On the principal eigenvalue of elliptic operators in $\R^N$ and applications

Two generalizations of the notion of principal eigenvalue for elliptic operators in $\R^N$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and ``limit periodic'' operators. These results apply to questions of existence and uniqueness for some semi-linear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures.

[1]  P. Kuchment,et al.  Integral representations and Liouville theorems for solutions of periodic elliptic equations , 2000, math/0007051.

[2]  S. Varadhan,et al.  The principal eigenvalue and maximum principle for second‐order elliptic operators in general domains , 1994 .

[3]  M. Safonov,et al.  Harnack's inequality for elliptic equations and the Hölder property of their solutions , 1983 .

[4]  Andreas E. Kyprianou,et al.  Local extinction versus local exponential growth for spatial branching processes , 2004 .

[5]  Henri Berestycki,et al.  Analysis of the periodically fragmented environment model : I – Species persistence , 2005, Journal of mathematical biology.

[6]  R. Pinsky.  TRANSIENCE, RECURRENCE AND LOCAL EXTINCTION PROPERTIES OF THE SUPPORT FOR SUPERCRITICAL FINITE MEASURE-VALUED DIFFUSIONS' , 1996 .

[7]  V. A. Kondrat'ev,et al.  On Positive Solutions of Elliptic Equations , 1971 .

[8]  J. Lions,et al.  Un théorème de Liouville pour des équations elliptiques à coefficients périodiques , 1989 .

[9]  N. Krylov,et al.  A CERTAIN PROPERTY OF SOLUTIONS OF PARABOLIC EQUATIONS WITH MEASURABLE COEFFICIENTS , 1981 .

[10]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[11]  Henri Berestycki,et al.  Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts , 2005 .

[12]  J. Moser,et al.  On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus , 1992 .

[13]  M. A. Krasnoselʹskii,et al.  Positive Linear Systems, the Method of Positive Operators , 1989 .

[14]  R. Pinsky.  Positive Harmonic Functions and Diffusion: References , 1995 .

[15]  Henri Berestycki,et al.  Liouville-type results for semilinear elliptic equations in unbounded domains , 2007 .

[16]  L. Hansen,et al.  Long Term Risk: An Operator Approach , 2006 .

[17]  Henri Berestycki,et al.  Analysis of the periodically fragmented environment model : I - Influence of periodic heterogeneous environment on species persistence. , 2005 .

[18]  R. Pinsky.  Second Order Elliptic Operators with Periodic Coefficients: Criticality Theory, Perturbations, and Positive Harmonic Functions , 1995 .

[19]  S Bochner,et al.  A NEW APPROACH TO ALMOST PERIODICITY. , 1962, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Henri Berestycki,et al.  Le nombre de solutions de certains problèmes semi-linéaires elliptiques , 1981 .

[21]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[22]  R. Pinsky,et al.  On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching , 1999 .

[23]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .