Treven Wall

We consider the existence and several properties of the minima of the energy functional

I[u] := ∫Ω |∂u/∂x2|2 + |∂u/∂x1 + f(u)|2 dx,

operating on complex-valued functions u ∈ H1(Ω), where Ω is a Lipschitz domain in R² and f : C → C is nonlinear but sufficiently well-behaved. We show that these minima solve the Euler-Lagrange equations for I [·], which are naturally quasilinear. The title result, assuming Ω is bounded, is

Theorem. Given Ψ ∈ H1(Ω), let u ∈ H1(Ω) be a minimizer of I [·] on the set A := {w : (w – Ψ) ∈ H01(Ω)}. Then u converges to the trace of Ψ non-tangentially almost everywhere (with respect to surface measure on ∂Ω).

We also show that for Ω = O × (0, ∞), where O ⊂ R is a bounded open interval, solutions to the Euler-Lagrange equation for I [·] exist and converge to the boundary value at O ×{0} non-tangentially almost everywhere (with respect to Lebesgue measure on O).