Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C∞ Semigroups

• Main Authors: Barraza Martínez, B. (Author), González Ospino, J. (Author), Grau Acuña, R. (Author), Hernández Monzón, J. (Author)
• Format: Article
• Language: English
• Published: MDPI 2022
• Subjects: analytic semigroups; C∞-semigroups; Fourier multipliers; Λ-ellipticity
• Online Access: View Fulltext in Publisher

 We consider Fourier multiplier systems on Rn with components belonging to the standard Hörmander class Sm1,0(Rn), but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector Λ ⊂ C (introduced by Denk, Saal, and Seiler) we show the generation of both C∞ semigroups and analytic semigroups (in a particular case) on the Sobolev spaces Wkp (Rn, Cq) with k ∈ N0, 1 ≤ p < ∞ and q ∈ N. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.