Uniqueness for then-dimensional half space Dirichlet problem

Abstract

In JR n , we prove uniqueness for the Dirichlet problem in the half space x n > 0, with continuous data, under the growth condition u = o(\x\sec Ί θ) as \x\ ->• oo (x n = |#|cos#, 7 G ffi). Under the natural integral condition for convergence of the Poisson integral with Dirichlet data, the Poisson integral will satisfy this growth condition with 7 = n -1. A PhragmenLindelδf principle is established under this same growth condition. We also consider the Dirichlet problem with data of higher order growth, including polynomial growth. In this case, if u = o(|#| N+1 sec 7 θ) (7 G M, N > 1), we prove solutions are unique up to the addition of a harmonic polynomial of degree N that vanishes when x n = 0.

On the Phragmén–Lindelöf principle for second-order elliptic equations

“…the maximum principle holds for ∆ in the half-plane considering functions with growth o(|x|) and this result can be extended to higher dimensions in the half-space. It should be mentioned that Siegel [12] in two variables and Siegel-Talvila [13] in higher dimensions improved the admissible growth to o(|x|/ sin γ θ), γ ∈ R, when |x| cos θ is the projection of x on the hyperplane which is the boundary of the half-space. It is also worth to remark that, in the case of an infinite open connected cone Σ such thatΣ = R n , BerestyckiCaffarelli-Nirenberg [1] stated a maximum principle for ∆ in which the admissible growth is related to the principal eigenvalue of the Laplace-Beltrami operator −∆ S with homogeneous Dirichlet boundary conditions in the region Σ ∩ S n−1 .…”

The Dirichlet problem on quadratic surfaces

“…Note that the uniqueness result in the theorem below fails on some nondegenerate quadratic surfaces, so the hypothesis that q is nonhyperbolic cannot be deleted. For example, on the quadratic surface defined by {x ∈ R n : x To obtain uniqueness results on half-spaces, even in the class of polynomial solutions, a growth condition on the solutions is needed (see [7]). However, the theorem below shows that we have unique polynomial solutions on our quadratic surfaces without the requirement of a growth condition.…”

Growth properties for modified Poisson integrals in a half space

“…In this paper, we consider functions f satisfying R n−1 |f (y)| p (1 + |y|) −γ dy < ∞ (1.4) for 1 p < ∞ and a real number γ. To obtain the Dirichlet solution for the boundary data f , as in [13,14] and [19], we use the following modified kernel function defined by…”