论文标题
弱解决方案涉及各向异性扩散率的非常奇异的椭圆方程的连续可不同性
Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity
论文作者
论文摘要
在本文中，我们考虑了一个非常奇异的椭圆方程，涉及各向异性扩散操作员，包括一拉普拉斯，并受到$ p $ -laplacian型扩散操作员的扰动，$ 1 <p <\ p <\ infty $。这个方程式在分析上似乎很难在一个方面进行处理，该方程在梯度消失的地方。我们的主要目的是证明即使在整个方面，薄弱的解决方案也是不断可区分的。在这里，知道梯度在刻面附近被截断时是否连续。为了肯定地，我们考虑一个近似问题，并使用包括De Giorgi的截断和冻结系数方法在内的标准方法。
In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi's truncation and freezing coefficient methods.
