Existence of solutions for gradient coupled Dirichlet systems

A Italo, per i suoi 3/4 di secolo

¹ Istituto Lombardo & Sapienza Università di Roma, Italy
² Sapienza Università di Roma, Italy

^* boccardo@mat.uniroma1.it; orsina@mat.uniroma1.it

Received: 16 March 2023
Accepted: 3 February 2024

Abstract

In this paper, we prove existence of weak solutions in W^1,2₀ (Ω) ∩L∞(Ω) for the gradient coupled Dirichlet system

{u ∈ W^1,2₀ (Ω): –div(M(x)∇u) + u + a(x)∇u · ∇ψ = f(x),

{ψ ∈ W^1,2₀ (Ω): –div(M(x)∇ψ) + ψ + a(x)∇u · ∇ψ = g(x).

We also prove that if f (x), g(x) ≥ 0 (of course ≢ 0 a.e.), then u(x), ψ(x) ≥ 0 and the sets {u = 0} and {ψ = 0} have zero Lebesgue measure.