Title: Fractional De Giorgi conjecture in dimension 2 via complex-plane methods Show Chinese title

Title: 通过复平面方法证明二维分数德乔治猜想

Abstract: We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in $\mathbb{R}^2$ for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the $s$-harmonic extension of the fractional Laplacian in the half-space $\mathbb{R}^{3}_+$ introduced by L. Caffarelli and L. Silvestre in 2007. We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space $\mathbb{R}^{n+1}_+$ under Neumann boundary conditions. This generalizes the $s$-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace. Show Chinese abstract

Abstract: 我们提供了对Allen-Cahn方程在$\mathbb{R}^2$中分数阶De Giorgi猜想的新证明，适用于所有指数范围。 我们的证明结合了A. Farina于2003年引入的方法与L. Caffarelli和L. Silvestre于2007年在半空间$\mathbb{R}^{3}_+$中引入的分数拉普拉斯算子的$s$调和扩展。 我们还提供了在半空间$\mathbb{R}^{n+1}_+$上，在Neumann边界条件下的一类加权椭圆偏微分方程的有限能量弱解的表示公式。 这推广了分数拉普拉斯算子的$s$调和扩展，并使我们能够将扩展空间中的一个一般问题与迹上的非局部问题联系起来。