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Traces and embeddings of anisotropic function spaces

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Abstract

In this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities. To a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in Sobolev and mixed derivative embeddings with fixed integrability. We apply the general results to prove maximal \(L^p\)-\(L^q\)-regularity for the linearized, fully inhomogeneous two-phase Stefan problem with Gibbs–Thomson correction.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)

    MATH  Google Scholar 

  2. Amann, H.: Linear and Quasilinear Parabolic Problems. Abstract Linear Theory, vol. 1. Birkhäuser, Boston (1995)

    Book  Google Scholar 

  3. Amann, H.: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186, 5–56 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amann, H.: Nonlocal quasilinear parabolic equations. Uspekhi Mat. Nauk. 60(6(366)), 21–32 (2005)

    Article  MathSciNet  Google Scholar 

  5. Amann, H.: Anisotropic function spaces and maximal regularity for parabolic problems. Part 1: Function spaces. In: Jindrich Necas Center for Mathematical Modeling Lecture Notes, vol. 6, Prague (2009)

  6. Berkolajko, M.Z.: On traces of generalized spaces of differentiable functions with mixed norm. Sov. Math. Dokl. 30, 60–64 (1984)

    MATH  Google Scholar 

  7. Cannone, M.: Harmonic analysis tools for solving the incompressible Navier–Stokes equations. In: Handbook of Mathematical Fluid Dynamics, Vol. III, pp. 161–244. North-Holland, Amsterdam (2004)

  8. Clément, Ph., Li, S.: Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3(special issue), 17–32 (1993/94)

  9. Denk, R., Hieber, M., Prüss, J.: \(R\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788) (2003)

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

  11. Denk, R., Kaip, M.: Operator Theory: Advances and Applications. General parabolic mixed order systems in \(L_p\) and applications, vol. 239. Birkhäuser, Basel (2013)

    Google Scholar 

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

  13. Denk, R., Saal, J., Seiler, J.: Inhomogeneous symbols, the Newton polygon, and maximal \(L^p\)-regularity. Russ. J. Math. Phys. 15(2), 171–191 (2008)

  14. Di Blasio, G.: Linear parabolic evolution equations in \(L^p\)-spaces. Ann. Math. Pura Appl. IV. Ser. 138, 55–104 (1984)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)

  17. Grafakos, L.: Graduate Texts in Mathematics. Modern Fourier analysis, vol. 250, 2nd edn. Springer, New York (2009)

    Google Scholar 

  18. Haase, M.H.A.: Operator Theory: Advances and Applications. The functional calculus for sectorial operators, vol. 169. Birkhäuser Verlag, Basel (2006)

    Google Scholar 

  19. Han, Y.-S., Meyer, Y.: A characterization of Hilbert spaces and the vector-valued Littlewood–Paley theorem. Methods Appl. Anal. 3(2), 228–234 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. University Press, Cambridge (1952)

    MATH  Google Scholar 

  21. Johnsen, J., Sickel, W.: On the trace problem for lizorkin-triebel spaces with mixed norms. Math. Nachr. 281(5), 669–696 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kaip, M.: General parabolic mixed order systems in \(L_p\) and applications. PhD thesis, University of Constance, Konstanz (2012)

  23. Köhne, M., Prüss, J., Wilke, M.: On quasilinear parabolic evolution equations in weighted \(L_p\)-spaces. J. Evol. Equ. 10(2), 443–463 (2010)

  24. Kunstmann, P.C., Weis, L.W.: Maximal \(L^p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^{\infty }\)-functional calculus. Functional analytic methods for evolution equations. In: Lecture Notes in Mathematics, Vol. 1855, pp. 65–311. Springer, Berlin (2004)

  25. Lunardi, A.: Interpolation Theory, 2nd edn. Scuola Normale Superiore di Pisa, Pisa (2009)

    MATH  Google Scholar 

  26. McConnell, T.R.: On Fourier multiplier transformations of Banach-valued functions. Trans. Am. Math. Soc. 285(2), 739–757 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meyries, M., Schnaubelt, R.: Interpolation, embeddings and traces for anisotropic fractional Sobolev spaces with temporal weights. J. Funct. Anal. 262, 1200–1229 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Meyries, M., Schnaubelt, R.: Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions. Math. Nachr. 285, 1032–1051 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Meyries, M., Veraar, M.C.: Sharp embedding results for spaces of smooth functions with power weights. Stud. Math. 208(3), 257–293 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Stochastic maximal \(L^{p}\)-regularity. Ann. Probab. 40(2), 788–812 (2012)

