Maximal \(L_p\)\(L_q\) regularity for the quasi-steady elliptic problems


In this paper, we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal \(L_p\)\(L_q\) regularity for these problems and give examples that our results are applicable. The Lopatinskii–Shapiro and the asymptotic Lopatinskii–Shapiro conditions are important to get boundedness of solution operators.

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  1. 1.

    R. Denk, M. Hieber and J. Prüss, \(mathcal{R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166(788), viii+114 (2003)

  2. 2.

    R. Denk, M. Hieber and J. Prüss, Optimal \(L^p\)\(L^q\)-estimates for parabolic problems with inhomogeneous boundary data, Math. Z., 257(1),193–224 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    R. Denk, J. Prüss and R. Zacher, Maximal \(L_p\)-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal. 255(11), 3149–3187 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    G. Dore and A. Venni, \(H^\infty \) functional calculus for sectorial and bisectorial operators, Studia Math. 166, 221–241 (2005)

    MathSciNet  Article  Google Scholar 

  5. 5.

    J. Escher, J. Prüss and G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction, J. Reine Angew, Math. 563, 1–52 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    J. Escher and G. Simonett, Classical solutions to Hele–Shaw models with surface tension, Adv. Diff. Equ. 2, 619–642 (1997)

  7. 7.

    C. G. Gal, A Cahn–Hilliard model in bounded domains with permeable walls. Math. Methods Appl. Aci. 29, 2009–2036 (2006)

  8. 8.

    N. Kajiwara, Global well-posedness for a Cahn–Hilliard equation on bounded domains with permeable and non-permeable walls in maximal regularity spaces. Adv. Math. Sci. Appl. 27 (2), 277–298 (2018)

  9. 9.

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

  10. 10.

    P.C. Kunstmann and L. Weis, Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus, In Functional analytic methods for evolution equations, vol. 1855 of Lecture Notes in Math. Splinger, Berlin, 65–311 (2004)

  11. 11.

    J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, vol. 105, Birkhäuser/Splinger, Cham (2016)

    Google Scholar 

  12. 12.

    H. Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel (1983)

    Google Scholar 

  13. 13.

    L. Weis, Operator-valued Fourier multiplier theorems and maximal \(L_p\)-regularity, Math. Ann. 319(4), 735–758 (2001)

    MathSciNet  Article  Google Scholar 

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The first author was supported by the Program for Leading Graduate Schools, MEXT, Japan. Second author was supported by JSPS KAKENHI Grant No. 19K23408.

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Correspondence to Naoto Kajiwara.

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Furukawa, K., Kajiwara, N. Maximal \(L_p\)\(L_q\) regularity for the quasi-steady elliptic problems. J. Evol. Equ. (2020).

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  • Maximal \(L_p\)\(L_q\) regularity
  • Quasi-steady problems