AI 中文总结
本文研究非齐次p-拉普拉斯方程解的梯度的齐次函数的更高正则性，利用内在尺度、Schauder估计等方法，得到了相关函数的正则性结果及多孔介质方程的正则性准则。
AI 中文摘要
本文研究非齐次p-拉普拉斯方程div(|Du|^{p-2}Du)=f的解的梯度的齐次函数的更高正则性。尽管解在其临界点处不必属于C²类，但其梯度是局部赫尔德连续的。假设Du∈C^{0,α}_{loc}且α≤1/(p-1)，设Φ在原点外光滑且为正m次齐次函数，我们证明当m>k/α时，Φ(Du)∈C^k_{loc}，且其所有阶数不超过k的导数在临界点处均为零。证明使用了内在尺度r≃|Du|^{1/α}、归一化一致椭圆方程的Schauder估计，以及临界点处的延拓引理。我们还在梯度满足适当赫尔德假设的条件下，针对自治各向异性方程、椭圆型和抛物型p-拉普拉斯系统得到了相应结果。最后，相同的论证给出了多孔介质方程非负解的高次幂的C^k正则性准则。
英文摘要
In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally Hölder continuous. Suppose that $Du\in C^{0,α}_{\rm loc}$ with $α\le 1/(p-1)$, and let $Φ$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $Φ(Du)\in C^k_{\rm loc}$ whenever $m>k/α$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/α}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate Hölder assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.
