| Title:
|
Extension theory for Sobolev spaces on open sets with Lipschitz boundaries (English) |
| Author:
|
Burenkov, Viktor I. |
| Language:
|
English |
| Journal:
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Nonlinear Analysis, Function Spaces and Applications |
| Volume:
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Vol. 6 |
| Issue:
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1998 |
| Year:
|
|
| Pages:
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1-49 |
| . |
| Category:
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math |
| . |
| MSC:
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46E35 |
| MSC:
|
47A30 |
| idZBL:
|
Zbl 0964.46014 |
| idMR:
|
MR1777711 |
| . |
| Date available:
|
2009-10-08T09:46:49Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702471 |
| . |
| Reference:
|
[1] Adams R. A.: Sobolev spaces.Academic Press, New York 1975. Zbl 0314.46030, MR 0450957 |
| Reference:
|
[2] Babich V. M.: On the extension of functions.(Russian). Uspekhi Mat. Nauk 8 (1953), 111–113. MR 0056675 |
| Reference:
|
[3] Besov O. V., in V. P. Il,’ Kudryavtsev L. D., Lizorkin P. I., skii S. M. Nikol,’ : Embedding theory for classes of differentiable functions of several variables.(Russian). Proc. Sympos. in honour of the 60th birthday of academician S. L. Sobolev, Inst. Mat. Sibirsk. Otdel. Akad. Nauk. SSSR, Nauka, Moscow 1970, 38–63. |
| Reference:
|
[4] Besov O. V., in V. P. Il,’ skii S. M. Nikol,’ : Integral representation of functions and embedding theorems.(Russian). 1st ed., Nauka, Moscow 1975; 2nd ed., Nauka, Moscow 1996 (Russian); English transl. of 1st ed., Vols. 1, 2, Wiley, 1979. MR 0430771 |
| Reference:
|
[5] Burago, Yu. D., ya V. G. Maz,’ : Some problems of the potential theory and function theory for domains with irregular boundaries.Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov 3 (1967), 1–152 (Russia); English transl.: Seminars in Math., V. A. Steklov Math. Inst., Leningrad 3 (1969). |
| Reference:
|
[6] Burenkov V. I.: Some properties of classes of differentiable functions in connection with embedding and extension theorems.(Russian). Ph.D. thesis, Moscow, Steklov Math. Inst. (1966), 145 pp. |
| Reference:
|
[7] Burenkov V. I.: Embedding and extension theorems for classes of differentiable functions of several variables defined on the whole space.(Russian). Itogi Nauki i Tekhniki: Mat. Anal. 1965, VINITI, Moscow 1966, 71–155; English transl.: Progress in Math. 2, Plenum Press, 1968. MR 0206698 |
| Reference:
|
[8] Burenkov V. I.: On regularized distance.(Russian). Trudy MIREA. Issue 67. Mathematics (1973), 113–117. MR 0499023 |
| Reference:
|
[9] Burenkov V. I.: On the density of infinitely differentiable functions in Sobolev spaces for an arbitrary open set.(Russian). Trudy Mat. Inst. Steklov 131 (1974), 39–50; English transl.: Proc. Steklov Inst. Math. 131 (1974). Zbl 0313.46033, MR 0367642 |
| Reference:
|
[10] Burenkov V. I.: On the extension of functions with preservation and with deterioration of the differential properties.(Russian). Dokl. Akad. Nauk SSSR 224 (1975), 269–272; English transl.: Soviet Math. Dokl. 16 (1975). Zbl 0351.46022, MR 0405083 |
| Reference:
|
[11] Burenkov V. I.: On a certain method for extending differentiable functions.(Russian). Trudy Mat. Inst. Steklov 140 (1976), 27–67; English transl.: Proc. Steklov Inst. Math. 140 (1976). |
| Reference:
|
[12] Burenkov V. I.: On the extension of functions with preservation of semi-norm.(Russian). Dokl. Akad. Nauk SSSR 228 (1976), 779–782; English transl.: Soviet Math. Dokl. 17 (1976). MR 0412796 |
| Reference:
|
[13] Burenkov V. I.: On partitions of unity.(Russian). Trudy Mat. Inst. Steklov 150 (1979), 24–30; English transl.: Proc. Steklov Inst. Math. 150 (1979). Zbl 0417.46036, MR 0544002 |
| Reference:
|
[14] Burenkov V. I.: Investigation of spaces of differentiable functions with irregular domain.(Russian). D.Sc. thesis, Steklov Math. Inst., Moscow 1982, 312 pp. |
