The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values

Potential Analysis

Volumn 25, Issue 3, 2006, Pages 205-222

  Harjulehto, Petteri ^a   Hästö, Peter ^b   Koskenoja, Mika ^a   Varonen, Susanna ^a  

^a UNIVERSITY OF HELSINKI   (Finland)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Dirichlet energy integral; Poincar inequality; Sobolev capacity; Variable exponent Sobolev space; Zero boundary values

Indexed keywords

EID: 33749465843     PISSN: 09262601     EISSN: 1572929X     Source Type: Journal    
DOI: 10.1007/s11118-006-9023-3     Document Type: Article

References (44)

1. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin Heidelberg New York (1996)
2. Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121-140 (2001)
3. Alkhutov, Yu. A.: The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with nonstandard growth condition (Russian). Differ. Uravn. 33(12), 1651-1660, 1726 (1997);
4. translation in Diff. Equ. 33(12), 1653-1663 (1997)
5. Alkhutov, Yu. A.: On the Hölder continuity of p(x)-harmonic functions, (Russian). Mat. Sb. 196(2), 3-28 (2005);
6. translation in Sb. Math. 196(1-2), 147-171 (2005)
7. Coscia, A., Mingione, G.: Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris, Ser. 1 Math. 328(4), 363-368 (1999)
8. p(.). Math. Inequal. Appl. 7(2), 245-254 (2004)
9. p(.) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197-220 (2003)
10. Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory. Interscience, New York (1958)
11. k,p(x) (Ω). Proc. R. Soc. London, Ser. A 437, 229-236 (1992)
12. Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent. Stud. Math. 143, 267-293 (2000)
13. Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent, II. Math. Nachr. 246-247, 53-67 (2002)
14. p(x) spaces. Z. Anal. Anwend. 21, 681-690 (2002)
15. k,p(x). J. Math. Anal. Appl. 262, 749-760 (2001)
16. Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295-318 (1999)
17. Fan, X., Zhao, D.: The quasi-minimizers of integral functionals with m(x) growth conditions. Nonlinear Anal. 39, 807-816 (2000)
18. Fan, X., Zhang, Q.: Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843-1852 (2003)
19. Harjulehto, P., Hästö, P.: A capacity approach to Poincaré inequalities and Sobolev imbedding in variable exponent Sobolev spaces. Rev. Mat. Complut. 17, 129-146 (2004)
20. Harjulehto, P., Hästö, P., Koskenoja, M.: The Dirichlet energy integral on intervals in variable exponent Sobolev spaces. Z. Anal. Anwend. 22(4), 911-923 (2003)
21. n). J. Funct. Spaces Appl. 1(1), 17-33 (2003)
22. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)
23. k(Ω). Comment. Math. Prace Mat. 21, 315-324 (1980)
24. Hästö, P.: On the density of continuous functions in variable exponent Sobolev space. Rev. Mat. Iberoamericana, to appear
25. Hästö, P.: Counter-examples of regularity in variable exponent Sobolev spaces. In: The p-Harmonic Equation and Recent Advances in Analysis (Manhattan, KS, 2004), Contemp. Math. vol. 367, pp. 133-143. American Mathematical Society, Providence, Rhode Island (2005)
26. Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn., Math. 23, 261-262 (1998)
27. Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12, 233-247 (2000)
28. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, London (1980)
29. 1,p(x). Czechoslov. Math. J. 41(116), 592-618 (1991)
30. Maĺ, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equation, Mathematical Surveys and Monographs, vol. 51. American Mathematical Society, Providence, Rhode Island (1997)
31. Marcellini, P.: Regularity and existance of solutions of elliptic equations with p, q-growth conditions. J. Differ. Equ. 50(1), 1-30 (1991)
32. Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin Heidelberg New York (1983)
33. Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3, 200-212 (1931)
34. p(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math. 19, 369-371 (2001)
35. Rákosník, J.: Sobolev inequality with variable exponent. In: Mustonen, V., Rákosník, J. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis, pp. 220-228. Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague (2000)
36. Rudin, W.: Functional Analysis, TMH Edition, 14th Reprint. Tata McGraw-Hill, New Delhi (1990)
37. Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin Heidelberg New York (2000)
38. n) (Russian). Dokl. Ross. Acad Nauk 369(4), 451-454 (1999);
39. translation in Dokl. Math. 60, 382-385 (1999)
40. Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45(3), 1021-1050 (2001)
41. p(t) ([0, 1]). Mat. Zametki 26(4), 613-632 (1979)
42. s. Uch. Zap. Dagestan Gos. Univ. 7, 25-37 (1961)
43. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk. SSSR, Ser. Mat. 50, 675-710, 877 (1986)
44. Zhikov, V.V.: On Lavrentiev's phenomenon. Russ. J. Math. Phys. 3(2), 249-269 (1995)