带二次强迫的分数阶障碍问题的椭球正性集
Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing
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中文总结 AI 辅助
该研究证明带二次强迫的分数阶障碍问题的衰减粘性解正性集为椭球，并结合已有分类结果验证了Fernández-Real和Yu关于三次全局薄障碍解的相关猜想。
中文摘要 AI 辅助
设n≥1，0<s<1，c>0，且A为正定对称矩阵。我们证明了ℝⁿ中方程min{u, (-Δ)^s u - (c - ⟨Ax,x⟩)} = 0的唯一衰减粘性解具有形式u(x)=K max{1-⟨Bx,x⟩,0}^{1+s}，其中K>0，B为某正定对称矩阵。特别地，其正性集为椭球。当s=1/2且n≥2时，该结果结合Fernández-Real和Yu的分类结论，可推出具有非空有界正性集的三次全局薄障碍解的正性集为椭球，这证明了Fernández-Real和Yu在非空有界正性情形下的猜想。
英文摘要
Let \(n\ge1\), \(0<s<1\), \(c>0\), and let \(A\) be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of \[ \min\{u,\,(-Δ)^s u-(c-\langle Ax,x\rangle)\}=0 \qquad\text{in }\mathbb R^n \] has the form \[ u(x)=K\max\{1-\langle Bx,x\rangle,0\}^{1+s} \] for some \(K>0\) and some positive definite symmetric matrix \(B\). In particular, its positivity set is an ellipsoid. For \(s=1/2\) and \(n\ge2\), this result, combined with the classification of Fernández-Real and Yu, implies that cubic global thin obstacle solutions with nonempty bounded positivity set have ellipsoidal positivity sets. This proves the conjecture of Fernández-Real and Yu in the nonempty bounded-positivity case.
发表机构
- Research Institute for Science and Engineering, Waseda University（早稻田大学理工学术研究院）
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