| Reference:
|
[15] Burenkov V. I.: On estimates of the norms of extension operators.(Russian). 9-th All-Union school on the theory of operators in function spaces. Ternopol. Abstracts (1984), 19–20. |
| Reference:
|
[16] Burenkov V. I.: Extension of functions with preservation of Sobolev semi-norm.(Russian). Trudy Mat. Inst. Steklov 172 (1985), 81–95; English transl.: Proc. Steklov Inst. Math. 172 (1985). MR 0810420 |
| Reference:
|
[17] Burenkov V. I.: Extension theorems for Sobolev spaces.In: Proc. of the conference “Functional analysis, partial differential equations and applications” in honour of V. Maz’ya, held in Rostock, 31. 08.–4. 09. 1998 (to appear). MR 1747873 |
| Reference:
|
[18] Burenkov V. I.: On sharp constants in the inequalities for the norms of intermediate derivatives on a finite interval, II.(Russian). Trudy Mat. Inst. Steklov 173 (1986), 38–49; English transl.: Proc. Steklov Inst. Math. 173 (1986). MR 0864833 |
| Reference:
|
[19] Burenkov V. I.: Sobolev spaces on domains.B.G. Teubner, Stuttgart 1998. Zbl 0893.46024, MR 1622690 |
| Reference:
|
[20] Burenkov V. I.: Compactness of embeddings for Sobolev and more general spaces and extensions with preservation of some smoothness.To appear. |
| Reference:
|
[21] Burenkov V. I., Fain B. L.: On the extension of functions in Sobolev spaces from a strip with deteriorations of class.(Russian). Deposited in VINITY Ac. Sci. USSR, No 2511–74 (1975), 12 pp. |
| Reference:
|
[22] Burenkov V. I., Fain B. L.: On the extension of functions in anisotropic spaces with preservation of class.(Russian). Dokl. Akad. Nauk SSSR 228 (1976), 525–528; English transl.: Soviet Math. Dokl. 17 (1976). MR 0410359 |
| Reference:
|
[23] Burenkov V. I., Fain B. L.: On the extension of functions in anisotropic classes with preservation of the class.(Russian). Trudy Mat. Inst. Steklov 150 (1979), 52–66; English transl.: Proc. Steklov Inst. Math. 150 (1979). MR 0544004 |
| Reference:
|
[24] Burenkov V. I., dman M. L. Gol,’ 1979, On the extension of functions in $L_p$ (Russian). Trudy Mat. Inst. Steklov : 150.(,))., 31–51; English transl.: Proc. Steklov Inst. Math. 150 (197,9 MR 0544003 |
| Reference:
|
[25] Burenkov V. I., Gorbunov A. L.: Sharp estimates for the minimal norm of an extension operator for Sobolev spaces.(Russian). Dokl. Akad. Nauk SSSR 330 (1993), 680–682; English transl.: Soviet Math. Dokl. 47 (1993). MR 1242163 |
| Reference:
|
[26] Burenkov V. I., Gorbunov A. L.: Sharp estimates for the minimal norm of an extension operator for Sobolev spaces.(Russian). Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), 1–44. MR 1440311 |
| Reference:
|
[27] Burenkov V. I., Kalyabin G. A.: Lower estimates of the norms of extension operators for Sobolev spaces on the halfline.To appear in Math. Nachr. Zbl 0986.46021, MR 1784635 |
| Reference:
|
[28] Burenkov V. I., Popova E. M.: On improving extension operators with the help of the operators of approximation with preservation of the boundary values.(Russian). Trudy Mat. Inst. Steklov 173 (1986), 50–54; English transl.: Proc. Steklov Inst. Math. 173 (1986). MR 0864834 |
| Reference:
|
[29] Burenkov V. I., Schulze B.-W., Tarkhanov N. N.: Extension operators for Sobolev spaces commuting with a given transform.Glasgow Math. J. 40 (1998), 291–296. Zbl 0913.35160, MR 1630199 |
| Reference:
|
[30] Calderón A. P.: Lebesgue spaces of differentiable functions and distributions.Proc. Sympos. Pure Math. IV (1961), 33–49. Zbl 0195.41103, MR 0143037 |
| Reference:
|
[31] Chua S. K.: Extension theorems on weighted Sobolev spaces.Indiana. Univ. Math. 117 (1992), 1027–1076. Zbl 0767.46025, MR 1206339 |
| Reference:
|
[32] Fain B. L.: The extension of functions from an infinite cylinder.(Russian). Trudy Mat. Inst. Steklov 140 (1976), 277–284; English transl.: Proc. Steklov Inst. Math. J. 140 (1979). Zbl 0397.46026, MR 0626997 |
