Theorems of Phragmèn-Lindelöf type for quasilinear elliptic equations

A Phragmèn–Lindelöf theorem and the behavior at infinity of solutions of non-hyperbolic equations

“…The construction of barriers for Q is somewhat similar to the constructions of barriers for the operator Q # given in [13] and [22].…”

“…Notice that E(ω, y, z, q) = −(q 2 + ω 2 1 ) for ω = (ω 1 , ω 2 ) ∈ S 1 . The Dirichlet problem (13) here is k ω (y) = (k ω (y)) 2 + ω 2 1 with k ω (−1) = k ω (1) = 0 and its solution is…”

“…Notice that (8) requires k ω (y) = cosh(1) for all y ∈ [−1, 1]. However this constant function is not a solution of (13). In fact, it is impossible to obtain a function k 1 (y) which satisfies (9) and (11) when α is sufficiently small.…”

“…If b/a nn have appropriate limits at infinity (i.e., (7)), φ ∈ C 0 (R n ), ω ∈ S n−2 is a direction of Ω at infinity (i.e., (6)) and Assumptions 1 and 2 in §2 are satisfied, we will prove that every bounded solution f ∈ C 2 (Ω) ∩ C 0 (Ω) of the Dirichlet problem Qf = 0 in Ω (2) and f = φ on ∂Ω (3) satisfies f (x, y) → k ω (y) for X = (x, y) ∈ Ω (4) as |X| → ∞ with x |x| → ω, where k ω is a solution of a related boundary value problem (e.g., (13)). …”

CMC capillary surfaces at reentrant corners

“…As a solution of a boundary value problem for a quasilinear elliptic equation with positive genre (g = 2), the possibility of obtaining such conditions should seem remote; solutions of equations like (1a) can behave near a point on the boundary in significantly different ways than can solutions of equations such as Laplace's equation or a Poisson equation, as illustrated by [Jin and Lancaster 1999] and [Serrin 1969]. Let us assume is a domain in ‫ޒ‬ 2 whose boundary has a corner of size 2α at a point, which we may temporarily take to be the origin O = (0, 0) for convenience; we may assume the domain is oriented so that the rays θ = α and θ = − α are tangent to ∂ at O and so that the directions θ ∈ (−α, α) are the interior directions to from O, where (r, θ ) denotes polar coordinates about O.…”