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Harmonic and Trace Inequalities in Lipschitz Domains

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Frontiers in Functional Equations and Analytic Inequalities

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Abstract

We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore–Penrose inverse, and of a special inner product. We show that our trace inequalities are particularly useful to prove harmonic inequalities, which serve as powerful tools to characterize the harmonic functions on Sobolev spaces of non-integer order.

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References

  1. R.A. Adams, J.J.F. Fournier, Sobolev Spaces (Elsevier/Academic Press, Amsterdam 2003)

    MATH  Google Scholar 

  2. E. Carlen, Trace inequalities and quantum entropy: an introductory course, in Entropy and the Quantum. Contemporary Mathematics, vol. 529 (American Mathematical Society, Providence, RI, 2010), pp. 73–140

    Google Scholar 

  3. C.D. Collins, J.L. Taylor, Eigenvalue convergence on perturbed Lipschitz domains for elliptic systems with mixed general decompositions of the boundary, J. Differ. Equ. 265(12), 6187–6209 (2018)

    Article  MathSciNet  Google Scholar 

  4. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)

    Article  MathSciNet  Google Scholar 

  5. X. Chen, R. Jiang, D. Yang, Hardy and Hardy-Sobolev spaces on strongly Lipschitz domains and some applications. Anal. Geom. Metr. Spaces 4, 336–362 (2016)

    MathSciNet  MATH  Google Scholar 

  6. B. Dacorogna, Introduction to the Calculus of Variations, 3rd edn. (Imperial College Press, London, 2015)

    MATH  Google Scholar 

  7. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Functional and Variational Methods, vol. 2 (Springer, Berlin, 1988)

    Google Scholar 

  8. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)

    Google Scholar 

  9. E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)

    MathSciNet  MATH  Google Scholar 

  10. V.K. Gupta, P. Sharma, Hypergeometric inequalities for certain unified classes of multivalent harmonic functions. Appl. Appl. Math. 13(1), 315–332 (2018)

    MathSciNet  MATH  Google Scholar 

  11. M. Hayajneh, S. Hayajneh, F. Kittaneh, On some classical trace inequalities and a new Hilbert-Schmidt norm inequality. Math. Inequal. Appl. 21(4), 1175–1183 (2018)

    MathSciNet  MATH  Google Scholar 

  12. M. Kohr, W.L. Wendland, Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds. Calc. Var. Partial Differ. Equ. 57(6), 165 (2018)

    Google Scholar 

  13. J.-Ph. Labrousse, Inverses généralisés d’opérateurs non bornés. Proc. Amer. Math. Soc. 115(1), 125–129 (1992)

    MathSciNet  MATH  Google Scholar 

  14. C.R. Loga, An extension theorem for matrix weighted Sobolev spaces on Lipschitz domains. Houston J. Math. 43(4), 1209–1233 (2017)

    MathSciNet  MATH  Google Scholar 

  15. G. Maze, U. Wagner, A note on the weighted harmonic-geometric-arithmetic means inequalities. Math. Inequal. Appl. 15(1), 15–26 (2012)

    MathSciNet  MATH  Google Scholar 

  16. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000)

    MATH  Google Scholar 

  17. J. Nečas, Les méthodes directes en théorie des équations elliptiques (Masson et Cie, Éditeurs, Paris, 1967)

    Google Scholar 

  18. S.M. Nikol’skiı̆, P.I. Lizorkin, Inequalities for harmonic, spherical and algebraic polynomials. Dokl. Akad. Nauk SSSR 289(3), 541–545 (1986)

    Google Scholar 

  19. M. Prats, Sobolev regularity of the Beurling transform on planar domains. Publ. Mat. 61(2), 291–336 (2017)

    Article  MathSciNet  Google Scholar 

  20. L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3 (Springer, Berlin, 2007)

    Google Scholar 

  21. S. Touhami, A. Chaira, D.F.M. Torres, Functional characterizations of trace spaces in Lipschitz domains. Banach J. Math. Anal. 13, 407–426 (2019)

    Article  MathSciNet  Google Scholar 

  22. B. Wei, W. Wang, Some inequalities for general L p-harmonic Blaschke bodies. J. Math. Inequal. 10(1), 63–73 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is part of the first author’s Ph.D. project, which is carried out at Moulay Ismail University, Meknes. It was essentially finished during a visit of Touhami to the Department of Mathematics of University of Aveiro, Portugal, November 2018. The hospitality of the host institution and the financial support of Moulay Ismail University, Morocco, and CIDMA, Portugal, are here gratefully acknowledged. Torres was partially supported by the Portuguese Foundation for Science and Technology (FCT) through CIDMA, project UID/MAT/04106/2019.

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Authors and Affiliations

  1. Moulay Ismail University, Faculté des Sciences, Laboratoire de Mathématiques et leures Applications, Equipe EDP et Calcul Scientifique, Meknes, Morocco

    Soumia Touhami & Abdellatif Chaira

  2. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Delfim F.  M. Torres

Authors
  1. Soumia Touhami
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  2. Abdellatif Chaira
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  3. Delfim F.  M. Torres
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Corresponding author

Correspondence to Delfim F.  M. Torres .

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Editors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA

    George A. Anastassiou

  2. Pedagogical Department E. E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, Athens, Greece

    John Michael Rassias

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Touhami, S., Chaira, A., Torres, D.F. . (2019). Harmonic and Trace Inequalities in Lipschitz Domains. In: Anastassiou, G., Rassias, J. (eds) Frontiers in Functional Equations and Analytic Inequalities. Springer, Cham. https://doi.org/10.1007/978-3-030-28950-8_30

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