发表机构
School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院与高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对奇异分数阶p-拉普拉斯方程，通过内在过剩衰减论证等方法，证明了特定参数范围内有界弱解的梯度Hölder正则性，为相关未解决问题提供了部分解答。
AI 中文摘要
设n≥2，1<p<2，α_loc(n,p)是局部p-调和函数梯度的容许内部Hölder指数。我们证明，对任意0<α<α_loc(n,p)，存在s_*=s_*(n,p,α)<1，使得当s∈(s_*,1)时，B_2内(-Δ_p)^s u=0的每个有界弱解都属于C^{1,α}(B_{1/2})。该证明依赖于内在的过剩衰减论证：我们引入斜率归一化的Bregman能量，同时在有界斜率和大斜率区域建立紧性，对应的爆破极限分别为平移局部p-能量的极小值者和一致椭圆常系数方程的解。随后将这两个局部族的一致平坦度改进估计转移到分数阶方程，通过将平均外通量与进入上、下De Giorgi截断的单侧尾部分离，并结合尺度不变的环形L^p界与插值来闭合迭代。该尾部论证适用于整个结构范围sp>p-1，而s接近1的假设仅用于紧性步骤。该结果为奇异范围内的未解决C^{1,α}问题提供了部分解答。
英文摘要
Let $n\ge2$, $1<p<2$, $0<s<1$, and $sp>p-1$. We prove that every globally bounded fractional $p$-harmonic function is locally $C^{1,α}$ for some $α=α(n,p,s)>0$. This settles the open problem of interior gradient Hölder regularity in the singular range throughout the natural first-order regime $sp>p-1$. The proof combines an affine-invariant improvement-of-flatness argument with a Liouville theorem for globally Lipschitz entire solutions. In the large-slope regime, the shifted Bregman energies converge to an anisotropic stable form of order $sp-p+2>1$. In the bounded-slope regime, the Liouville theorem follows from rigidity of extremal secants, a recurrent blow-down argument, and a directional Morrey-Kato estimate for the singular linearized kernel. An affine Campanato argument controls the variation of the best affine approximations across scales. These estimates yield a scale-invariant decay of the affine excess and hence the local $C^{1,α}$ estimate.
