arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 2302.08254math.AP
收藏

变系数竞争变分椭圆型方程组的最优一致界

Optimal uniform bounds for competing variational elliptic systems with variable coefficients

Manuel Dias, Hugo Tavares

备注：
50 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设$Ω\subset \mathbb{R}^N$为开集。在这项工作中，我们考虑以下梯度椭圆系统\[ -\text{div}(A(x)\nabla u_{i,β}) = f_i(x,u_{i,β}) + a(x)β|u_{i, β}|^{γ-1}u_{i, β} \mathop{\sum_{j=1}^l}_{j\neq i} |u_{j, β}|^{γ+ 1}, \]的解决方案，$i=1,\ldots, l$。我们工作在竞争的情况下，即$β<0$。在对$A$，$a$，$f_i$和指数$γ$的适当假设下，证明了正解族$\{u_β\}_{β<0}=\{(u_{1,β},\ldots, u_{l,β})\}_{β<0}$的一致$L^\infty$ -界蕴含着一致Lipschitz界（这是最优的）.证明中的要点之一是适当的推广Almgren的和Alt-Caffarelli-Friedman的单调性公式的解决方案，这样的系统。我们的工作推广了以前的结果，其中的情况下$A(x)=Id$（即运营商是拉普拉斯算子）进行了治疗。

英文摘要：

Let $Ω\subset \mathbb{R}^N$ be an open set. In this work we consider solutions of the following gradient elliptic system \[ -\text{div}(A(x)\nabla u_{i,β}) = f_i(x,u_{i,β}) + a(x)β|u_{i, β}|^{γ-1}u_{i, β} \mathop{\sum_{j=1}^l}_{j\neq i} |u_{j, β}|^{γ+ 1}, \] for $i=1,\ldots, l$. We work in the competitive case, namely $β<0$. Under suitable assumptions on $A$, $a$, $f_i$ and on the exponent $γ$, we prove that uniform $L^\infty$-bounds on families of positive solutions $\{u_β\}_{β<0}=\{(u_{1,β},\ldots, u_{l,β})\}_{β<0}$ imply uniform Lipschitz bounds (which are optimal). One of the main points in the proof are suitable generalizations of Almgren's and Alt-Caffarelli-Friedman's monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case $A(x)=Id$ (i.e. the operator is the Laplacian) was treated.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