On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint. (August 2022) - Taheri, Ali
Abstract: 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: div { A ( | x |, | u | 2, | ∇ u | 2 ) ∇ u } + B ( | x |, | u | 2, | ∇ u | 2 ) u = div { P ( x ) [ cof ∇ u ] } in Ω, det ∇ u = 1 in Ω, u = φ on ∂ Ω, where Ω ⊂ R n ( n ≥ 2 ) is a bounded domain, u = ( u 1, …, u n ) is a vector-map and φ is a prescribed boundary condition. Moreover P is a hydrostatic pressure associated with the constraint det ∇ u ≡ 1 and A = A ( | x |, | u | 2, | ∇ u | 2 ), B = B ( | x |, | u | 2, | ∇ 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 draw upon intimate links with the Lie group 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 system, will be shown to have a decisive role on the structure and multiplicity of these solutions.