Estimates for fundamental solutions of parabolic equations in non-divergence form

Hongjie Dong; Seick Kim; Sungjin Lee

doi:10.1016/j.jde.2022.09.007

Estimates for fundamental solutions of parabolic equations in non-divergence form

Hongjie Dong, Seick Kim, Sungjin Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.

Original language	English
Pages (from-to)	557-591
Number of pages	35
Journal	Journal of Differential Equations
Volume	340
DOIs	https://doi.org/10.1016/j.jde.2022.09.007
Publication status	Published - 2022 Dec 15

Bibliographical note

Funding Information:
H. Dong was partially supported by the Simons Foundation, grant no. 709545, a Simons fellowship, grant no. 007638, and the NSF under agreement DMS-2055244.S. Kim was partially supported by the National Research Foundation of Korea under agreements NRF-2019R1A2C2002724 and NRF-2022R1A2C1003322.

Publisher Copyright:
© 2022 Elsevier Inc.

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1016/j.jde.2022.09.007

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title = "Estimates for fundamental solutions of parabolic equations in non-divergence form",

abstract = "We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.",

author = "Hongjie Dong and Seick Kim and Sungjin Lee",

note = "Funding Information: H. Dong was partially supported by the Simons Foundation, grant no. 709545, a Simons fellowship, grant no. 007638, and the NSF under agreement DMS-2055244.S. Kim was partially supported by the National Research Foundation of Korea under agreements NRF-2019R1A2C2002724 and NRF-2022R1A2C1003322. Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

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doi = "10.1016/j.jde.2022.09.007",

language = "English",

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AU - Dong, Hongjie

AU - Kim, Seick

AU - Lee, Sungjin

N1 - Funding Information: H. Dong was partially supported by the Simons Foundation, grant no. 709545, a Simons fellowship, grant no. 007638, and the NSF under agreement DMS-2055244.S. Kim was partially supported by the National Research Foundation of Korea under agreements NRF-2019R1A2C2002724 and NRF-2022R1A2C1003322. Publisher Copyright: © 2022 Elsevier Inc.

PY - 2022/12/15

Y1 - 2022/12/15

N2 - We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.

AB - We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.

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SN - 0022-0396

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JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

Estimates for fundamental solutions of parabolic equations in non-divergence form

Abstract

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