| Title:
|
Some results in the theory of Orlicz spaces and applications to variational problems (English) |
| Author:
|
Cianchi, Andrea |
| Language:
|
English |
| Journal:
|
Nonlinear Analysis, Function Spaces and Applications |
| Volume:
|
Vol. 6 |
| Issue:
|
1998 |
| Year:
|
|
| Pages:
|
50-92 |
| . |
| Category:
|
math |
| . |
| MSC:
|
35B45 |
| MSC:
|
35J20 |
| MSC:
|
46B70 |
| MSC:
|
46E30 |
| MSC:
|
46E35 |
| MSC:
|
46M35 |
| MSC:
|
47B38 |
| MSC:
|
47G10 |
| MSC:
|
49J10 |
| MSC:
|
49N60 |
| idZBL:
|
Zbl 0965.46017 |
| idMR:
|
MR1777712 |
| . |
| Date available:
|
2009-10-08T09:47:01Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702470 |
| . |
| Reference:
|
[A1] Adams R. A. : Sobolev spaces.Academic Press, New York 1975. Zbl 0314.46030, MR 0450957 |
| Reference:
|
[A2] Adams R. A. : On the Orlicz-Sobolev imbedding theorem.J. Funct. Anal. 24 (1977), 241–257. Zbl 0344.46077, MR 0430770 |
| Reference:
|
[BR] Bennett C., Rudnick K. : On Lorentz-Zygmund spaces.Dissertationes Mathematicae, CLXXV, Warszawa 1980. Zbl 0456.46028, MR 0576995 |
| Reference:
|
[BS] Bennett C., Sharpley R. : Interpolation of operators.Pure and Applied Mathematics, Vol. 129, Academic Press, Boston 1988. Zbl 0647.46057, MR 0928802 |
| Reference:
|
[BL] Bhattacharya T., Leonetti F. : A new Poincaré inequality and its applications to the regularity of minimizers of integral functionals with nonstandard growth.Nonlinear Anal. 17 (1991), 833–839. MR 1131493 |
| Reference:
|
[BK] Bloom S., Kerman R. : Weighted $L_{\Phi }$ integral inequalities for operators of Hardy type.Studia Math. 110 (1994), 35–52. MR 1279373 |
| Reference:
|
[BZ] Brothers J. E., Ziemer W. P. : Minimal rearrangements of Sobolev functions.J. Reine Angew. Math. 384 (1988), 153–179. Zbl 0633.46030, MR 0929981 |
| Reference:
|
[Ca] Calderón A. P. : Spaces between $L^1$ and $L^{\infty }$ and the theorem of Marcinkiewicz.Studia Math. 26 (1966), 273–299. MR 0203444 |
| Reference:
|
[Ci1] Cianchi A. : On relative isoperimetric inequalities in the plane.Boll. Un. Mat. Ital. (7) 3-B (1989), 289–325. Zbl 0674.49030, MR 0997998 |
| Reference:
|
[Ci2] Cianchi A. : Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems.Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 23 (1996), 575–608. Zbl 0877.46023, MR 1440034 |
| Reference:
|
[Ci3] Cianchi A. : A sharp embedding theorem for Orlicz–Sobolev spaces.Indiana Univ. Math. J. 45 (1996), 39–65. Zbl 0860.46022, MR 1406683 |
| Reference:
|
[Ci4] Cianchi A. : Strong and weak type inequalities for some classical operators in Orlicz spaces.To appear in J. London Math. Soc. Zbl 0940.46015, MR 1721824 |
| Reference:
|
[Ci5] Cianchi A. : Hardy inequalities in Orlicz spaces.To appear in Trans. Amer. Math. Soc. Zbl 0920.46021, MR 1433113 |
| Reference:
|
[Ci6] Cianchi A. : Boundedness of solutions to variational problems under general growth conditions.Comm. Partial Differential Equations 22 (1997), 1629–1646. Zbl 0892.35048, MR 1469584 |
| Reference:
|
[Ci7] Cianchi A. : An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces.J. Funct. Anal. 153 (1998), 357–381. Zbl 0913.46028, MR 1614590 |