| Reference:
|
[33] Fain B. L.: On extension of functions in Sobolev spaces for irregular domains with preservation of the smoothness exponent.(Russian). Dokl. Akad. Nauk SSSR 285 (1985), 296–301; English transl.: Soviet Math. Dokl. 30 (1985). MR 0820855 |
| Reference:
|
[34] Garofalo N., Nhieu D. M.: Lipschitz continuity, global smoothness approximation and extension theorems for Sobolev functions in Carnot-Carathéodory spaces.Preprint, Purdue Univ. (1996). MR 1631642 |
| Reference:
|
[35] dshtein V. M. Gol,’ : Extension of functions with first generalized derivatives from planar domains.(Russian). Dokl. Akad. Nauk SSSR 257 (1981), 268–271; English transl.: Soviet Math. Dokl. 23 (1981). MR 0610166 |
| Reference:
|
[36] dshtein V. M. Gol,’ Reshetnyak, Yu. G.: Foundations of the theory of functions with generalized derivatives and quasiconformal mappings.(Russian). Nauka, Moscow 1983; English transl.: Reidel, Dordrecht 1989. MR 0738784 |
| Reference:
|
[37] dshtein V. M. Gol,’ Sitnikov V. N.: On extension of functions in the class $W_p^1$ across Hölder boundaries.(Russian). In “Embedding theorems and their applications”. Trudy Semin. S. L. Sobolev, Novosibirsk 1 (1982), 31–43. |
| Reference:
|
[38] dshtein V. M. Gol,’ yanov S. K. Vodop,’ : Prolongement des fonctions de classe $L_p^1$ et applications quasiconformes.C. R. Acad Sci. Paris Sér. A 290 (1983), 453–456. |
| Reference:
|
[39] Herron D. A., Koskela P.: Uniform and Sobolev extension domains.Proc. Amer. Math. Soc. 114, 2 (1992), 483–489. Zbl 0757.30022, MR 1075947 |
| Reference:
|
[40] Hestenes M. R.: Extension of the range of a differentiable function.Duke Math. J. 8 (1941), 183–192. Zbl 0024.38602, MR 0003434 |
| Reference:
|
[41] Jones P. W.: Quasiconformal mappings and extendability of functions in Sobolev spaces.Acta Math. 147 (1981), 71–88. Zbl 0489.30017, MR 0631089 |
| Reference:
|
[42] Kalyabin G. A.: The least norm estimates for certain extension operators from convex planar domains.In: Conference in Mathematical Analysis and Applications in Honour of L. I. Hedberg’s Sixtieth Birthday. Abstracts. Linköping 1996, pp. 55–56. |
| Reference:
|
[43] Konovalov V. N.: A criterion for extension of Sobolev spaces $W_\infty ^{(r)}$ on bounded planar domains.(Russian). Dokl. Akad. Nauk SSSR 289 (1986), 36–39; English transl.: Soviet Math. Dokl. 33 (1986). MR 0852285 |
| Reference:
|
[44] Kudryavtsev L. D., skii S. M. Nikol,’ : Spaces of differentiable functions of several variables and the embedding theorems.(Russian). Contemporary problems in mathematics. Fundamental directions. V. 26 (1988). Analysis – 3. VINITI, Moscow, 5–157. MR 1178111 |
| Reference:
|
[45] Kufner A.: Weighted Sobolev spaces.John Wiley and Sons, Chichester 1985. Zbl 0579.35021, MR 0802206 |
| Reference:
|
[46] Kufner A., John O., Fučík S.: Function spaces.Academia, Prague & Noordhoff International Publishing, Leyden 1977. MR 0482102 |
| Reference:
|
[47] Lieb E. H., Loss M.: Analysis.Amer. Math. Soc. 1997. |
| Reference:
|
[48] ya V. G. Maz,’ : Sobolev spaces.(Russian). LGU, Leningrad 1984; English transl.: Springer-Verlag, Springer Series in Soviet Mathematics, Berlin 1985. MR 0807364 |
| Reference:
|
[49] ya V. G. Maz,’ Poborchii S. V.: On extension of functions in Sobolev spaces to the exterior of a domain with a peak vertex on the boundary.(Russian). Dokl. Akad. Nauk SSSR 275 (1984), 1066–1069; English transl.: Soviet Math. Dokl. 29 (1984). MR 0745847 |
| Reference:
|
[50] ya V. G. Maz,’ Poborchii S. V.: Extension of functions in Sobolev spaces to the exterior of a domain with a peak vertex on the boundary I.(Russian). Czechoslovak Math. J. 36 (1986), 634–661. MR 0863193 |
| Reference:
|
[51] ya V. G. Maz,’ Poborchii S. V.: Extension of functions in Sobolev spaces to the exterior of a domain with a peak vertex on the boundary II.(Russian). Czechoslovak Math. J. 37 (1987), 128–150. MR 0875135 |
| Reference:
|
