Banach Journal of Mathematical Analysis

Functional characterizations of trace spaces in Lipschitz domains

Soumia Touhami, Abdellatif Chaira, and Delfim F. M. Torres

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces Hs(Ω) involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary Ω of a bounded Lipschitz domain ΩRd and applying Moore–Penrose pseudoinverse properties together with a special inner product on H1(Ω). We also establish generalized results of the Moore–Penrose pseudoinverse.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 2 (2019), 407-426.

Dates
Received: 21 August 2018
Accepted: 30 November 2018
First available in Project Euclid: 13 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1550048427

Digital Object Identifier
doi:10.1215/17358787-2018-0044

Mathematical Reviews number (MathSciNet)
MR3927880

Zentralblatt MATH identifier
07045465

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

Keywords
Lipschitz domains trace spaces trace operators Moore–Penrose pseudoinverse

Citation

Touhami, Soumia; Chaira, Abdellatif; Torres, Delfim F. M. Functional characterizations of trace spaces in Lipschitz domains. Banach J. Math. Anal. 13 (2019), no. 2, 407--426. doi:10.1215/17358787-2018-0044. https://projecteuclid.org/euclid.bjma/1550048427


Export citation

References

  • [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amsterdam) 140, Elsevier, Amsterdam, 2003.
  • [2] G. Auchmuty, Spectral characterization of the trace spaces $H^{s}(\partial \Omega)$, SIAM J. Math. Anal. 38 (2006), no. 3, 894–905.
  • [3] G. Auchmuty, The S.V.D. of the Poisson kernel, J. Fourier Anal. Appl. 23 (2017), no. 6, 1517–1536.
  • [4] G. Auchmuty, Steklov representations of Green’s functions for Laplacian boundary value problems, Appl. Math. Optim. 77 (2018), no. 1, 173–195.
  • [5] M. Bakonyi and H. J. Woerdeman, On the strong Parrott completion problem, Proc. Amer. Math. Soc. 117 (1993), no. 2, 429–433.
  • [6] J. Behrndt and T. Micheler, Elliptic differential operators on Lipschitz domains and abstract boundary value problems, J. Funct. Anal. 267 (2014), no. 10, 3657–3709.
  • [7] J. Cao and Y. Xue, Perturbation analysis of the Moore-Penrose metric generalized inverse with applications, Banach J. Math. Anal. 12 (2018), no. 3, 709–729.
  • [8] A. P. Choudhury, A. Hussein, and P. Tolksdorf, Nematic liquid crystals in Lipschitz domains, SIAM J. Math. Anal. 50 (2018), no. 4, 4282–4310.
  • [9] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626.
  • [10] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, II: Functional and Variational Methods, Springer, Berlin, 1988.
  • [11] R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.
  • [12] C. Foias and A. Tannenbaum, A strong Parrott theorem, Proc. Amer. Math. Soc. 106 (1989), no. 3, 777–784.
  • [13] E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili, Rend. Semin. Mat. Univ. Padova 27 (1957), 284–305.
  • [14] G. Geymonat, Trace theorems for Sobolev spaces on Lipschitz domains: Necessary conditions, Ann. Math. Blaise Pascal 14 (2007), no. 2, 187–197.
  • [15] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985.
  • [16] C. W. Groetsch, Inclusions and identities for the Moore-Penrose inverse of a closed linear operator, Math. Nachr. 171 (1995), 157–164.
  • [17] E. Ko, On operators with similar positive parts, J. Math. Anal. Appl. 463 (2018), no. 1, 276–293.
  • [18] J.-Ph. Labrousse, Inverses généralisés d’opérateurs non bornés, Proc. Amer. Math. Soc. 115 (1992), no. 1, 125–129.
  • [19] J.-Ph. Labrousse and M. Mbekhta, Les opérateurs points de continuité pour la conorme et l’inverse de Moore-Penrose, Houston J. Math. 18 (1992), no. 1, 7–23.
  • [20] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, I, Trav. Rech. Math. 17, Dunod, Paris, 1968.
  • [21] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge, 2000.
  • [22] L. Molnár, Busch-Gudder metric on the cone of positive semidefinite operators and its isometries, Integral Equations Operator Theory 90 (2018), no. 2, art. ID 20.
  • [23] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris, 1967.
  • [24] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007.
  • [25] M. Warma, On a fractional $(s,p)$-Dirichlet-to-Neumann operator on bounded Lipschitz domains, J. Elliptic Parabol. Equ. 4 (2018), no. 1, 223–269.
  • [26] Z.-Q. Xiang, Canonical dual $K$-g-Bessel sequences and $K$-g-frame sequences, Results Math. 73 (2018), no. 1, art. ID 17.
  • [27] A. Yamada, Parrott’s theorem and bounded solutions of a system of operator equations, Complex Anal. Oper. Theory 11 (2017), no. 4, 961–976.