| Reference:
|
[Ci8] Cianchi A. : A fully anisotropic Sobolev inequality.Preprint no. 15, 1998, Dip. Mat. ”U. Dini”, Univ. di Firenze. MR 1639390 |
| Reference:
|
[CF] Cianchi A., Fusco N. : Gradient regularity for minimizers under general growth conditions.J. reine angew. Math. 227 (1998), 166–186. MR 1670258 |
| Reference:
|
[CS] Cianchi A., Stroffolini B. : An extension of Hedberg’s convolution inequality and applications.To appear in J. Math. Anal. Appl. Zbl 0914.42014, MR 1652919 |
| Reference:
|
[D] Donaldson T. K. : Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces.J. Differential Equations 10 (1971), 507–528. Zbl 0218.35028, MR 0298472 |
| Reference:
|
[DT] Donaldson T. K., Trudinger N. S. : Orlicz-Sobolev spaces and imbedding theorems.J. Funct. Anal. 8 (1971), 52–75. Zbl 0216.15702, MR 0301500 |
| Reference:
|
[EGO] Edmunds D. E., Gurka P., Opic B. : Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces.Indiana Univ. Math. J. 44 (1995), 19–44. Zbl 0826.47021, MR 1336431 |
| Reference:
|
[EOP] Evans W. D., Opic B., Pick L. : Interpolation of operators on scales of generalized Lorentz-Zygmund spaces.Math. Nachr. 182 (1996), 127–181. Zbl 0865.46016, MR 1419893 |
| Reference:
|
[FLS] Fusco N., Lions P. L., Sbordone C. : Some remarks on Sobolev imbeddings in borderline cases.Proc. Amer. Math. Soc. 124 (1996), 561–565. MR 1301025 |
| Reference:
|
[FS] Fusco N., Sbordone C. : Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions.Comm. Pure Appl. Math. 43 (1990), 673–683. Zbl 0727.49021, MR 1057235 |
| Reference:
|
[Ge] Gehring F. W. : The $L^p$-integrability of the partial derivatives of a quasi conformal mapping.Acta Math. 130 (1973), 265–277. Zbl 0258.30022, MR 0402038 |
| Reference:
|
[GG] Giaquinta M., Giusti E. : Quasi-minima.Ann. Inst. H. Poincaré, Anal. Non Linéaire 1 (1984), 79–107. Zbl 0541.49008, MR 0778969 |
| Reference:
|
[Go1] Gossez J.-P. : Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients.Trans. Amer. Math. Soc. 190 (1974), 163–205. MR 0342854 |
| Reference:
|
[Go2] Gossez J.-P. : Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems.In: Nonlinear Analysis, Function Spaces and Applications, Vol. 1. S. Fučík and A. Kufner (eds.), Teubner, Leipzig 1979, 59–94. MR 0578910 |
| Reference:
|
[Go3] Gossez J.-P. : A strong nonlinear elliptic problem in Orlicz-Sobolev spaces.Proc. Sympos. Pure Math. 45 (1986), 455–462. MR 0843579 |
| Reference:
|
[GM] Gossez J.-P., Mustonen V. : Variational inequalities in Orlicz-Sobolev spaces.Nonlinear Anal. 11 (1987), 379–392. Zbl 0643.49006, MR 0881725 |
| Reference:
|
[GIS] Greco L., Iwaniec T., Sbordone C. : Variational integrals of nearly linear growth.Differential Integral Equations 10 (1997), 687–716. Zbl 0889.35026, MR 1741768 |
| Reference:
|
[GP] Gustavsson J., Peetre J. : Interpolation of Orlicz spaces.Studia Math. 60 (1977), 33–59. Zbl 0353.46019, MR 0438102 |
| Reference:
|
[HMT] Hempel J. A., Morris G. R., Trudinger N. S. : On the sharpness of a limiting case of the Sobolev imbedding theorem.Bull. Austral. Math. Soc. 3 (1970), 369–373. Zbl 0205.12801, MR 0280998 |
| Reference:
