove De Giorgi type results for this system for certain values of $\mathbf{s}$ and in lower dimensions, i.e. $n=2,3$. Just like the local case, the concepts of orientable systems and $H-$monotone solutions, established in [18], play the key role in proving symmetry results. In addition, we provide optimal energy estimates, a monotonicity formula, a Hamiltonian identity and various Liouville theorems.

arXiv comment20 pages. Comments are welcome