超定奇异问题与分数阶扭转
Overdetermined Singular Problems and Fractional Torsion
AI总结：
该研究针对带孤立不可移除内部奇点的分数阶拉普拉斯算子的Serrin型超定问题，分离出边界商的一阶切向抵消项，推导得Ω为球且u径向严格递减的结论，还给出扭转情形下解的具体表达式。
AI中文摘要：
我们研究有界开集内带孤立不可移除内部奇点的分数阶拉普拉斯算子的Serrin型超定问题。对0<s<1、N>2s且c<0时，在Ω\{0}上满足(-Δ)^s u=f(u)的正弱解，在R^N\Ω上满足u=0，在∂Ω上满足(∂_η)^s u=c；我们分离出边界商u/δ^s的一阶切向抵消项，该抵消项恰好满足闭合分数阶角论证所需的一阶抵消，且弱于要求u/δ^s在整个边界邻域内属于C^1的条件。在此条件下，Ω是中心在奇点处的球，u关于径向变量是径向且严格递减的，未假设极点处的逐点爆破速率。若Ω属于C^{2,α}类，则当s>1/2时该抵消自动成立；当f在零点附近为常数时，对所有s∈(0,1)均成立。特别地，在扭转情形f=1时，u(x)=τ_R(x)+kG_R(x,0)，其中τ_R和G_R分别是球的分数阶扭转函数和格林函数。
英文摘要：
We study a Serrin-type overdetermined problem for the fractional Laplacian in a bounded open set with an isolated non-removable interior singularity. For positive weak solutions of (-Delta)^s u=f(u) in Omega\{0}, u=0 in R^NΩ, and (partial_eta)^s u=c on partial Omega, with 0<s<1, N>2s and c<0, we isolate a first-order tangential cancellation of the boundary quotient u/delta^s that matches the first-order cancellation needed to close the fractional corner argument and is weaker than requiring u/delta^s in C^1 in a full boundary neighborhood. Under this condition, Omega is a ball centered at the singular point and u is radial and strictly decreasing in the radial variable. No pointwise blow-up rate at the pole is assumed. If Omega is of class C^{2,alpha}, the cancellation is automatic when s>1/2, and for every s in (0,1) when f is constant near zero. In particular, in the torsion case f=1, u(x)=tau_R(x)+kG_R(x,0), where tau_R and G_R are respectively the fractional torsion function and the Green function of the ball.
