On General Linear Degenerate Elliptic PDE Systems

Abstract

Let $\hat{I}\text{c◯}⋐{\mathbb{R}}^n$ be a strictly convex bounded domain. Suppose $\mathbf{A}:{\mathbb{R}}^{Nn}⟶{\mathbb{R}}^{Nn}$, $\mathbf{B}:{\mathbb{R}}^{Nn}⟶{\mathbb{R}}^N$, $\mathbf{C}:{\mathbb{R}}^N⟶{\mathbb{R}}^N$ are linear maps, where $\mathbf{A}$ is symmetric and non-negative definite. Given $f\in L^2(\hat{I}\text{c◯},{\mathbb{R}}^N)$, we consider the problem of existence of solutions $u:\hat{I}\text{c◯}⟶{\mathbb{R}}^N$ to the PDE system \[ \left\{ \begin{array}{rl} \displaystyle\sum_{β= 1}^{N}\sum_{i, j = 1}^{n} \mathbf{A}_{αi βj}\mathrm{D}_{ij}^{2}u_β + \sum_{β= 1}^{N} \sum_{i=1}^{n} \mathbf{B}_{αβi}\mathrm{D}_{i}u_β + \sum_{β= 1}^{N} \mathbf{C}_{αβ}u_β = f_α, &\text{ in $\hat{I}\text{c◯}$}, \\ u = 0,\ \,& \text{ on $\partial \hat{I}\text{c◯}$}. \end{array} \right. \] 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 $u\in L^2(\hat{I}\text{c◯},{\mathbb{R}}^N)$, 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.

15 pages