The Minimum Principle for Elliptic Systems

Abstract

The classical minimum principle for a second order elliptic operator L has recently been generalizied to a ‘best’ theorem which includes the eigenvalue case [3]. It states that if Lu + ≤cu 0 in D⊂R^m (D open and bounded, c = c(x)), u > 0 on Γ=∂D and if a function h with the properties Lh + ch ≤ 0 and h > 0 in D exists (h = 0 on Γ] allowed), then either u > 0 or u≡0 or u = −αh (a > 0). In the third case u is eigenfunction for the operator L + c and zero Dirichlet boundary data.

For an elliptic system with L = diag (L₁,…,L_n), u = (u₁,…,u_n)^T, c(x) = (c_ij(x)) this theorem remains valid in full generality (with inequalities defined componentwise) if the matrix c is essentially positive (i.e., c_ij:> 0 for i ¦ j) and irreducible.

Keywords:

• Elliptic systems
• minimum principle

AMS:

• 35J45