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On sectoriality of degenerate elliptic operators

Part of: Elliptic equations and systems Special classes of linear operators

Published online by Cambridge University Press:  18 August 2021

Tan Duc Do
Tan Duc Do*
Affiliation:
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam (doductan@tdmu.edu.vn)
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Abstract

Let $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$, where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

Keywords

degenerate elliptic operatorsectorial operatorholomorphic semigroupcontraction semigroupconsistent semigroup

MSC classification

Primary: 35J70: Degenerate elliptic equations 35J15: Second-order elliptic equations
Secondary: 47B44: Accretive operators, dissipative operators, etc.
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 689 - 710
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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