Imbedding theorems for Sobolev spaces on domains with peak and on Hölder domains

Maz′ya, V.; Poborchi, S.

doi:10.1090/S1061-0022-07-00962-4

Imbedding theorems for Sobolev spaces on domains with peak and on Hölder domains
HTML articles powered by AMS MathViewer

by V. G. Maz′ya and S. V. Poborchi
Translated by: S. V. Poborchi

St. Petersburg Math. J. 18 (2007), 583-605

DOI: https://doi.org/10.1090/S1061-0022-07-00962-4

Published electronically: May 29, 2007

PDF | Request permission

Abstract:

Necessary and sufficient conditions are obtained for the continuity and compactness of the imbedding operators $W_p^l(\Omega )\to L_q(\Omega )$ and $W_p^l(\Omega )\to C(\Omega )\cap L_\infty (\Omega )$ for a domain with an outward peak. More simple sufficient conditions are presented. Applications to the solvability of the Neumann problem for elliptic equations of order $2l$, $l\ge 1$, for a domain with peak are given. An imbedding theorem for Sobolev spaces on Hölder domains is stated.

References

S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. 4 (1938), no. 3, 471–497. (Russian)
S. L. Sobolev, Nekotorye primeneniya funkcional′nogo analiza v matematičeskoĭ fizike, Izdat. Leningrad. Gos. Univ., Leningrad, 1950 (Russian). MR 0052039
Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137 (Italian). MR 102740
O. V. Besov, Integral representations of functions and embedding theorems for a domain with a flexible horn condition, Trudy Mat. Inst. Steklov. 170 (1984), 12–30, 274 (Russian). Studies in the theory of differentiable functions of several variables and its applications, X. MR 790325
O. V. Besov, Sobolev’s embedding theorem for a domain with an irregular boundary, Mat. Sb. 192 (2001), no. 3, 3–26 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 3-4, 323–346. MR 1836304, DOI 10.1070/sm2001v192n03ABEH000548
Yu. G. Reshetnyak, Integral representations of differentiable functions in domains with a nonsmooth boundary, Sibirsk. Mat. Zh. 21 (1980), no. 6, 108–116, 221 (Russian). MR 601195
B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 52–68. MR 982072, DOI 10.1007/BFb0081242
S. Buckley and P. Koskela, Sobolev-Poincaré implies John, Math. Res. Lett. 2 (1995), no. 5, 577–593. MR 1359964, DOI 10.4310/MRL.1995.v2.n5.a5
Piotr Hajłasz and Pekka Koskela, Isoperimetric inequalities and imbedding theorems in irregular domains, J. London Math. Soc. (2) 58 (1998), no. 2, 425–450. MR 1668136, DOI 10.1112/S0024610798006346
T. Kilpeläinen and J. Malý, Sobolev inequalities on sets with irregular boundaries, Z. Anal. Anwendungen 19 (2000), no. 2, 369–380. MR 1768998, DOI 10.4171/ZAA/956
S. V. Poborchii, Some counterexamples to the embedding theorems for Sobolev spaces, Vestnik St. Petersburg Univ. Math. 31 (1998), no. 4, 46–55 (2000). MR 1794641
V. G. Maz′ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882–885. MR 0126152
V. G. Maz′ja, The continuity and boundedness of functions in S. L. Sobolev spaces, Problems of mathematical analysis, No. 4: Integral and differential operators. Differential equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 46–77, 143 (Russian). MR 0348481
V. G. Maz′ja, The summability of functions belonging to Sobolev spaces, Problems of mathematical analysis, No. 5: Linear and nonlinear differential equations, Differential operators (Russian), Izdat. Leningrad. Univ., Leningrad, 1975, pp. 66–98 (Russian). MR 0511931
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
V. G. Maz′ya and S. V. Poborchiĭ, Extension of functions in S. L. Sobolev classes to the exterior of a domain with the vertex of a peak on the boundary. II, Czechoslovak Math. J. 37(112) (1987), no. 1, 128–150 (Russian). MR 875135
—, Imbedding theorems for Sobolev spaces in domains with cusps, Linköping Univ., 1992, 34 pp. (Preprint/ LiTH–MAT–R–92–14).
Vladimir G. Maz′ya and Sergei V. Poborchi, Differentiable functions on bad domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR 1643072
I. G. Globenko, Some questions in the theory of imbedding for domains with singularities on the boundary, Mat. Sb. (N.S.) 57 (1962), 201–224 (Russian). MR 0143022
D. A. Labutin, Integral representation of functions and the embedding of Sobolev spaces on domains with zero angles, Mat. Zametki 61 (1997), no. 2, 201–219 (Russian, with Russian summary); English transl., Math. Notes 61 (1997), no. 1-2, 164–179. MR 1619998, DOI 10.1007/BF02355726
D. A. Labutin, Embedding of Sobolev spaces on Hölder domains, Tr. Mat. Inst. Steklova 227 (1999), no. Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18, 170–179 (Russian); English transl., Proc. Steklov Inst. Math. 4(227) (1999), 163–172. MR 1784315
D. A. Labutin, The unimprovability of Sobolev inequalities for a class of irregular domains, Tr. Mat. Inst. Steklova 232 (2001), no. Funkts. Prostran., Garmon. Anal., Differ. Uravn., 218–222 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(232) (2001), 211–215. MR 1851450
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Masatoshi Fukushima and Matsuyo Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps, Probab. Theory Related Fields 106 (1996), no. 4, 521–557. MR 1421991, DOI 10.1007/s004400050074
L. E. Fraenkel, Formulae for high derivatives of composite functions, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 2, 159–165. MR 486377, DOI 10.1017/S0305004100054402
V. D. Stepanov, Two-weight estimates for Riemann-Liouville integrals, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 645–656 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 669–681. MR 1072699, DOI 10.1070/IM1991v036n03ABEH002039
S. V. Poborchii, On solvability of the Neumann problem for elliptic equations of high order, Vestnik St. Petersburg Univ. Math. 31 (1998), no. 3, 57–60 (1999). MR 1794685