论文标题
在快速扩散案例中，抛物线托架$ p $ laplace方程的超级平均功能
Supercaloric functions for the parabolic $p$-Laplace equation in the fast diffusion case
论文作者
论文摘要
我们将一类普遍的超施加措施，即所谓的$ p $ - 超定量功能，用于抛物线$ p $ laplace方程。这类函数被定义为较低的半连续函数，在密集集中有限并满足抛物线比较原理。他们的属性相对较好地理解了$ p \ geq 2 $，但是在快速扩散案例$ 1 <p <2 $中，知之甚少。每个有限的$ p $ - 苏联功能都属于天然Sobolev空间，并且是整个范围的抛物线$ p $ laplace方程的较弱的超级溶液。我们的主要结果表明，无界的$ p $ - 柔和函数分为两个相互排他性的类，对该功能进行了尖锐的局部整合性估计，并且在超临界情况下，其弱梯度$ \ frac {2n} {n+1} {n+1} <p <2 $。 Barenblatt溶液和无限点源溶液表明，两种替代方案都发生了。 Barenblatt Solutions在亚临界情况下不存在$ 1 <p \ leq \ frac {2n} {n+1} $，并且该理论尚未很好地理解。
We study a generalized class of supersolutions, so-called $p$-supercaloric functions, to the parabolic $p$-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $p\geq 2$, but little is known in the fast diffusion case $1<p<2$. Every bounded $p$-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic $p$-Laplace equation for the entire range $1<p<\infty$. Our main result shows that unbounded $p$-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $\frac{2n}{n+1}<p<2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $1<p\leq \frac{2n}{n+1}$ and the theory is not yet well understood.
