The three divergence free matrix fields problem

Article type: Research Article

Authors: Palombaro, Mariapia^; | Ponsiglione, Marcello

Affiliations: Dipartimento di Matematica, Università “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy E‐mail: palombar@mat.uniroma1.it | S.I.S.S.A., Via Beirut 2‐4, 34014 Trieste, Italy E‐mail: ponsigli@sissa.it

Note: [] Corresponding author.

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(Ai−Aj)=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.

Keywords: differential inclusions, phase transitions, homogenization

Journal: Asymptotic Analysis, vol. 40, no. 1, pp. 37-49, 2004