A note on properties of the restriction operator on Sobolev spaces

• Autorzy: Hewett D. P. , Moiola A.

Identyfikatory

DOI

10.1515/jaa-2017-0001

• Języki publikacji: EN

Abstrakty

EN

In our companion paper [3] we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset Ω of Rⁿ, defined as closed subspaces of the classical Bessel potential spaces H^s(Rⁿ) for s ϵ R. These spaces are mapped by the restriction operator to certain spaces of distributions on Ω. In this note we make some observations about the relation between these spaces of global and local distributions. In particular, we study conditions under which the restriction operator is or is not injective, surjective and isometric between given pairs of spaces.We also provide an explicit formula for minimal norm extension (an inverse of the restriction operator in appropriate spaces) in a special case.

Słowa kluczowe

EN

Bessel potential Sobolev spaces; non-Lipschitz domains; restriction operator; s-nullity; unitary realisations of dual spaces; minimal norm extension

PL

przestrzenie Sobolewa; operator zwężający; przestrzenie dualne

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2017
• Tom: Vol. 23, nr 1
• Strony: 1--8
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Hewett D. P.

• Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

autor

Moiola A.

• Department of Mathematics and Statistics, University of Reading, Whiteknights PO Box 220, Reading RG6 6AX, United Kingdom

Bibliografia

• [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
• [2] S. N. Chandler-Wilde, D. P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: Quantitative estimates and counterexamples, Mathematika 61 (2015), 414-443.
• [3] S. N. Chandler-Wilde, D. P. Hewett and A. Moiola, Sobolev spaces on non-Lipschitz subsets of Rn with application to boundary integral equations on fractal screens, Integral Equations and Operator Theory 87 (2017), no. 2, 179-224.
• [4] L. Grafakos, Classical Fourier Analysis, Springer, New York, 2008.
• [5] D. P. Hewett and A. Moiola, On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space, Anal. Appl. (2016), DOI 10.1142/ S021953051650024X.
• [6] T. Kato, Perturbation Theory for Linear Operators. Reprint of the Corr. Print. of the 2nd ed. 1980, Springer, Berlin, 1995.
• [7] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. I, Springer, Berlin, 1972.
• [8] V. G. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd ed., Springer, Berlin, 2011.
• [9] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).