arXiv 2206.030762022-06-07math.AP

一类具有不可压缩约束的散度型Nonlinear Elliptic方程组的多解性

On Multiple Solutions to a Family of Nonlinear Elliptic Systems in Divergence Form Coupled with an Incompressibility Constraint

Ali Taheri, Vahideh Vahidifar

备注：: 24 pages

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摘要：

本文的目的是证明一类带有逐点梯度约束的散度型非线性椭圆方程组：\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*}的多解存在性，其中$Ω\subset \mathbb{R}^n$（$n \ge 2$）是有界区域，$u=(u_1, \dots, u_n)$是向量映射，$φ$是给定的边界条件.此外，$\mathscr{P}$是与约束$\det \nabla u \equiv 1$和$\A = \A(|x|,|u|^2,|\nabla u|^2)$、$\B = \B(|x|,|u|^2,|\nabla u|^2)$相关联的流体静压，是满足在无穷远处的适当增长的充分正则标量值函数。该系统出现在不同的领域，例如，在连续介质力学、非线性弹性力学以及几何函数理论等方面有着重要的应用价值，并且清楚地了解解集的形式和结构具有重要的意义。这里构造的几何类型的解决方案借鉴了与李群${\bf SO}(n)$，其李指数和多维旋度算子作用于某些向量场的密切联系。最值得注意的是，来自偏微分方程的判别类型量$Δ=Δ(\A,\B)$将被证明对这些解的结构和多重性具有决定性作用。

英文摘要：

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.

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。

摘要：

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