Half-eigenvalues of elliptic operators

Research output: Contribution to journalArticle

Abstract

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rn, n = 2, and a, b ? L8(O). If the equation Lu = au+ - bu- + ?u (where ? ? R and u±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim ??8 f(x,?)/?, b(x) = lim ??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

Original languageEnglish
Pages (from-to)1439-1451
Number of pages13
JournalProceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume132
Issue number6
Publication statusPublished - 2002

Fingerprint

Elliptic Operator
Eigenvalue
Partial Differential Operators
Nontrivial Solution
Semilinear
Solvability
Bounded Domain

Cite this

@article{0adcb2015abd4991bf535b9986235ce8,
title = "Half-eigenvalues of elliptic operators",
abstract = "Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rn, n = 2, and a, b ? L8(O). If the equation Lu = au+ - bu- + ?u (where ? ? R and u±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carath{\'e}odory function such that, for a.e. x ? O, a(x) = lim ??8 f(x,?)/?, b(x) = lim ??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).",
author = "Rynne, {Bryan P.}",
year = "2002",
language = "English",
volume = "132",
pages = "1439--1451",
journal = "Proceedings of the Royal Society of Edinburgh, Section A: Mathematics",
issn = "0308-2105",
publisher = "Cambridge University Press",
number = "6",

}

Half-eigenvalues of elliptic operators. / Rynne, Bryan P.

In: Proceedings of the Royal Society of Edinburgh, Section A: Mathematics, Vol. 132, No. 6, 2002, p. 1439-1451.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Half-eigenvalues of elliptic operators

AU - Rynne, Bryan P.

PY - 2002

Y1 - 2002

N2 - Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rn, n = 2, and a, b ? L8(O). If the equation Lu = au+ - bu- + ?u (where ? ? R and u±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim ??8 f(x,?)/?, b(x) = lim ??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

AB - Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rn, n = 2, and a, b ? L8(O). If the equation Lu = au+ - bu- + ?u (where ? ? R and u±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim ??8 f(x,?)/?, b(x) = lim ??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

UR - http://www.scopus.com/inward/record.url?scp=0036979601&partnerID=8YFLogxK

M3 - Article

VL - 132

SP - 1439

EP - 1451

JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

SN - 0308-2105

IS - 6

ER -