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

Author

Listed:
  • Bienvenido Barraza Martínez

    (Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
    These authors contributed equally to this work.)

  • Jonathan González Ospino

    (Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
    These authors contributed equally to this work.)

  • Rogelio Grau Acuña

    (Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
    These authors contributed equally to this work.)

  • Jairo Hernández Monzón

    (Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Bienvenido Barraza Martínez & Jonathan González Ospino & Rogelio Grau Acuña & Jairo Hernández Monzón, 2022. "Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C ∞ Semigroups," Mathematics, MDPI, vol. 10(5), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:751-:d:759385
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    References listed on IDEAS

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    1. Raja, M. Mohan & Vijayakumar, V. & Udhayakumar, R., 2020. "A new approach on approximate controllability of fractional evolution inclusions of order 1 < r < 2 with infinite delay," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
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