The three divergence free matrix fields problem

Abstract

We prove that for any connected open set Ω⊂ $\mathbb{R}$ ⁿ and for any set of matrices K={A₁,A₂,A₃}⊂ $\mathbb{M}$ ^m×n, with m≥n and rank(A_i−A_j)=n for i≠j, there is no non‐constant solution B∈L^∞(Ω, $\mathbb{M}$ ^m×n), called exact solution, to the problem

Div B=0 in 𝒟′(Ω, $\mathbb{R}$ ^m) and B(x)∈K a.e.in Ω.

In contrast, Garroni and Nesi [10] exhibited an example of set K for which the above problem admits the so‐called approximate solutions. We give further examples of this type.

We also prove non‐existence of exact solutions when K is an arbitrary set of matrices satisfying a certain algebraic condition which is weaker than simultaneous diagonalizability.