关于一般线性退化椭圆型偏微分方程组
On General Linear Degenerate Elliptic PDE Systems
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中文总结 AI 辅助
研究严格凸有界区域\(\Omega\)上一般线性退化椭圆型偏微分方程组解的存在性，通过引入自然结构假设，证明存在唯一广义解\(u\in L^2(\Omega, \mathbb{R}^N)\)并满足部分正则性，扩展了早期工作至含低阶项情况。
中文摘要 AI 辅助
设\(\Omega \Subset \mathbb{R}^{n}\)为严格凸有界区域。假设\(\mathbf{A}: \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{Nn}\)，\(\mathbf{B}: \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{N}\)，\(\mathbf{C}: \mathbb{R}^{N} \longrightarrow \mathbb{R}^{N}\)为线性映射，其中\(\mathbf{A}\)对称且非负定。给定\(f \in L^2(\Omega, \mathbb{R}^N)\)，考虑偏微分方程组\(\left\{ \begin{array}{rl} \displaystyle\sum_{\beta = 1}^{N}\sum_{i, j = 1}^{n} \mathbf{A}_{\alpha i \beta j}\mathrm{D}_{ij}^{2}u_{\beta} + \sum_{\beta = 1}^{N} \sum_{i=1}^{n} \mathbf{B}_{\alpha \beta i}\mathrm{D}_{i}u_{\beta} + \sum_{\beta = 1}^{N} \mathbf{C}_{\alpha \beta}u_{\beta} = f_\alpha, &\text{ 在 }\Omega\text{ 内}, \\ u = 0,\ \,& \text{ 在 }\partial \Omega\text{ 上}. \end{array} \right.\)解\(u: \Omega \longrightarrow \mathbb{R}^N\)的存在性问题。此为线性退化椭圆型系统，在无严格秩 - 一凸性假设下此前未被研究过。一般情况下可能甚至没有分布解。通过引入一些自然结构假设，证明了存在适当定义的唯一广义解\(u\in L^2(\Omega, \mathbb{R}^N)\)，满足额外的部分正则性性质。本文将第一作者早期工作扩展到包含低阶项的情况。
英文摘要
Let $Ω\Subset \mathbb{R}^{n}$ be a strictly convex bounded domain. Suppose $\mathbf{A} : \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{Nn}$, $\mathbf{B}: \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{N}$, $\mathbf{C}: \mathbb{R}^{N} \longrightarrow \mathbb{R}^{N}$ are linear maps, where $\mathbf{A}$ is symmetric and non-negative definite. Given $f \in L^2(Ω, \mathbb{R}^N)$, we consider the problem of existence of solutions $u: Ω\longrightarrow \mathbb{R}^N$ to the PDE system \[ \left\{ \begin{array}{rl} \displaystyle\sum_{β= 1}^{N}\sum_{i, j = 1}^{n} \mathbf{A}_{αi βj}\mathrm{D}_{ij}^{2}u_β + \sum_{β= 1}^{N} \sum_{i=1}^{n} \mathbf{B}_{αβi}\mathrm{D}_{i}u_β + \sum_{β= 1}^{N} \mathbf{C}_{αβ}u_β = f_α, &\text{ in $Ω$}, \\ u = 0,\ \,& \text{ on $\partial Ω$}. \end{array} \right. \] This is a linear \textit{degenerate elliptic} system, and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not possess not even distributional solutions. By introducing some natural structural assumptions, we prove the existence of an appropriately defined unique generalised solution $u\in L^2(Ω, \mathbb{R}^N)$, satisfying additional partial regularity properties. This paper extends earlier work of the first appearing author [\textit{N. Katzourakis, On linear degenerate elliptic PDE systems with constant coefficients}, Adv.\ in Calc.Var.\ 9:3, 283-291 (2016)] to include lower-order terms.
