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Sectoriality of degenerate elliptic operators via p-ellipticity

  • Tan Duc Do EMAIL logo
Published/Copyright: December 15, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 35 Issue 1

Abstract

Let Ω ⊂ ℝ d be open and c k ⁢ l ∈ L ∞ ⁢ ( Ω , ℂ ) with Im ⁡ c k ⁢ l = Im ⁡ c l ⁢ k for all k , l ∈ { 1 , … , d } . Assume that C = ( c k ⁢ l ) 1 ≤ k , l ≤ d satisfies ( C ⁢ ( x ) ⁢ ξ , ξ ) ∈ Σ θ for all x ∈ Ω and ξ ∈ ℂ d , where Σ θ is the closed sector with vertex 0 and semi-angle θ in the complex plane. We emphasize that Ω is an arbitrary domain and C need not be symmetric. We show that C is (degenerate) p-elliptic for all p ∈ ( 1 , ∞ ) with | 1 - 2 p | < cos ⁡ θ in the sense of Carbonaro and Dragičević. As a consequence, we obtain the consistent holomorphic extension for the C 0 -semigroup generated by the second-order differential operator in divergence form associated with C. The core property for this operator is also investigated.

Keywords: Degenerate elliptic operator; sectorial operator; holomorphic semigroup; contraction semigroup; consistent semigroup; core property
MSC 2010: 35J70; 35J15; 47B44

Communicated by Siegfried Echterhoff

Acknowledgements

I wish to thank Tom ter Elst and the referee for giving detailed and valuable comments.

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Received: 2021-08-02
Revised: 2022-09-19
Published Online: 2022-12-15
Published in Print: 2023-01-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Tameness of Margulis space-times with parabolics
  3. Sectoriality of degenerate elliptic operators via p-ellipticity
  4. Comparing Hecke eigenvalues of Siegel eigenforms
  5. Isomorphisms of Orlicz spaces
  6. The Engel graph of a finite group
  7. Nonnegative solutions of a fractional differential inequality on Grushin spaces and nilpotent Lie groups
  8. Representations of some associative pseudoalgebras
  9. On the convex components of a set in ℝ n
  10. Spectrality of homogeneous Moran measures on ℝ n
  11. Supercharacters of discrete algebra groups
  12. Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition
https://doi.org/10.1515/forum-2021-0199
Keywords for this article
Degenerate elliptic operator; sectorial operator; holomorphic semigroup; contraction semigroup; consistent semigroup; core property

Articles in the same Issue

  1. Frontmatter
  2. Tameness of Margulis space-times with parabolics
  3. Sectoriality of degenerate elliptic operators via p-ellipticity
  4. Comparing Hecke eigenvalues of Siegel eigenforms
  5. Isomorphisms of Orlicz spaces
  6. The Engel graph of a finite group
  7. Nonnegative solutions of a fractional differential inequality on Grushin spaces and nilpotent Lie groups
  8. Representations of some associative pseudoalgebras
  9. On the convex components of a set in ℝ n
  10. Spectrality of homogeneous Moran measures on ℝ n
  11. Supercharacters of discrete algebra groups
  12. Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition
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