  31. Prüss, J.: Maximal regularity for evolution equations in \(L_p\)-spaces. Conf. Semin. Mat. Univ. Bari 2002(285), 1–39 (2003)

  32. Prüss, J., Shimizu, S., Shibata, Y., Simonett, G.: On well-posedness of incompressible two-phase flows with phase transitions: the case of equal densities. Evol. Equ. Control Theory 1(1), 171–194 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted \(L_p\)-spaces. Arch. Math. 82(5), 415–431 (2004)

  34. Schmeisser, H.-J., Sickel, W.: Traces, Gagliardo–Nirenberg inequalities and Sobolev type embeddings for vector-valued function spaces. Jenaer Schriften zur Mathematik und Informatik (2001)

  35. Seeley, R.: Interpolation in \(L^{p}\) with boundary conditions. Stud. Math. 44, 47–60 (1972)

  36. Shimizu, S.: Local solvability of free boundary problems for the two-phase Navier–Stokes equations with surface tension in the whole space. Progr. Nonlinear Differ. Equ. Appl. 80, 647–686 (2011)

    Google Scholar 

  37. Sobolevskii, P.E.: Fractional powers of coercively positive sums of operators. Sov. Math. Dokl. 16, 1638–1641 (1975)

    MATH  Google Scholar 

  38. Stein, E.M.: Princeton Mathematical Series, Monographs in Harmonic Analysis, III. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, vol. 43. Princeton University Press, Princeton (1993)

    Google Scholar 

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

    Google Scholar 

  40. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)

    MATH  Google Scholar 

  41. Triebel, H.: Monographs in Mathematics. Theory of function spaces. III, vol. 100. Birkhäuser Verlag, Basel (2006)

    Google Scholar 

  42. Turesson, B.O.: Lecture Notes in Mathematics. Nonlinear potential theory and weighted Sobolev spaces, vol. 1736. Springer-Verlag, Berlin (2000)

    Google Scholar 

  43. Weidemaier, P.: Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed \(L_p\)-norm. Electron. Res. Announc. Am. Math. Soc. 8, 47–51 (2002)

  44. Weidemaier, P.: Lizorkin–Triebel spaces of vector-valued functions and Sharp trace theory for functions in Sobolev spaces with a mixed \(L_p\)-norm in parabolic problems. Mat. Sb. 196(6), 3–16 (2005)

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

  46. Zacher, R.: Quasilinear parabolic problems with nonlinear boundary conditions. PhD thesis. Martin-Luther Universität Halle-Wittenberg, Halle (2003)

  47. Zacher, R.: Maximal regularity of type \(L_p\) for abstract parabolic Volterra equations. J. Evol. Equ. 5(1), 79–103 (2005)

  48. Zimmermann, F.: On vector-valued Fourier multiplier theorems. Stud. Math. 93(3), 201–222 (1989)

    MATH  Google Scholar 

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Acknowledgments

The authors thank the anonymous referees for helpful suggestions which lead to improvements of the results and the presentation of the paper.

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Author notes

  1. Martin Meyries

    Present address: Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 , Halle (Saale), Germany

Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76128 , Karlsruhe, Germany

    Martin Meyries

  2. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA , Delft, The Netherlands

    Mark C. Veraar

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  1. Martin Meyries
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Correspondence to Martin Meyries.

Additional information

M. Meyries was supported by Deutsche Forschungsgemeinschaft (DFG), project ME 3848/1-1. M. C. Veraar was supported by a VENI subsidy 639.031.930 of the Netherlands Organisation for Scientific Research (NWO)

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Meyries, M., Veraar, M.C. Traces and embeddings of anisotropic function spaces. Math. Ann. 360, 571–606 (2014). https://doi.org/10.1007/s00208-014-1042-6

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