Mathematics (Feb 2022)

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and <i>C</i><sup>∞</sup> Semigroups

  • Bienvenido Barraza Martínez,
  • Jonathan González Ospino,
  • Rogelio Grau Acuña,
  • Jairo Hernández Monzón

DOI
https://doi.org/10.3390/math10050751
Journal volume & issue
Vol. 10, no. 5
p. 751

Abstract

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We consider Fourier multiplier systems on Rn with components belonging to the standard Hörmander class S1,0mRn, 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 WpkRn,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.

Keywords