Moser's estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients. (December 2019)

• Record Type: Journal Article
• Authors: Anceschi, Francesca; Polidoro, Sergio; Ragusa, Maria Alessandra
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: 35K70 -- 35Q84; Kolmogorov equations -- Moser's estimates -- Weak solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.07.001
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We prove L loc ∞ estimates for positive solutions to the following degenerate second order partial differential equation of Kolmogorov type with measurable coefficients of the form ∑ i, j = 1 m 0 ∂ x i a i j ( x, t ) ∂ x j u ( x, t ) + ∑ i, j = 1 N b i j x j ∂ x i u ( x, t ) − ∂ t u ( x, t ) + + ∑ i = 1 m 0 b i ( x, t ) ∂ i u ( x, t ) − ∑ i = 1 m 0 ∂ x i a i ( x, t ) u ( x, t ) + c ( x, t ) u ( x, t ) = 0 where ( x, t ) = ( x 1, …, x N, t ) = z is a point of R N + 1, and 1 ≤ m 0 ≤ N . ( a i j ) is a uniformly positive symmetric matrix with bounded measurable coefficients, ( b i j ) is a constant matrix. We apply the Moser's iteration method to prove the local boundedness of the solution u under minimal integrability assumption on the coefficients.