The Largest Eigenvalue of a Convex Function, Duality, and a Theorem of Slodkowski

First, we provide an exposition of a theorem due to Slodkowski regarding the largest “eigenvalue” of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for C 2 functions corresponds to the largest eigenvalue of the Hessian. The...

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Veröffentlicht in:The Journal of geometric analysis Jg. 26; H. 4; S. 3027 - 3055
1. Verfasser: Dellatorre, Matthew
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.10.2016
Springer Nature B.V
Schlagworte:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Convexity
Differential Geometry
Dirichlet problem
Dynamical Systems and Ergodic Theory
Eigenvalues
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Lower bounds
Mathematics
Mathematics and Statistics
Theorems
52A41
35J60
Dualilty
26B25
Convex analysis
Legendre transform
estimates
ISSN:1050-6926, 1559-002X
Online-Zugang:Volltext
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Zusammenfassung:First, we provide an exposition of a theorem due to Slodkowski regarding the largest “eigenvalue” of a convex function. In his work on the Dirichlet problem, Slodkowski introduces a generalized second-order derivative which for C 2 functions corresponds to the largest eigenvalue of the Hessian. The theorem allows one to extend an a.e. lower bound on this largest “eigenvalue” to a bound holding everywhere. Via the Dirichlet duality theory of Harvey and Lawson, this result has been key to recent progress on the fully non-linear, elliptic Dirichlet problem. Second, using the Legendre–Fenchel transform we derive a dual characterization of this largest eigenvalue in terms of convexity of the conjugate function. This dual characterization offers further insight into the nature of this largest eigenvalue and allows for an alternative proof of a necessary bound for the theorem.
Bibliographie:ObjectType-Article-1
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-015-9660-0