论文标题
$ c^1 $ - 变性扩散方程
$C^1$-regularity for degenerate diffusion equations
论文作者
论文摘要
我们证明，如果方程式的退化法可以满足可集成性标准，即否定的椭圆形PDE的任何解决方案都是$ c^1 $的类别。 $σ^{ - 1} \在l^1 \ left（\ frac {1}λ{\ bf d}λ\ right）$。证明是基于构造一系列收敛的切线超平面，该平面平面平面近似于$ u（x）$，接近$ x_0 $，订单$ \ text {o}（o}（| x-x-x_0 |）$。对这种超平面的明确控制通过构造进行了，从而对$ {c}^1 $ - 定期解决方案进行了通用估计。在证明中需要的主要新成分中，我们开发了一种替代递归算法，用于重新估算溶液的重新归一化。这种新方法基于一种量身定制的技术，以防止通过该过程构建的堕落定律序列自身变性。
We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $σ^{-1} \in L^1\left (\frac{1}λ {\bf d}λ\right )$. The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate $u(x)$, near $x_0$, by an error of order $\text{o}(|x-x_0|)$. Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the ${C}^1$--regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for the renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.
