The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions. (English) Zbl 1164.46322

Summary: We present a detailed proof of the density of the set \(C^\infty (\overline \Omega )\cap V\) in the space of test functions \(V\subset H^1(\Omega )\) that vanish on some part of the boundary  \(\partial \Omega \) of a bounded domain  \(\Omega \).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] R. A. Adams: Sobolev Spaces. Academic Press, New York-San Francisco-London, 1975.
[2] O.V. Besov: On some families of functional spaces. Imbedding and continuation theorems. Doklad. Akad. Nauk SSSR 126 (1959), 1163–1165. (In Russian.) · Zbl 0097.09701
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.
[4] P. Doktor: On the density of smooth functions in certain subspaces of Sobolev space. Commentat. Math. Univ. Carol. 14 (1973), 609–622. · Zbl 0268.46036
[5] A. Kufner, O. John, and S. Fučík: Function Spaces. Academia, Praha, 1977.
[6] P. I. Lizorkin: Boundary properties of functions from ”weight” classes. Sov. Math. Dokl. 1 (1960), 589–593; transl. from Dokl. Akad. Nauk SSSR 132 (1960), 514–517. (In Russian.) · Zbl 0106.30802
[7] J. Nečas: Les mèthodes directes en thèorie des èquations elliptiques. Academia, Praha, 1967.
[8] V. I. Smirnov: A Course in Higher Mathematics V. Gosudarstvennoje izdatelstvo fiziko-matematičeskoj literatury, Moskva, 1960. (In Russian.)
[9] S.V. Uspenskij: An imbedding theorem for S. L. Sobolev’s classes W p r of fractional order. Sov. Math. Dokl. 1 (1960), 132–133; traslation from Dokl. Akad. Nauk SSSR 130 (1960), 992–993. · Zbl 0103.08205
[10] A. Ženíšek: Sobolev Spaces and Their Applications in the Finite Element Method. VUTIUM, Brno, 2005; see also A. Ženíšek: Sobolev Spaces. VUTIUM, Brno, 2001. (In Czech.)
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