论文标题
在多个解决方案的非线性椭圆形系统中，发散形式与不可压缩的约束结合
On Multiple Solutions to a Family of Nonlinear Elliptic Systems in Divergence Form Coupled with an Incompressibility Constraint
论文作者
论文摘要
本文的目的是证明存在多种解决方案，用于以差异形式的非线性椭圆系统家族，以及一个点式的梯度约束：\ begin {align*} \ left \ left \ oken {arnay} {array} {ll} {ll} {ll} {ll} {ll} {ll} {ll} \ dive \ dive \ \ a { U \} + \ b（| x |，| U |^2，| \ nabla u |^2） ＆\ text {in} \ω，\\ u =φ\＆\ text {on} \ \partialΩ，\ end end {array} \ right。 \ end {align*}其中$ω\ subset \ mathbb {r}^n $（$ n \ ge 2 $）是一个有界域，$ u =（u_1，\ dots，u_n）$是矢量映射，$φ$是规定的边界条件。此外，$ \ Mathscr {p} $是与约束$ \ det \ nabla u \ equiv 1 $和$ \ a = \ a = \ a（| x |，| U |^2，| \ nabla u |^2）$，$ \ \ b = \ b = \ b（| x |，| x | U |^nabla |^nabla |^2，|满足无穷大的合适增长。该系统出现在不同领域，例如连续力学和非线性弹性，以及几何函数理论，仅举几例，并且对解决方案集的形式和结构有清晰的理解。此处构建的几何类型借鉴了与Lie group $ {\ bf so}（n）$的紧密链接，其Lie指数和在某些向量字段上作用的多维卷曲操作员。最值得注意的是，从PDE提示的判别类型$δ=δ（\ a，\ b）$将显示在这些解决方案的结构和多重性上具有决定性的作用。
The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll} \dive\{\A(|x|,|u|^2,|\nabla u|^2) \nabla u\} + \B(|x|,|u|^2,|\nabla u|^2) u = \dive \{ \mcP(x) [{\rm cof}\,\nabla u] \} \quad &\text{ in} \ Ω, \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ Ω, \\ u =φ\ &\text{ on} \ \partial Ω, \end{array} \right. \end{align*} where $Ω\subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $φ$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det \nabla u \equiv 1$ and $\A = \A(|x|,|u|^2,|\nabla u|^2)$, $\B = \B(|x|,|u|^2,|\nabla u|^2)$ are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draws upon intimate links with the Lie group ${\bf SO}(n)$, its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably a discriminant type quantity $Δ=Δ(\A,\B)$, prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.
