Bounded \(H^\infty \)-calculus for a class of nonlocal operators: the bidomain operator in the \(L_q\)-setting

Abstract

The bidomain operator \({{\mathbb {A}}}\), a nonlocal operator, is studied in a bounded domain \(\Omega \subset {{\mathbb {R}}}^d\) with boundary \(\partial \Omega \) of class \(C^{2-}\) within the \(L_q\)-setting for \(1<q<\infty \). Assuming a fairly general framework, it is shown that this operator is sectorial, invertible on functions with mean zero, and admits a bounded \({H}^\infty \)-calculus with \({H}^\infty \)-angle 0. In particular, it has the property of maximal \(L_p-L_q\)-regularity and the fractional power domains \({{\mathcal {D}}}({{\mathbb {A}}}^\alpha )\) are identified. Furthermore, it is shown that \(-{{\mathbb {A}}}\) generates a strongly continuous, compact, analytic and exponentially stable semigroup on \(L_{q,0}(\Omega )\) with q-independent, purely countable point spectrum \(\sigma ({{\mathbb {A}}})\).

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References

  1. 1.

    Ambrosio, L., Colli-Franzone, P., Savaré, G.: On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model. Interfaces Free Bound 2, 213–266 (2000)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bourgault, Y., Coudière, Y., Pierre, C.: Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal. Real World Appl. 10, 458–482 (2009)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Colli-Franzone, P., Guerri, L., Tentoni, S.: Mathematical modeling of the excitation process in myocardium tissues: Influence of fiber rotation on wavefront progagation and potential field. Math. Biosci. 110, 155–235 (1990)

    Article  Google Scholar 

  4. 4.

    Colli-Franzone, P., Pavarino, L., Scacchi, S.: Mathematical Cardiac Electrophysiology. Springer, New York (2014)

    Google Scholar 

  5. 5.

    Colli-Franzone, P., Savaré, G.: Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level. In: Evolution equations, Semigroups and Functional Analysis. Program Nonlinear Differential Equations Application, vol. 50, Birkhäuser, Basel, pp. 49–78, (2000)

  6. 6.

    Denk, R., Dore, G., Hieber, M., Prüss, J., Venni, A.: New thoughts on old results of R.T. Seeley. Math. Ann. 328, 545–583 (2004)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Denk, R., Hieber, M., Prüss, J.: \({\cal{R}}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Memoirs of the American Mathematical Society 166, (2003)

  8. 8.

    Denk, R., Hieber, M., Prüss, J.: Optimal \(L^p\)-\(L^q\)-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 193–224 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Duong, X.T., Robinson, D.: Semigroup kernels, Poisson bounds, and holomorphis functional calculus. J. Funct. Anal. 142, 89–125 (1996)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Giga, Y., Kajiwara, N.: On a resolvent estimate for bidomain operators and its applications. J. Math. Anal. Appl. 459, 528–555 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hieber, M., Prüss, J.: Heat kernels and maximal \(L^p-L^q\)-estimates for parabolic evolution equations. Comm. Partial Diff. Equations 22, 1647–1669 (1997)

    Article  Google Scholar 

  13. 13.

    Hieber, M., Prüss, J.: On the bidomain problem with FitzHugh-Nagumo transport. Arch. Math. 111, 313–327 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hieber, M., Saal, J.: The Stokes equation in the \(L^q\)-setting: wellposedness and regularity properties. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 117-206, Springer, Cham, (2018)

  15. 15.

    Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Sapces, vol. II. Springer, New York (2017)

    Google Scholar 

  16. 16.

    Kalton, N., Weis, L.: The \(H^\infty \)-calculus and sums of closed operators. Math. Ann. 321, 319–345 (2001)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, second edition, Grundlehren der Mathematischen Wissenschaften, Vol. 132 (1976)

  18. 18.

    Keener, J., Sneyd, J.: Mathematical Physiology. Interdisciplinary Applied Mathematics. Springer, New York (1998)

    Google Scholar 

  19. 19.

    Kunstmann, P., Weis, L.: Maximal \(L^p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. In: Springer LNM 1855 (M. Ianelli, R. Nagel, S. Piazzera, eds.), pp. 65–311 (2004)

  20. 20.

    Kunstmann, P., Weis, L.: New criteria for the \(H^\infty \)-calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. 17, 387–409 (2017)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Mori, Y., Matano, H.: Stability of front solutions of the bidomain equation. Comm. Pure Appl. Math. 69, 2364–2426 (2016)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Pennacchio, M., Savaré, G., Colli-Franzone, P.: Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37, 1333–1370 (2005)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Prüss, J.: Maximal regularity for evolution equations in \(L_p\)-spaces. Conf. Semin. Mat. Univ. Bari 285, 1–39 (2003)

    Google Scholar 

  24. 24.

    Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted \(L_p\)-spaces. Arch. Math. 82, 415–431 (2004)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics 105, Birkhäuser, (2016)

  26. 26.

    Veneroni, M.: Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl 10, 849–868 (2009)

    MathSciNet  Article  Google Scholar 

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Correspondence to Matthias Hieber.

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Hieber, M., Prüss, J. Bounded \(H^\infty \)-calculus for a class of nonlocal operators: the bidomain operator in the \(L_q\)-setting. Math. Ann. 378, 1095–1127 (2020). https://doi.org/10.1007/s00208-019-01916-2

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Mathematics Subject Classification

  • 35K50
  • 42B20
  • 92C35