|
[H] Hunt R. A. : An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces.Bull. Amer. Math. Soc. 70 (1964), 803–807. Zbl 0145.38203, MR 0169037 |
| Reference:
|
[KR] skii M. A. Krasnosel,’ Rutickii, Ya. B. : Convex functions and Orlicz spaces.P. Noordhoff, Groningen 1961. MR 0126722 |
| Reference:
|
[Kr] Kraynek W. T. : Interpolation of sublinear operators in generalized Orlicz and Hardy-Orlicz spaces.Studia Math. 43 (1972), 93–123. MR 0312245 |
| Reference:
|
[K] Kufner A. : Weighted Sobolev spaces.Teubner, Leipzig 1980. Zbl 0455.46034, MR 0664599 |
| Reference:
|
[KJF] Kufner A., John O., Fučík S. : Function Spaces.Noordhoff Int. Publ., Leyden 1977. MR 0482102 |
| Reference:
|
[LU] Ladyzhenskaya O. A., tseva N. N. Ural,’ : Linear and quasilinear elliptic equations.Academic Press, New York 1968. MR 0244627 |
| Reference:
|
[LM] Landes R., Mustonen V. : Pseudo monotone mappings in Sobolev-Orlicz spaces and nonlinear boundary value problems on unbounded domains.J. Math. Anal. Appl. 88 (1982), 25–36. Zbl 0492.35015, MR 0661399 |
| Reference:
|
[L] Lieberman G. M. : The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations.Comm. Partial Differential Equations 16 (1991), 311–361. Zbl 0742.35028, MR 1104103 |
| Reference:
|
[Mal] Maligranda L. : Orlicz spaces and interpolation.Seminários de Matemática, Univ. Estadual de Campinas 1989. Zbl 0874.46022, MR 2264389 |
| Reference:
|
[Mar] Marcellini P. : Regularity for elliptic equations with general growth conditions.J. Differential Equations 105 (1993), 296–333. Zbl 0812.35042, MR 1240398 |
| Reference:
|
[Mark] Marcinkiewicz J. : Sur l’interpolation d’operateurs.C. R. Acad. Sci. Paris (1939), 1272–1273. |
| Reference:
|
[MP1] Mascolo E., Papi G. : Local boundedness of minimizers of integrals of the calculus of variations.Ann. Mat. Pura Appl. 167 (1994), 323–339. Zbl 0819.49023, MR 1313560 |
| Reference:
|
[MP2] Mascolo E., Papi G. : Harnack inequality for minimizers of integral functionals with general growth conditions.Nonlinear Differential Equations Appl. 3 (1996), 232–244. Zbl 0855.49027, MR 1385885 |
| Reference:
|
[M] ya V. G. Maz,’ : Sobolev spaces.Springer-Verlag, Berlin 1985. MR 0817985 |
| Reference:
|
[MT] Mustonen V., Tienari M. : An eigenvalue problem for generalized Laplacian in Orlicz-Sobolev spaces.To appear in Proc. Royal Soc. Edinburgh A. Zbl 0926.46030, MR 1669197 |
| Reference:
|
[O] Neil R. O,’ : Fractional integration in Orlicz spaces.Trans. Amer. Math. Soc. 115 (1965), 300–328. MR 0194881 |
| Reference:
|
[OK] Opic B., Kufner A. : Hardy-type inequalities.Longman Scientific & Technical, Harlow 1990. Zbl 0698.26007, MR 1069756 |
| Reference:
|
[Pe] Peetre J. : A new approach in interpolation spaces.Studia Math. 34 (1970), 23–42. Zbl 0188.43602, MR 0264390 |
| Reference:
|
[Po] Pokhozhaev S. I. : On the imbedding Sobolev theorem for $pl=n$.(Russian.) Doklady conference, Section Math., Moscow Power Institute (1965), 158–170. |
| Reference:
|
[Pu] nik E. I. Pustyl,’ : On optimal interpolation and some interpolation properties of Orlicz spaces.Soviet Math. Dokl. 27 (1983). MR 0698510 |
| Reference:
