半线性椭圆方程的稳定解在维数$9$之前是光滑的

IF 5.4 3区 材料科学 Q2 CHEMISTRY, PHYSICAL
X. Cabré, A. Figalli, Xavier Ros-Oton, J. Serra
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引用次数: 55

摘要

在本文中,我们证明了以下长期存在的猜想:双线性椭圆方程的稳定解在维数$n\leq9$上是有界的(因此是光滑的)。这个结果只在$n\leq4$中成立,是最优的:$\log(1/|x|^2)$是$n\geq10$的$W^{1,2}$奇异稳定解。这个猜想的证明是一个新的普遍估计的结果:我们证明,在维数$n\leq9$中,稳定解仅根据其$L^1$范数是有界的,与非线性无关。此外,在每个维度上,我们建立了Morrey空间中梯度的一个更高的可积结果和解的最优可积结果。从一系列经典例子中可以看出,我们所有的结果都是尖锐的。此外,作为推论,我们得到了Gelfand问题的极值解在每个维度上都是$W^{1,2}$,并且它们在维度$n\leq9$上是光滑的。这回答了Brezis和Brezis Vazquez提出的两个著名的公开问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Stable solutions to semilinear elliptic equations are smooth up to dimension $9$
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. This result, that was only known to be true for $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n \leq 9$. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.
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来源期刊
ACS Applied Energy Materials
ACS Applied Energy Materials Materials Science-Materials Chemistry
CiteScore
10.30
自引率
6.20%
发文量
1368
期刊介绍: ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.
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