Phragmèn-Lindelöf and decay rate theorems for solutions of non-hyperbolic Dirichlet problems

Zhiren Jin Wichita State University, USA

                    e-mail zhiren@math.twsu.edu

Kirk Lancaster Wichita State University, USA

                   e-mail lancaster@math.twsu.edu

Let M > 0, S_{M} = {X = (x_{1}, …, x_{n}) ∈ R^{n} | |x_{n}| < M } and Ω be an open subset of S_{M}; for simplicity, we set Ω = S_{M} here. Suppose (a_{ij}) is any n × n (symmetric) matrix with entries in C^{1}(Ω × R^{n+1}) which is positive semidefinite and has trace one on Ω×R^{n+1} and satisfies a_{nn}(X, z, P) ≥ σ_{1}(|P |) for X ∈ Ω, z ∈ R and P ∈ R^{n} with |P | ≥ 1, where σ_{1} is any positive function in C^{0}([1, ∞)). Let b ∈ C^{1}(Ω × R^{n+1}) and Q be the non-hyperbolic operator defined by

Qu(X) = ∑_{i,j=1}^{n} a_{ij}(X, u(X), Du(X))D_{ij}u(X) + b(X, u(X), Du(X)).

We write elements X = (x_{1}, …, x_{n}) of R^{n} as (x, y), where x = (x_{1}, …, x_{n−1}) and y = x_{n}. Let ω ∈ S^{n−2} and assume there exist O open in S^{n−2} with ω ∈ O and E ∈ C^{1}(O × [−M, M] × R^{2}) such that E(\frac{x}{|x|}, y, z, q) is nonincreasing in z and

\frac{b(x, y, z, p, q)}{a_{nn}(x, y, z, p, q)} → E(σ, y, z, q)

as |x| → ∞ with \frac{x}{|x|} → σ and |p| → 0 uniformly for |y| < M, σ ∈ O, and z, q ∈ R. Let ϕ ∈ C^{0}(∂Ω) and suppose there exists k ∈ C^{2}([−M, M]) such that k′′(y) + E(ω, y, k(y), k′(y)) = 0 for |y| < M and ϕ(x, ±M) → k(±M) uniformly as |x| → ∞ and \frac{x}{|x|} → ω. Theorem Suppose f ∈ C^{2}(Ω) ∩ C^{0}(\overline{Ω}) ∩ L^{∞}(Ω) satisfies Qf = 0 in Ω and f = ϕ on ∂Ω. Then f(x_{j}, y_{j}) → k(y_{j}) as j → ∞ uniformly for sequences \{(x_{j}, y_{j})\} in \overline{Ω} with |x_{j}| → ∞ and \frac{x_{j}}{|x_{j}|} → ω as j → ∞. Estimates of the rate (in terms of |x|) at which |f(x, y)−k(y)| converges to zero will also be given.

Presented by Kirk Lancaster