On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint
On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint
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DOI：
10.1016/j.na.2022.112889
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
A. Taheri;Vahideh Vahidifar
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文献类型：
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作者：
A. Taheri;Vahideh Vahidifar
本文的目的是证明散度形式的非线性椭圆系统族与逐点梯度约束相结合的多个解的存在性： \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} \ \Omega , \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right。 \end{align*} 其中()是有界域，是向量图，是规定的边界条件。此外，还有与约束相关的静水压力 $\A = \A(|x|,|u|^2,|\nabla u|^2)$、$\B = \B(|x|,|u|^2,|\nabla u|^2)$ 是足够规则的标量值函数，满足无穷大的适当增长。该系统出现在不同的领域，例如连续介质力学和非线性弹性，以及几何函数理论等，清楚地理解解集的形式和结构具有重要意义。这里构建的几何类型的解利用了与李群、其李指数和作用于某些向量场的多维旋度算子的密切联系。最值得注意的是，由偏微分方程引发的判别式量 $\Delta=\Delta(\A,\B)$ 将被证明对这些解的结构和多重性具有决定性作用。
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} \ \Omega , \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where() is a bounded domain,is a vector-map andis a prescribed boundary condition. Moreoveris a hydrostatic pressure associated with the constraintand $\A = \A(|x|,|u|^2,|\nabla u|^2)$, $\B = \B(|x|,|u|^2,|\nabla u|^2)$ 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, its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably a discriminant type quantity $\Delta=\Delta(\A,\B)$, prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.