A weak solution to a perturbed one-Laplace system by p-Laplacian is continuously differentiable
A weak solution to a perturbed one-Laplace system by p-Laplacian is continuously differentiable
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DOI：
10.1007/s00208-022-02539-w
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发表时间：
2022-08
影响因子：
1.4
通讯作者：
Shuntaro Tsubouchi
中科院分区：
文献类型：
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作者：
Shuntaro Tsubouchi
在本文中，我们的目的是证明一个拉普拉斯方程组的弱解的连续可微性。这个方程的主要困难是均匀椭圆性在小平面附近破裂，在那里梯度消失。我们想证明弱解的导数即使在小平面上也是连续的。这可以通过估计雅可比矩阵的赫尔德连续性乘以其模在零附近截断。为了证明这一估计，我们考虑一个近似的系统，并使用标准的方法，包括德Giorgi的截断和冻结系数参数。
In this paper we aim to show continuous differentiability of weak solutions to a one-Laplace system perturbed byp-Laplacian with. The main difficulty on this equation is that uniform ellipticity breaks near a facet, the place where a gradient vanishes. We would like to prove that derivatives of weak solutions are continuous even across the facets. This is possible by estimating Hölder continuity of the Jacobian matrix multiplied with its modulus truncated near zero. To show this estimate, we consider an approximated system, and use standard methods including De Giorgi’s truncation and freezing coefficient arguments.