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

Abstract

We prove a Phragmèn-Lindelöf theorem which yields the behavior at infinity of bounded solutions of Dirichlet problems for non-hyperbolic (e.g., elliptic, parabolic) quasilinear second-order partial differential equations in terms of particular solutions of appropriate ordinary differential equations.

Phragmén–Lindelöf theorems in cylinders

“…Theorem 2.7 is related primarily to [13, theorem 2.2] and [14, theorem 2.6], and secondarily to [12, theorem 2.5]. The results of [13,14] require a n,n in [13] or min k=1,...,m a n,n k in [14] to have a positive lower bound (e.g. [13, eqn (14)], [14, eqn (2.6)]), while theorem 2.7 requires only the weaker condition (2.16); when m = 1 (so that C m M = S M ), theorem 2.7 and [13, theorem 2.2] are essentially the same.…”

Phragmén–Lindelöf theorems in slabs for some systems of non-hyperbolic second-order quasi-linear equations

“…The literature on the existence of solutions of Dirichlet problems for systems of (nonlinear) ordinary di¬erential equations is extensive (see, for example, [2,4{6, 11{13]). Assuming assumption 2.3 is true, lemma 4.1 of [9] can be used to show that assumption 2.4 is satis ed.…”

“…As in [9], we will be unable to use barriers speci cally tailored to our operator as occurred in the construction in [8, x 7]. De ne the functions …”

“…[7,15]). The goal of this note is to extend the results in [9] by using (scalar) comparison principles to establish the asymptotic behaviour at in nity of solutions of certain types of non-hyperbolic second-order systems. We will present a Phragm en{Lindel of theorem for unbounded open subsets « of R n that satisfy, for some xed M > 0, « » S M d ef = fX = (x 1 ; : : : ; x n ) 2 R n j jx n j < M g:…”

“…If we could ignore the`coupling terms' u 2 (@u 1 =@x) 2 in (1.1) and (u 1 ) 2 @u 2 =@x in (1.2), then these limits are easily seen to both be cos(y) for jyj 6 1 2 º (e.g. [9]). The statement of our main result (theorem 2.6) has the e¬ect of ignoring these coupling terms (e.g.…”