Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Author:

Polidoro Sergio1,Rebucci Annalaura1,Stroffolini Bianca2

Affiliation:

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

Abstract

<p style='text-indent:20px;'>We study the regularity properties of the second order linear operator in <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb {R}}}^{N+1} $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $\end{document}</tex-math></inline-formula> are real valued matrices with constant coefficients, with <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> symmetric and strictly positive. We prove that, if the operator <inline-formula><tex-math id="M4">\begin{document}$ {\mathscr{L}} $\end{document}</tex-math></inline-formula> satisfies Hörmander's hypoellipticity condition, and <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a Dini continuous function, then the second order derivatives of the solution <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> to the equation <inline-formula><tex-math id="M7">\begin{document}$ {\mathscr{L}} u = f $\end{document}</tex-math></inline-formula> are Dini continuous functions as well. We also consider the case of Dini continuous coefficients <inline-formula><tex-math id="M8">\begin{document}$ a_{jk} $\end{document}</tex-math></inline-formula>'s. A key step in our proof is a Taylor formula for classical solutions to <inline-formula><tex-math id="M9">\begin{document}$ {\mathscr{L}} u = f $\end{document}</tex-math></inline-formula> that we establish under minimal regularity assumptions on <inline-formula><tex-math id="M10">\begin{document}$ u $\end{document}</tex-math></inline-formula>.</p>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Analysis,General Medicine

Cited by 7 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3