[52] ya V. G. Maz,’ Poborchii S. V.: Extension of functions in Sobolev spaces on parameter dependent domains.Math. Nachr. 178 (1996), 5–41. MR 1380702 |
| Reference:
|
[53] ya V. G. Maz,’ Poborchii S. V.: Differentiable functions on bad domains.World Scientific Publishing, Singapore 1997. MR 1643072 |
| Reference:
|
[54] skii S. M. Nikol,’ : On the solutions of the polyharmonic equation by a variational method.(Russian). Dokl. Akad. Nauk SSSR 88 (1953), 409–411. |
| Reference:
|
[55] skii S. M. Nikol,’ : On embedding, extension and approximation theorems for differentiable functions of several variables.(Russian). Uspekhi Mat. Nauk 16 (1961), 63–114; English transl.: Russian Math. Surveys 16 (1961). |
| Reference:
|
[56] skii S. M. Nikol,’ : Approximation of functions of several variables and embedding theorems.1st ed., Nauka, Moscow 1969 (Russian); 2nd ed., Nauka, Moscow 1977 (Russian); English transl. of 1st ed., Springer-Verlag, Berlin 1975. |
| Reference:
|
[57] Popova E. M.: On improving extension operators with the help of the operators of approximation with preservation of the boundary values.(Russian). In: “Function spaces and applications to differential equations”, Peoples’ Friendship University of Russia, Moscow (1992), 154–165. |
| Reference:
|
[58] Rychkov V. S.: On restrictions and extensions of Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains.To appear. MR 1721827 |
| Reference:
|
[59] Seeley R. T.: Extension of $C^\infty $-functions defined in halfspace.Proc. Amer. Math. Soc. 15 (1964), 625–626. Zbl 0127.28403, MR 0165392 |
| Reference:
|
[60] Sobolev S. L.: Applications of functional analysis in mathematical physics.1st ed.: LGU, Leningrad 1950 (Russian). 2nd ed.: NGU, Novosibirsk 1963 (Russian). 3rd ed.: Nauka, Moscow 1988 (Russian). English translation of 3rd ed.: Translations of Mathematical Monographs, 90, Amer. Math. Soc., Providence, RI 1991. MR 0165337 |
| Reference:
|
[61] Sobolev S. L.: Introduction to the theory of cubature formulae.(Russian). Nauka, Moscow 1974. MR 0478560 |
| Reference:
|
[62] Sobolev S. L., skii S. M. Nikol,’ : Embedding theorems.(Russian). Vol. 1, Proc. Fourth All-Union Math. Congress, 1961, Nauka, Leningrad 1963, 227–242; English transl.: Amer. Math. Soc. Transl. (2), 87 (1970). MR 0171188 |
| Reference:
|
[63] Stein E. M.: Singular integrals and differentiability properties of functions.Princeton Univ. Press, Princeton 1970. Zbl 0207.13501, MR 0290095 |
| Reference:
|
[64] Triebel H.: Theory of function spaces.Birkhäuser, Basel 1983; and Akad. Verlag. Geest & Portig, Leipzig 1983. Zbl 0546.46028, MR 0781540 |
| Reference:
|
[65] Triebel H.: Theory of function spaces. II.Birkhäuser, Basel 1992. Zbl 0763.46025, MR 1163193 |
| Reference:
|
[66] Uspenskii S. V., Demidenko G. V., Perepelkin V. G.: Embedding theorems and their applications to differential equations.(Russian). Nauka, Novosibirsk 1984. |
| Reference:
|
[67] yanov S. K. Vodop,’ dshtein V. M. Gol,’ Latfullin T. G.: A criterion for extension of functions in class $L_2^1$ from unbounded planar domains.(Russian). Sibirsk. Mat. Zh. 34 (1979), 416–419; English transl.: Siberian Math. J. 34 (1979). |
| Reference:
|
[68] yanov S. K. Vodop,’ dshtein V. M. Gol,’ Reshetnyak, Yu. G.: On geometric properties of functions with first generalized derivatives.(Russian). Uspekhi Mat. Nauk 34 (1979), no. 1(205), 3–74; English transl.: Russian Math. Surveys 20 (1985). |
| Reference:
|
[69] Whitney H.: Analytic extension of differentiable functions defined in closed sets.Trans. Amer. Math. Soc. 36 (1934), 63–89. MR 1501735 |
| Reference:
|
[70] Ziemer W. P.: Weakly differentiable functions.Springer-Verlag, New York 1989. Zbl 0692.46022, MR 1014685 |
| Reference:
|
[71] Zobin N. : Whitney’s problem: extendability of functions and intrinsic metric.C. R. Acad. Sci. Paris 320, 1 (1995), 781–786. Zbl 0826.46017, MR 1326682 |
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