On the contraction properties for weak solutions to linear elliptic equations with 𝐿²-drifts of negative divergence

Lee, Haesung

doi:10.1090/proc/16672

On the contraction properties for weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence
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by Haesung Lee;

Proc. Amer. Math. Soc. 152 (2024), 2051-2068

DOI: https://doi.org/10.1090/proc/16672

Published electronically: March 26, 2024

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Abstract:

We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the $L^r$-contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian $C_0$-resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an $L^1$-stability result through an extended version of the $L^1$-contraction property.

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