论文标题
分数Sobolev规则性，用于强烈退化抛物线方程的解决方案
Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
论文作者
论文摘要
我们进行调查始于[2]，介绍了弱解的规律性，对强烈退化的抛物线方程\ [u_ {t} - \ Mathrm {div} \ left [（\ vert du \ vert-1）_ {+} du \ vert} \ right] = f \，\，\，\，\，\，\，\，\，\，\，\，\，\，\ mathrm {In} $ n \ geq2 $，$ p \ geq2 $和$ \ left（\，\ cdot \，\ right）_ {+} $代表积极零件。在这里，我们假设$ f \ in l_ {loc}^{p'} \ left（0，t; b_ {p'，p'，\ int}^α\ left（ω\ right）\ right）$，削弱了右侧的假设。这导致我们为解决方案的空间梯度$ du $的功能获得了更高的分数可不同性结果。此外，我们就空间变量建立了$ du $的更高总和。上述方程式的主要新颖性是结构函数仅满足以原点为中心的单位球以外的标准椭圆状和生长条件。我们想指出的是，一方面，可以将本文的主要结果视为[1]中包含的椭圆结果的抛物线，另一方面是[2]中建立的某些结果的分数版本。
We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right]=f\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that $f\in L_{loc}^{p'}\left(0,T;B_{p',\infty,loc}^α\left(Ω\right)\right)$, with $α\in(0,1)$ and $p'=p/(p-1)$. This leads us to obtain higher fractional differentiability results for a function of the spatial gradient $Du$ of the solutions. Moreover, we establish the higher summability of $Du$ with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [1], and on the other hand as the fractional version of some results established in [2].
