arXiv 1502.026182015-02math.AP

关于常系数线性退化椭圆PDE系统

On Linear Degenerate Elliptic PDE Systems with Constant Coefficients

Nikos Katzourakis

备注：: Journal: Advances in Calculus of Variations, a uniqueness assertion has been added, 11 pages, 2 figures

摘要：

设$\textbf{A}$是$\mathbb{R}^{Nn}$上的对称凸二次型，$Ω\Subset \mathbb{R}^n$是有界凸域.考虑当$ f\in L^2(Ω,\mathbb{R}^N)$满足条件时，问题\[ \tag{1} \label{1} \begin{split} \sum_{β=1}^N\sum_{i,j=1}^n \textbf{A}_{αi βj}\, D^2_{ij}u_β\,=\, f_α, \text{ in }Ω,\quad \ u\,=\, 0, \text{ on }\partial Ω, \end{split} \]解$u: Ω\subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$的存在性问题. \eqref{1}是退化椭圆的，在没有严格秩一凸性假设的情况下，以前没有考虑过它。一般来说，它甚至可能没有分配解决方案。通过引入一个适用于\eqref{1}的分布的推广，我们证明了它的存在性、部分正则性和唯一性.边界条件的满足也是一个问题，由于低正则性的解决方案。研究\eqref{1}和证明方法的动机来自作者[K4，arXiv：1501.06164]最近关于完全非线性系统的广义解的工作。

英文摘要：

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $Ω\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: Ω\subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the problem \[ \tag{1} \label{1} \begin{split} \sum_{β=1}^N\sum_{i,j=1}^n \textbf{A}_{αi βj}\, D^2_{ij}u_β\,=\, f_α, \text{ in }Ω,\quad \ u\,=\, 0, \text{ on }\partial Ω, \end{split} \] when $ f\in L^2(Ω,\mathbb{R}^N)$. \eqref{1} is degenerate elliptic and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not have even distributional solutions. By introducing an extension of distributions adapted to \eqref{1}, we prove existence, partial regularity and by imposing an extra condition uniqueness as well. The satisfaction of the boundary condition is also an issue due to the low regularity of the solution. The motivation to study \eqref{1} and the method of the proof arose from recent work of the author [K4, arXiv:1501.06164] on generalised solutions for fully nonlinear systems.

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