The three divergence free matrix fields problem

Original · EN

We prove that for any connected open set Ω⊂ ⁿ and for any set of matrices K={A₁,A₂,A₃}⊂ Mᵐ× ⁿ, with m≥ n and rank(Aᵢ-Aⱼ)=n for i≠ j, there is no non-constant solution B∈ L∞(Ω,Mᵐ× ⁿ), called exact solution, to the problem Div B=0 in D'(Ω,ᵐ) and B(x)∈ K a.e. in Ω. In contrast, A. Garroni and V. Nesi GN 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.