Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms

Markus Kunze; Marianna Porfido; Abdelaziz Rhandi

doi:10.3934/dcdss.2023091

Article Contents

2024, Volume 17, Issue 5&6: 2141-2172. Doi: 10.3934/dcdss.2023091

This issue Previous Article New semigroups from old: An approach to Feller boundary conditions Next Article Tikhonov regularization of a nonhomogeneous first-order evolution equation

Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms

1.
Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
2.
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (Sa), Italy

^*Corresponding author: Abdelaziz Rhandi
^*Corresponding author: Abdelaziz Rhandi

Received: September 2022

Revised: March 2023

Early access: April 2023

Published: May 2024

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Abstract

Global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order elliptic operators in $ \mathbb{R}^d $ was obtained in the case of bounded diffusion coefficients in [13, Section 5]. In this paper we generalize these results to the case of unbounded diffusions. Our technique is based on an approximation procedure and a De Giorgi type regularity result. We like to point out, that such an approximation procedure cannot be applied to the result in [13], since the constants in the estimates obtained in [13] depend on the infinity norm of the diffusion coefficients.

Keywords:

Parabolic equation with unbounded coefficients,
heat kernel estimates,
estimates of the derivative,
one-parameter semigroup.

Mathematics Subject Classification: 35K08, 35K10, 47D06.

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References

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