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

Bienvenido Barraza Martínez, Jonathan González Ospino, Rogelio Grau Acuña and Jairo Hernández Monzón
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Bienvenido Barraza Martínez: Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
Jonathan González Ospino: Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
Rogelio Grau Acuña: Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
Jairo Hernández Monzón: Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia

Mathematics, 2022, vol. 10, issue 5, 1-19

Abstract: We consider Fourier multiplier systems on R n with components belonging to the standard Hörmander class S 1 , 0 m R n , 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 W p k R n , C q with k ∈ N 0 , 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.

Keywords: C ? -semigroups; analytic semigroups; Fourier multipliers; ?-ellipticity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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