|
[Ra] Rao M. M. : Interpolation, ergodicity and martingales.J. Math. Mech. 16 (1966), 543–567. Zbl 0166.11704, MR 0207960 |
| Reference:
|
[RR] Rao M. M., Ren Z. D. : Theory of Orlicz spaces.Marcel Dekker Inc., New York 1991. Zbl 0724.46032, MR 1113700 |
| Reference:
|
[Ri] Riordan W. J. : On the interpolation of operators.PhD Dissertation, Univ. Chicago 1956 (see also Notices Amer. Math. Soc. 5 (1958), 590). |
| Reference:
|
[Sa] Sawyer E. T. : Boundedness of classical operators on classical Lorentz spaces.Studia Math. 96 (1990), 145–158. Zbl 0705.42014, MR 1052631 |
| Reference:
|
[Sb] Sbordone C. : On some integral inequalities and their applications to the Calculus of Variations.Boll. Un. Mat. Ital., Analisi Funzionale e Applicazioni, Ser VI 5 (1986), 73–94. Zbl 0678.49008, MR 0897186 |
| Reference:
|
[S] Stampacchia G. : On some regular multiple integral problems in the calculus of variations.Comm. Pure Appl. Math. 16 (1963), 383–421. Zbl 0138.36903, MR 0155209 |
| Reference:
|
[Stre] Stredulinsky E. W. : Higher integrability from reverse Hölder inequalities.Indiana Univ. Math. J. 29 (1980), 407–413. Zbl 0442.35064, MR 0570689 |
| Reference:
|
[Stri] Strichartz R. S. : A note on Trudinger’s extension of Sobolev’s inequalities.Indiana Univ. Math. J. 21 (1972), 841–842. Zbl 0241.46028, MR 0293389 |
| Reference:
|
[Ta1] Talenti G. : Best constant in Sobolev inequality.Ann. Mat. Pura Appl. 110 (1976), 353–372. Zbl 0353.46018, MR 0463908 |
| Reference:
|
[Ta2] Talenti G. : Elliptic equations and rearrangements.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 697–718. Zbl 0341.35031, MR 0601601 |
| Reference:
|
[Ta3] Talenti G. : An embedding theorem.Essays of Math. Analysis in honour of E. De Giorgi, Birkhäuser, Boston 1989. Zbl 0702.46020, MR 1034035 |
| Reference:
|
[Ta4] Talenti G. : Boundedness of minimizers.Hokkaido Math. J. XIX (1990), 259–279. Zbl 0723.58015, MR 1059170 |
| Reference:
|
[Ta5] Talenti G. : Inequalities in rearrangement invariant function spaces.In: Nonlinar Analysis, Function Spaces and Applications, vol. 5. M. Krbec, A. Kufner, B. Opic and J. Rákosník (eds.), Prometheus Publishing House, Prague 1994, 177–230. Zbl 0872.46020, MR 1322313 |
| Reference:
|
[To1] Torchinsky A. : Interpolation of operators and Orlicz classes.Studia Math. 59 (1976), 177–207. MR 0438105 |
| Reference:
|
[To2] Torchinsky A. : Real variable methods in harmonic analysis.Academic Press, San Diego 1986. Zbl 0621.42001, MR 0869816 |
| Reference:
|
[Tr] Trudinger N. S. : On imbeddings into Orlicz spaces and some applications.J. Math. Mech. 17 (1967), 473–483. Zbl 0163.36402, MR 0216286 |
| Reference:
|
[Y] Yudovich V. I. : Certain estimates connected with integral operators and solutions of elliptic equations.Dokl. Akad. Nauk SSSR 138 (1961), 805–808. MR 0140822 |
| Reference:
|
[Zi] Ziemer W. P.: Weakly differentiable functions.Springer-Verlag, New York 1989. Zbl 0692.46022, MR 1014685 |
| Reference:
|
[Z] Zygmund A. : On a theorem of Marcinkiewicz concerning interpolation of operations.J. Math. Pures Appl. 35 (1956), 223–248. Zbl 0070.33701, MR 0080887 |
| . |
