Abstract
We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent.
Similar content being viewed by others
References
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin Heidelberg New York (1996)
Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)
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); translation in Diff. Equ. 33(12), 1653–1663 (1997)
Alkhutov, Yu. A.: On the Hölder continuity of \(p(x)\)-harmonic functions, (Russian). Mat. Sb. 196(2), 3–28 (2005); translation in Sb. Math. 196(1–2), 147–171 (2005)
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)
Diening, L.: Maximal function on generalized Lebesgue spaces \({L}^{\,p(\cdot)}\). Math. Inequal. Appl. 7(2), 245–254 (2004)
Diening, L., Růžička, M.: Calderón–Zygmund operators on generalized Lebesgue spaces \({L}^{\,p(\cdot)}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)
Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory. Interscience, New York (1958)
Edmunds, D.E., Rákosník, J.: Density of smooth functions in \(W^{k,p(x)}(\Omega)\). Proc. R. Soc. London, Ser. A 437, 229–236 (1992)
Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent. Stud. Math. 143, 267–293 (2000)
Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent, II. Math. Nachr. 246–247, 53–67 (2002)
Edmunds, D.E., Meskhi, A.: Potential-type operators in \(L\sp {\,p(x)}\) spaces. Z. Anal. Anwend. 21, 681–690 (2002)
Fan, X., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces \(W^{\,k,p(x)}\). J. Math. Anal. Appl. 262, 749–760 (2001)
Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295–318 (1999)
Fan, X., Zhao, D.: The quasi-minimizers of integral functionals with \(m(x)\) growth conditions. Nonlinear Anal. 39, 807–816 (2000)
Fan, X., Zhang, Q.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)
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)
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)
Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: Sobolev capacity on the space \(W^{1,p(\cdot)}({{\rm{I}\kern-2pt{R}}^n})\). J. Funct. Spaces Appl. 1(1), 17–33 (2003)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)
Hudzik, H.: The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces \(W_M^k(\Omega)\). Comment. Math. Prace Mat. 21, 315–324 (1980)
Hästö, P.: On the density of continuous functions in variable exponent Sobolev space. Rev. Mat. Iberoamericana, to appear
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)
Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn., Math. 23, 261–262 (1998)
Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12, 233–247 (2000)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, London (1980)
Kováčik, O., Rákosník, J.: On spaces \(L^{\,p(x)}\) and \(W^{1,p(x)}\). Czechoslov. Math. J. 41(116), 592–618 (1991)
Malý, 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)
Marcellini, P.: Regularity and existance of solutions of elliptic equations with \(p,q\)-growth conditions. J. Differ. Equ. 50(1), 1–30 (1991)
Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin Heidelberg New York (1983)
Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3, 200–212 (1931)
Pick, L., Růžička, M.: An example of a space \(L^{{p{\left( x \right)}}} \) on which the Hardy–Littlewood maximal operator is not bounded. Expo. Math. 19, 369–371 (2001)
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)
Rudin, W.: Functional Analysis, TMH Edition, 14th Reprint. Tata McGraw-Hill, New Delhi (1990)
Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin Heidelberg New York (2000)
Samko, S.: Denseness of \(C^\infty_0({{\rm{I}\kern-2pt{R}}^n})\) in the generalized Sobolev spaces \(W^{\,m,p(x)}({{\rm{I}\kern-2pt{R}}^n})\) (Russian). Dokl. Ross. Acad Nauk 369(4), 451–454 (1999); translation in Dokl. Math. 60, 382–385 (1999)
Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45(3), 1021–1050 (2001)
Sharapudinov, I.I.: The topology of the space \(\mathcal{L}^{p(t)}([0,1])\). Mat. Zametki 26(4), 613–632 (1979)
Tsenov, I.V.: Generalization of the problem of best approximation of a function in the space \(L^s\). Uch. Zap. Dagestan Gos. Univ. 7, 25–37 (1961)
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)
Zhikov, V.V.: On Lavrentiev's phenomenon. Russ. J. Math. Phys. 3(2), 249–269 (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Harjulehto, P., Hästö, P., Koskenoja, M. et al. The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values. Potential Anal 25, 205–222 (2006). https://doi.org/10.1007/s11118-006-9023-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-006-9023-3