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

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

doi:10.3390/math10050751

Abstract

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.

Highlights

Elliptic systems of partial differential equations were introduced in 1955 by A
They introduced the formulation of parameter–ellipticity with respect to a subsector Λ ⊂ C, which is motivated by a notion of parameter–ellipticity introduced by Denk, Menniken, and Volevich in [5] and connected with the so-called Newton polygon associated with the system
They showed that their formulation of ellipticity is equivalent to the given by Koževnikov in [3] and that this condition implies the existence of a bounded H∞-calculus for their pseudodifferential systems in suitable scales of Sobolev spaces with 1 < p < ∞, of Lp-maximal regularity

Summary

Introduction

Elliptic systems of partial differential equations were introduced in 1955 by A. We will consider certain Fourier multiplier systems on Rn, similar but not necessarily with the exact structure of a Douglis–Nirenberg system, with components belonging to the standard Hörmander class S1m,0(Rn), but with limited regularity (see Definition 2), and using the notion of parameter–ellipticity with respect to a subsector Λ ⊂ C given in [4], we will establish (in Theorem 1) the generation of C∞ semigroups and analytic semigroups (in a particular case) on the Sobolev spaces Wpk(Rn, Cq) with k ∈ N0 and 1 ≤ p < ∞ giving a direct proof For this direct proof of our main result we use the approach based on oscillatory integrals and kernel estimates for them (as in [6]), taking advantage of the fact that the associated symbols to the pseudodifferential operators are matrices valued and the entries of these matrices are symbols of order greater than 1/2 and are independent of the spatial variable.

Fourier Multiplier Systems

Λ-Elliptic Fourier Multipliers Systems

Examples

Conclusions