Harnack inequality for a class of strongly degenerate elliptic operators formed by Hörmander vector fields

2011 ◽  
Vol 135 (3-4) ◽  
pp. 361-380 ◽  
Author(s):  
G. Di Fazio ◽  
M. S. Fanciullo ◽  
P. Zamboni
Keyword(s):  
Harnack Inequality ◽  
Vector Fields ◽  
Elliptic Operators ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Strongly Degenerate ◽  
Hörmander Vector Fields

scholarly journals High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengcheng Niu ◽  
Kelei Zhang
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Carnot Group ◽  
A Priori ◽  
Elliptic Operators ◽  
High Order ◽  
Carnot Groups ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Horizontal Vector

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

scholarly journals A class of strongly degenerate elliptic operators

1989 ◽  
Vol 39 (2) ◽  
pp. 177-200 ◽  
Author(s):  
Duong Minh Duc
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Elliptic Operators ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic Operators ◽  
Singular Elliptic Equations ◽  
Degenerate Elliptic ◽  
Weighted Poincaré Inequality ◽  
Strongly Degenerate

Using a weighted Poincaré inequality, we study (ω1,…,ωn)-elliptic operators. This method is applied to solve singular elliptic equations with boundary conditions in W1,2. We also obtain a result about the regularity of solutions of singular elliptic equations. An application to (ω1,…,ωn)-parabolic equations is given.

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

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.

scholarly journals Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

2015 ◽  
Vol 103 (5) ◽  
pp. 1276-1293 ◽  
Author(s):  
Henri Berestycki ◽  
Italo Capuzzo Dolcetta ◽  
Alessio Porretta ◽  
Luca Rossi
Keyword(s):  
Maximum Principle ◽  
Principal Eigenvalue ◽  
Elliptic Operators ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic

scholarly journals Spectral properties of degenerate elliptic operators with matrix coefficients

2013 ◽  
Vol 5 (4) ◽  
pp. 37-48 ◽  
Author(s):  
Makhmadrakhim Gafurovich Gadoev ◽  
Sulaimon Abunasrovich Iskhokov
Keyword(s):  
Spectral Properties ◽  
Elliptic Operators ◽  
Degenerate Elliptic Operators ◽  
Matrix Coefficients ◽  
Degenerate Elliptic
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