Sharpness of Sobolev Inequalities for a~Class of Irregular Domains

D. A. Labutin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 218-222.

Résumé

Recently, O. V. Besov proved the embedding $W^{m}_p(\Omega)\subset L_q(\Omega)$ for the Sobolev spaces of higher orders $m=2,3,\ldots $ over a domain $\Omega\subset\mathbb R^n$ satisfying $s$-John condition. We show that the number $q$ obtained by Besov in this embedding is maximal over the class of $s$-John domains. An unimprovable embedding of the Sobolev spaces $W^1_p(\Omega )$ was found earlier in works of Hajłasz and Koskela and of Kilpeläinen and Malý.

Bibliographie

[1] Adams D. R., Hedberg L. I., Function spaces and potential theory, Springer, Berlin, 1996 | MR

[2] Besov O. V., “Integralnye predstavleniya funktsii i teoremy vlozheniya na oblastyakh s usloviem gibkogo roga”, Tr. MIAN, 170, 1984, 12–30 | MR | Zbl

[3] Besov O. V., “Teorema vlozheniya Soboleva dlya oblasti s neregulyarnoi granitsei”, Dokl. RAN, 373:2 (2000), 151–154 | MR | Zbl

[4] Besov O. V., “Teorema vlozheniya Soboleva dlya oblasti s neregulyarnoi granitsei”, Mat. sb., 192:3 (2001), 3–26 | MR | Zbl

[5] Besov O. V., “Vlozhenie anizotropnogo prostranstva Soboleva na oblasti s usloviem gibkogo roga”, Tr. MIAN, 181, 1988, 3–14 | MR | Zbl

[6] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[7] Bojarski B., “Remarks on Sobolev imbedding inequalities”, Lect. Notes Math., 1351, 1988, 52–68 | MR

[8] Boman J., $L^p$ estimates for very strongly elliptic systems, Rept 29. Dept. Math. Univ. Stockholm, 1982

[9] Globenko I. G., “Nekotorye voprosy teorii vlozheniya dlya oblastei s osobennostyami na granitse”, Mat. sb., 57 (1962), 201–224 | MR | Zbl

[10] Goldshtein V. M., Reshetnyak Yu. G., Vvedenie v teoriyu funktsii s obobschennymi proizvodnymi i kvazikonformnye otobrazheniya, Nauka, M., 1983 | MR

[11] Varopoulos N., Saloff-Coste L., Coulhon T., Analysis and geometry on groups, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[12] Buckley S., Koskela P., Lu G., “Boman equals John”, XVI Rolf Nevanlinna Colloquium (Joensuu, 1995), W. de Gruyter, Berlin, 1996, 91–99 | MR | Zbl

[13] Hajłasz P., Koskela P., “Isoperimetric inequalities and imbedding theorems in irregular domains”, J. London Math. Soc., 58 (1998), 425–450 | DOI | MR | Zbl

[14] Kilpeläinen T., Malý J., “Sobolev inequalities on sets with irregular boundaries”, Ztschr. Anal. und Anwend., 19:2 (2000), 369–380 | MR | Zbl

[15] John F., “Rotation and strain”, Commun. Pure and Appl. Math., 4 (1961), 391–414 | DOI | MR

[16] Maz'ya V., Netrusov Yu., “Some counterexamples for the theory of Sobolev spaces on bad domains”, Pot. Anal., 4 (1995), 47–65 | DOI | MR

[17] Garofalo N., Nhieu D.-M., “Isoperimetric and Sobolev inequalities for Carnot–Caratheodory spaces and the existence of minimal surfaces”, Commun. Pure and Appl. Math., 49 (1996), 1081–1144 | 3.0.CO;2-A class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[18] Hajłasz P., Koskela P., “Sobolev met Poincare”, Mem. Amer. Math. Soc., 145 (2000), 1–101 | MR

[19] Koskela P., “Sobolev spaces and quasiconformal mappings”, Proc. III Europ. Congr. Math., Birkhäuser, Basel etc., 2000

[20] Kurant R., Gilbert D., Metody matematicheskoi fiziki, T. 1, Gostekhteorizdat, M., L., 1951

[21] Labutin D. A., “Integralnoe predstavlenie funktsii i vlozhenie prostranstv Soboleva na oblastyakh s nulevymi uglami”, Mat. zametki, 61 (1997), 164–179 | MR | Zbl

[22] Labutin D. A., “Vlozhenie prostranstv Soboleva na gëlderovykh oblastyakh”, Tr. MIAN, 227, 1999, 170–179 | MR | Zbl

[23] Mazya V. G., Prostranstva S. L. Soboleva, Izd-vo LGU, L., 1985 | MR

[24] Mazya V. G., “Klassy oblastei, mer i emkostei v teorii prostranstv differentsiruemykh funktsii”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 26, VINITI, M., 1988, 159–228 | MR

[25] Buckley S. M., Koskela P., “Criteria for imbeddings of Sobolev–Poincare type”, Intern. Math. Res. Not., 1996, no. 18, 881–901 | DOI | MR | Zbl

[26] Buckley S., Koskela P., “Sobolev–Poincare implies John”, Math. Res. Lett., 2 (1995), 577–593 | MR | Zbl

[27] Maz'ya V. G., Poborchii S. V., Differentiable functions on bad domains, World Sci., River Edge, NJ, 1997 | MR

[28] Nikodym O., “Sur une classe de fonctions considérémes le problée de Dirichlet”, Fund. Math., 21 (1933), 129–150 | Zbl

[29] Reshetnyak Yu. G., “Integralnoe predstavlenie differentsiruemykh funktsii v oblastyakh s negladkoi granitsei”, Sib. mat. zhurn., 21 (1980), 108–116 | MR | Zbl

[30] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 | MR

[31] Sobolev S. L., “Ob odnoi teoreme funktsionalnogo analiza”, Mat. sb., 46 (1938), 471–497

[32] Smith W., Stegenga D. A., “Hölder domains and Poincaré domains”, Trans. Amer. Math. Soc., 319 (1990), 67–100 | DOI | MR | Zbl