Operators of order 2$ n $ with interior degeneracy

Genni Fragnelli; Jerome A. Goldstein; Rosa Maria Mininni; Silvia Romanelli

doi:10.3934/dcdss.2020128

Article Contents

2020, Volume 13, Issue 12: 3417-3426. Doi: 10.3934/dcdss.2020128

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Operators of order 2$ n $ with interior degeneracy

1.
Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy
2.
Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, USA

^* Corresponding author: Rosa Maria Mininni
^* Corresponding author: Rosa Maria Mininni

Dedicated to Gisèle Ruiz Goldstein, outstanding mathematician, with great admiration and friendship on her 60th birthday

Received: February 28, 2019

Early access: November 2019

Published: August 2020

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Abstract

We consider a differential operator of order 2$ n $ of the type $ A_n u = (-1)^n (a u^{(n)})^{(n)} $, where $ a(x)>0 $ in $ [0, 1]\setminus\{x_0\} $ and $ a(x_0) = 0 $. We show that, for any $ n\in{\mathbb{N}} $, the operator $ -A_n $ generates a contractive analytic semigroup of angle $ \pi/2 $ on $ L^2 (0, 1) $. Note that the domain of $ A_n $ depends on the type of degeneracy of $ a $. Our theorems extend some previous results in [3] where $ n = 1 $.

Keywords:

Linear differential operators of order 2n,
interior degeneracy,
operators in divergence form,
Dirichlet boundary conditions,
analytic semigroups.

Mathematics Subject Classification: Primary: 47D06, 35K65; Secondary: 47B25, 47N20.

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References

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