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# On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint

journal contribution

posted on 2023-06-10, 02:53 authored by Ali TaheriAli Taheri, Vahideh VahidifarThe 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,|

abla u|^2)

abla u\} + \B(|x|,|u|^2,|

abla u|^2) u = \dive \{ \mcP(x) [{\rm cof}\,

abla u] \} \quad &\text{ in} \ \Omega , \\ \text{det}\,

abla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det

abla u \equiv 1$ and $\A = \A(|x|,|u|^2,|

abla u|^2)$, $\B = \B(|x|,|u|^2,|

abla 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 $\Delta=\Delta(\A,\B)$, prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.

abla u|^2)

abla u\} + \B(|x|,|u|^2,|

abla u|^2) u = \dive \{ \mcP(x) [{\rm cof}\,

abla u] \} \quad &\text{ in} \ \Omega , \\ \text{det}\,

abla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det

abla u \equiv 1$ and $\A = \A(|x|,|u|^2,|

abla u|^2)$, $\B = \B(|x|,|u|^2,|

abla 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 $\Delta=\Delta(\A,\B)$, prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.

## History

## Publication status

- Published

## File Version

- Accepted version

## Journal

Nonlinear Analysis Theory Methods and Applications## ISSN

0362-546X## Publisher

Elsevier## External DOI

## Volume

221## Article number

a112889## Department affiliated with

- Mathematics Publications

## Research groups affiliated with

- Analysis and Partial Differential Equations Research Group Publications

## Full text available

- Yes

## Peer reviewed?

- Yes