Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Sergio Polidoro; Annalaura Rebucci; Bianca Stroffolini

doi:10.3934/cpaa.2022023

Article Contents

2022, Volume 21, Issue 4: 1385-1416. Doi: 10.3934/cpaa.2022023

This issue Previous Article Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth Next Article On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

1.
Università degli Studi di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, 41125 Modena, Italy
2.
Università di Napoli Federico II, Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Via Claudio 25, 80125 Napoli, Italy

^*Corresponding author
^*Corresponding author

Received: August 31, 2021

Early access: January 2022

Published: April 2022

This research was partially supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

We study the regularity properties of the second order linear operator in $ {{\mathbb {R}}}^{N+1} $:

$ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k = 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum\limits_{j,k = 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} $

where $ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $ are real valued matrices with constant coefficients, with $ A $ symmetric and strictly positive. We prove that, if the operator $ {\mathscr{L}} $ satisfies Hörmander's hypoellipticity condition, and $ f $ is a Dini continuous function, then the second order derivatives of the solution $ u $ to the equation $ {\mathscr{L}} u = f $ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $ a_{jk} $'s. A key step in our proof is a Taylor formula for classical solutions to $ {\mathscr{L}} u = f $ that we establish under minimal regularity assumptions on $ u $.

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Mathematics Subject Classification: 35K70, 35K65, 35B65.

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