$www.math.cornell.edu$

Treven Wall

Ph.D. (2007) Cornell University

First Position

Research fellow at University of Edinburgh

Dissertation

A Fatou Theorem for a Class of Quasi-linear Elliptic Partial Differential Equations

Advisor:

Leonard Gross

Research Area:

Partial differential equations and harmonic analysis

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

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

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).

Last modified: June 7, 2007