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Solvability of a fourth order elliptic problem in a bounded sector, part II

Rabah Labbas, Stéphane Maingot, Alexandre Thorel

Abstract

After different variables and functions changes, the generalized dispersal problem, recalled in (1) below and considered in part I, see Labbas, Maingot and Thorel [14], leads us to consider, to study and to invert the sum of linear operators (4) below in a suitable Banach space by using two strategies: namely the theory of sums of operators in Banach spaces as developed by Da Prato-Grisvard [4] and successfully improved by Dore-Venni [5].

Solvability of a fourth order elliptic problem in a bounded sector, part II

Abstract

After different variables and functions changes, the generalized dispersal problem, recalled in (1) below and considered in part I, see Labbas, Maingot and Thorel [14], leads us to consider, to study and to invert the sum of linear operators (4) below in a suitable Banach space by using two strategies: namely the theory of sums of operators in Banach spaces as developed by Da Prato-Grisvard [4] and successfully improved by Dore-Venni [5].
Paper Structure (14 sections, 16 theorems, 229 equations, 1 figure)

This paper contains 14 sections, 16 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Definitions and prerequisites
  3. The class of Bounded Imaginary Powers of operators
  4. Recall on the sum of linear operators
  5. Spectral study of operators
  6. Study of operator $\mathcal{L}_{1,\mu}$
  7. Study of operator $\mathcal{L}_2$
  8. Case $\lambda = 0$ : Invertibility of $\mathcal{A}$
  9. Case $\lambda < 0$ : Spectral study of $\mathcal{A}$
  10. Study of the sum $\mathcal{L}_{1,\mu}+\mathcal{L}_2$
  11. Invertibility of the closure of the sum
  12. Convexity inequalities
  13. Back to the abstract problem
  14. Proof of Theorem

Key Result

Theorem 1.1

Let $\mathcal{F} \in L^p(0,+\infty;X)$ and assume that Then, there exists $\rho_0 > 0$ such that for all $\rho \in (0,\rho_0]$, the abstract equation has a unique classical solution $V \in L^p(0,+\infty;X)$, that is In particular, $\mathcal{L}_{1,\mu} + \mathcal{L}_2$ is closed and $V \in D(\mathcal{L}_{1,\mu} + \mathcal{L}_2)$.

Figures (1)

  • Figure 1: In this figure, $\Pi_\mu$ is the entire uncolored area.

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6: Da Prato and Grisvard daprato-grisvard, Grisvard grisvard 2
  • Corollary 2.7
  • proof
  • Proposition 3.1
  • ...and 28 more