Wolff potential estimates for elliptic equations with nonstandard growth and applications

Forum Mathematicum

Volumn 22, Issue 6, 2010, Pages 1061-1087

  Lukkari, Teemu ^a   Maeda, Fumi Yuki ^b   Marola, Niko ^c  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b Helsinki University of Technology ^*  (Japan)

^c AALTO UNIVERSITY   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77953025062     PISSN: 09337741     EISSN: 14355337     Source Type: Journal    
DOI: 10.1515/FORUM.2010.057     Document Type: Article

References (42)

1. Acerbi E., Fusco N.: A transmission problem in the calculus of variations. Calc. Var. Partial Differential Equations 2 (1994), 1-16
2. Acerbi E., Mingione G.: Regularity results for a class of functionals with nonstand ard growth. Arch. Rational Mech.Anal. 156 (2001), 121-140
3. Acerbi E., Mingione G.: Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math. 584 (2005), 117-148
4. Adams D. R.: Weighted nonlinear potential theory. Trans. Amer. Math. Soc. 297(1) (1986), 73-94
5. AdamsD.R., Hedberg L. I.: Function Spaces and PotentialTheory.Grundlehren derMathematischenWissenschaften 314 [Fundamental Principles ofMathematical Sciences]. Springer-Verlag, Berlin 1996
6. Adams D. R., Meyers N. G.: Thinness and Wiener criteria for non-linear potentials. Indiana Univ. Math. J. 22 (1972/73), 169-197
7. AlkhutovYu.A., KrasheninnikovaO.V.:Continuity at boundary points of solutions of quasilinear elliptic equationswith nonstand ard growth. Izv. Ross.Akad.Nauk. Ser.Mat. 68(6) (2004), 3-60
8. Translation in Izv. Math. 68 (2004), 1063-1117
9. Antontsev S., Zhikov V.: Higher integrability for parabolic equations of p(x, t)-Laplacian type. Adv. Differential Equations 10 (2005), 1053-1080
10. Cruz-Uribe D., Fiorenza A., Neugebauer C. J.: The maximal function on variable Lp spaces. Ann.Acad. Sci. Fenn.Math. 28 (2003), 223-238
11. Diening L.: Maximal function on generalized Lebesgue spaces Lp(•). Math. Inequal. Appl. 7 (2004), 245-253
12. Evans L. C., Gariepy R. F.: Measure theory and fine properties of functions. CRC Press, Boca Raton, FL 1992
13. Fan X., Zhao D.: A class of De Giorgi type and Hölder continuity. NonlinearAnal. 36 (1999), 295-318
14. Harjulehto P.: Variable exponent Sobolev spaces with zero boundary values. Math. Bohem. 132(2) (2007), 125-136
15. Harjulehto P., Hästö P., Koskenoja M., Lukkari T., Marola N.: An obstacle problem and superharmonic functions with nonstand ard growth. NonlinearAnal. 67 (2007), 3424-3440
16. Harjulehto P., Hästö P., KoskenojaM., Varonen S.: Sobolev capacity on the spaceW1,p(•)(Rn). J. Funct. SpacesAppl. 1 (2003), 17-33
17. Harjulehto P., Hästö P., LatvalaV.: Minimizers of the variable exponent, non-uniformly convex Dirichlet energy. J.Math. PuresAppl. (9) 89 (2008), 174-197
18. Harjulehto P., Kinnunen J., Lukkari T.: Unbounded supersolutions of nonlinear equations with nonstand ard growth. Bound.Value Probl. 2007:Article ID 48348, 20 pages, 2007. Available at http://www.hindawi.com/GetArticle.aspx?doi= 10.1155/2007/48348
19. Harjulehto P., Latvala V.: Fine topology of variable exponent energy superminimizers. Ann. Acad. Sci. Fenn.Math. 33 (2008), 491-510
20. Hedberg L. I.: Non-linear potentials and approximation in the mean by analytic functions. Math. Z. 129 (1972), 299-319
21. Hedberg L. I., Wolff Th. H.: Thin sets in nonlinear potential theory.Ann. Inst. Fourier (Grenoble) 33(4) (1983), 161-187
22. Heinonen J., KilpeläinenT.: A-superharmonic functions and supersolutions of degenerate elliptic equations. Ark. Mat. 26 (1988), 87-105
23. Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, 1993
24. Hästö P.: Counter examples of regularity in variable exponent Sobolev spaces. Contemp. Math. 367 (2005), 133-143
25. Hästö P.: On the density of continuous functions in variable exponent Sobolev space. Rev.Mat. Iberoamericana 23 (2007), 215-237
26. KilpeläinenT., Malý J.: Degenerate elliptic equationswithmeasure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (1992), 591-613
27. KilpeläinenT., Malý J.: TheWiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172 (1994), 137-161
28. Kováčik O., Rákosník J.: On spaces Lp(x) and W1,p(x). CzechoslovakMath. J. 41 (116) (1991), 592-618
29. Labutin D. A.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111 (2002), 1-49
30. Lindqvist P.: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. 365 (1986), 67-79
31. Lukkari T.: Singular solutions of elliptic equations with nonstand ard growth. Math. Nachr. to appear. Preprint available at http://mathstat.helsinki. fi/analysis/varsobgroup/
32. Lukkari T.: Elliptic equationswith nonstand ard growth involving measures. Hiroshima Math. J. 38 (2008), 155-176
33. Malý J.: Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals. Manuscripta Math. 110 (2003), 513-525
34. Malý J., Ziemer W. P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. American Mathematical Society, 1997
35. Marcellini P.: Regularity and existence of solutions of elliptic equations with p, q-growth conditions. J. Differential Equations 90 (1991), 1-30
36. Mikkonen P.: On theWolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996), 1-71
37. Musielak J.: Orlicz Spaces and Modular Spaces. Springer-Verlag, Berlin 1983
38. n ). In: Direct and Inverse Problems of Mathematical Physics (Newark,DE, 1997). Int. Soc.Anal.Appl. Comput. 5, pp. 333-342. KluwerAcad. Publ., Dordrecht 2000
39. Trudinger N. S., Wang X.-J.: On the weak continuity of elliptic operators and applications to potential theory. Amer. J. Math. 124 (2002), 369-410
40. Zhikov V.V.: On Lavrentiev's phenomenon. Russian J. Math. Phys. 3 (1995), 249-269
41. Zhikov V.V.: On some variational problems. Russian J. Math. Phys. 5 (1997), 105-116
42. ZiemerW. P.: Weakly Differentiable Functions. Springer-Verlag, NewYork 1989