The three divergence free matrix fields problem
Mariapia Palombaro, Marcello Ponsiglione · Asymptotic Analysis · 2004
We prove that for any connected open set Ω⊂ $\mathbb{R}$ n and for any set of matrices K={A 1 ,A 2 ,A 3 }⊂ $\mathbb{M}$ 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.