Abstract
We study bounded trace maps on hypersurfaces for Sobolev spaces from a point of view that is fundamentally different from the one in the classical theory. This allows us to construct bounded trace maps under weak regularity assumptions on the hypersurfaces. In the case of bounded domains in \(\mathbf{R}^n\) we only require the continuity of the boundary. For hypersurfaces in the whole space \(\mathbf{R}^n\) we only assume that the hypersurfaces are Lebesgue measurable. As an application of our trace maps we consider the Dirichlet problem and we prove a coarea formula where the level sets are only assumed to be Lebesgue measurable hypersurfaces.
In Memory of Erik Balslev
Research partially supported by project PAPIIT-DGAPA UNAM IN103918 and by project SEP-CONACYT CB 2015, 254062. R. Weder—Fellow, Sistema Nacional de Investigadores.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
R.A. Adams, J.J.F. Fournier, Sobolev Spaces (Elsevier Science, Oxford, 2003)
S. Agmon, L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics. J. Anal. Math. 30, 1–38 (1976)
Yu. Burago, N. Kosovsky, Boundary trace for BV functions in regions with irregular boundary, in Analysis, Partial Differential Equations and Applications, The Vladimir Maz’ya Anniversary Volume. Operator Theory, Advances and Applications (Birkhaüser, Basel, 2009), p. 114
H. Federer, Geometric Measure Theory (Springer, Berlin, 1969)
W.H. Fleming, R.W. Rishel, An integral formula for total gradient variation. Arch. Math. 11, 218–222 (1960)
D. Gilberg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 2001)
L. Hörmander, The Analysis of Linear Partial Differential Operators II Differential Operators with Constant Coefficients (Springer, Berlin, 2005)
D. Jerison, C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)
A. Jonsson, H. Wallin, Functions spaces on subsets of \( R^n\), in Mathematical Reports, vol. 2 (Harwood Academic, London, 1984)
T. Kato, Perturbation Theory of Linear Operators (Springer, Berlin, 1995)
B. Levi, Sul prinzipio di Dirichlet. Rend. Palermo 22, 293–359 (1906)
J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I (Springer, Berlin, 1972)
J. Malý, D. Swanson, W.P. Ziemer, The coarea formula for Sobolev mappings. Trans. Am. Math. Soc. 355, 477–492 (2003)
V.G. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations (Springer, Berlin, 2011)
V.G. Maz’ya, S.V. Podorchi, Differentiable Functions on Bad Domains (World Scientific, Singapore, 2011)
J. Něcas, Les Méthodes Directes en Théorie des Équations Elliptiques (Masson, Paris, Academie, Prague, 1967)
M. Reed, B. Simon, Methods of Modern Mathematical Physics II Fourier Analysis, Self-adjointness (Academic, San Diego, 1975)
P.A. Shvartsman, Sobolev \(W^1_p-\)spaces on closed subsets of \(R^n\). Adv. Math. 220, 1842–1922 (2009)
R. Weder, Spectral analysis of strongly propagative systems. J. Reine Angew. Math. (Crelles) 354, 95–122 (1984)
R. Weder, Analyticity of the scattering matrix for wave propagation in crystals. J. Math. Pures et Appl. 64, 121–148 (1985)
D.R. Yafaev, Mathematical Scattering Theory Analytic Theory (AMS, Providence, Rhode Island, 2000)
Acknowledgements
This paper was partially written while I was visiting INRIA Saclay Île-de-France and ENSTA. I thank Anne-Sophie Bonnet-BenDhia and Patrick Joly for their kind hospitality. I thank Vladimir Maz’ya for his detailed information on the literature on trace maps and on the coarea formula.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Weder, R. (2021). Trace Maps Under Weak Regularity Assumptions. In: Albeverio, S., Balslev, A., Weder, R. (eds) Schrödinger Operators, Spectral Analysis and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-030-68490-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-68490-7_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-68489-1
Online ISBN: 978-3-030-68490-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)