Elliptic Operators on Manifolds with Singularities and K-Homology

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

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About the G-compactness of certain classes of second-order elliptic operators

Daghestan Electronic Mathematical Reports ◽

10.31029/demr.10.1 ◽

2018 ◽

pp. 1-12

Author(s):

Magomed Sirazhudinov ◽

Saida Dzhamaludinova

Keyword(s):

Elliptic Operators ◽

Second Order ◽

Second Order Elliptic Operators

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Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

Advances in Calculus of Variations ◽

10.1515/acv-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Anup Biswas ◽

Prasun Roychowdhury

Keyword(s):

Eigenvalue Problem ◽

Principal Eigenvalue ◽

Unbounded Domains ◽

Elliptic Operators ◽

Maximum Principles ◽

Generalized Eigenvalues ◽

Nonlinear Elliptic ◽

Generalized Eigenvalue ◽

General Convex ◽

Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

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Boundedness of solutions of variational inequalities with nonlinear degenerated elliptic operators of high order

Applicable Analysis ◽

10.1080/00036819708840560 ◽

1997 ◽

Vol 65 (3-4) ◽

pp. 225-249 ◽

Cited By ~ 27

Author(s):

Alexander Kovalevsky ◽

Francesco Nicolosi

Keyword(s):

Variational Inequalities ◽

Elliptic Operators ◽

High Order ◽

Boundedness Of Solutions

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Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

Geometric and Functional Analysis ◽

10.1007/s00039-021-00566-4 ◽

2021 ◽

Author(s):

Steve Hofmann ◽

José María Martell ◽

Svitlana Mayboroda ◽

Tatiana Toro ◽

Zihui Zhao

Keyword(s):

Carleson Measure ◽

Elliptic Operators ◽

Uniform Rectifiability

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Pseudo-monotonicity and degenerated or singular elliptic operators

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032184 ◽

1998 ◽

Vol 58 (2) ◽

pp. 213-221 ◽

Cited By ~ 9

Author(s):

P. Drábek ◽

A. Kufner ◽

V. Mustonen

Keyword(s):

Sobolev Spaces ◽

Differential Operators ◽

Weighted Sobolev Spaces ◽

Type Inequality ◽

Elliptic Operators ◽

Hardy Type Inequality ◽

Hardy Type ◽

Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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