zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. (English) Zbl 1120.46016

The authors consider the Dirichlet energy integral

${\int }_{{\Omega }}{|\nabla u\left(x\right)|}^{p\left(x\right)}\phantom{\rule{0.166667em}{0ex}}dx$

with variable $p\left(x\right)$, where ${\Omega }$ is a bounded domain in ${ℝ}^{n}$, which was studied by various authors. The existence and regularity of minimizers for this integral were studied by E. Acerbi and G. Minigone [Arch. Ration. Mech. Anal. 156, 121–140 (2001; Zbl 0984.49020)] for functions $u\in {W}^{1,1}\left({\Omega }\right)$ with boundary values in the classical sense, the minimizer being Hölder continuous in this case. The authors develop another approach to the study of this integral, based on the paper by N. Shanmugalingam [Ill. J. Math. 45, No. 3, 1021–1050 (2001; Zbl 0989.31003)].

They admit functions with boundary values in the Sobolev sense and minimize over functions in the variable exponent Sobolev space. To this end, they define the Sobolev space ${W}_{0}^{1,p\left(·\right)}$ with zero boundary values in terms of the Sobolev variable capacity introduced earlier by the authors. As a preliminary step, they also study a variable Poincaré $p\left(·\right)$-inequality and introduce a condition on $p\left(x\right)$ under which such an inequality is valid.

On this basis, they prove their main results in Theorems 5.2 and 5.3. In the former, they show that if $1<\text{ess}infp\left(x\right)\le \text{ess}supp\left(x\right)<\infty$ and $p\left(x\right)$ “is not too discontinuous”, then there exists a function $u\in {W}^{1,p\left(·\right)}\left({\Omega }\right)$ which minimizes the Dirichlet $p\left(·\right)$-integral with $u-w\in {W}_{0}^{1,p\left(·\right)}\left({\Omega }\right)$. The minimizer is unique in a certain sense. In the latter, they give a criterion for a function $u$ to be a minimizer. The results obtained are parallel to those known in the case of constant $p$.

MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 49J40 Variational methods including variational inequalities 31C45 Nonlinear potential theory, etc.
References:
 [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) · Zbl 0984.49020 · doi:10.1007/s002050100117 [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); translation in Diff. Equ. 33(12), 1653–1663 (1997) [4] Alkhutov, Yu. A.: On the Hölder continuity of $p\left(x\right)$ -harmonic functions, (Russian). Mat. Sb. 196(2), 3–28 (2005); translation in Sb. Math. 196(1–2), 147–171 (2005) [5] Coscia, A., Mingione, G.: Hölder continuity of the gradient of $p\left(x\right)$ -harmonic mappings. C. R. Acad. Sci. Paris, Ser. 1 Math. 328(4), 363–368 (1999) [6] Diening, L.: Maximal function on generalized Lebesgue spaces ${L}^{\phantom{\rule{0.166667em}{0ex}}p\left(·\right)}$ . Math. Inequal. Appl. 7(2), 245–254 (2004) [7] Diening, L., Ružička, M.: Calderón–Zygmund operators on generalized Lebesgue spaces ${L}^{\phantom{\rule{0.166667em}{0ex}}p\left(·\right)}$ and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003) · Zbl 1072.76071 · doi:10.1515/crll.2003.081 [8] Dunford, N., Schwartz, J.T.: Linear Operators. Part I. General Theory. Interscience, New York (1958) [9] Edmunds, D.E., Rákosník, J.: Density of smooth functions in ${W}^{k,p\left(x\right)}\left({\Omega }\right)$ . Proc. R. Soc. London, Ser. A 437, 229–236 (1992) · Zbl 0779.46027 · doi:10.1098/rspa.1992.0059 [10] Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent. Stud. Math. 143, 267–293 (2000) [11] Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent, II. Math. Nachr. 246–247, 53–67 (2002) · doi:10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T [12] Edmunds, D.E., Meskhi, A.: Potential-type operators in ${L}^{\phantom{\rule{0.166667em}{0ex}}p\left(x\right)}$ spaces. Z. Anal. Anwend. 21, 681–690 (2002) [13] Fan, X., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces ${W}^{\phantom{\rule{0.166667em}{0ex}}k,p\left(x\right)}$ . J. Math. Anal. Appl. 262, 749–760 (2001) · Zbl 0995.46023 · doi:10.1006/jmaa.2001.7618 [14] Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295–318 (1999) · Zbl 0927.46022 · doi:10.1016/S0362-546X(97)00628-7 [15] Fan, X., Zhao, D.: The quasi-minimizers of integral functionals with $m\left(x\right)$ growth conditions. Nonlinear Anal. 39, 807–816 (2000) · Zbl 0943.49029 · doi:10.1016/S0362-546X(98)00239-9 [16] Fan, X., Zhang, Q.: Existence of solutions for $p\left(x\right)$ -Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003) · Zbl 1146.35353 · doi:10.1016/S0362-546X(02)00150-5 [17] 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) [18] 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) · Zbl 1046.46027 · doi:10.4171/ZAA/1179 [19] Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: Sobolev capacity on the space ${W}^{1,p\left(·\right)}\left({\mathrm{I}\phantom{\rule{-2.0pt}{0ex}}\mathrm{R}}^{n}\right)$ . J. Funct. Spaces Appl. 1(1), 17–33 (2003) [20] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993) [21] Hudzik, H.: The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces ${W}_{M}^{k}\left({\Omega }\right)$ . Comment. Math. Prace Mat. 21, 315–324 (1980) [22] Hästö, P.: On the density of continuous functions in variable exponent Sobolev space. Rev. Mat. Iberoamericana, to appear [23] 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) [24] Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn., Math. 23, 261–262 (1998) [25] Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12, 233–247 (2000) · Zbl 0962.46021 · doi:10.1023/A:1008601220456 [26] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, London (1980) [27] Kováčik, O., Rákosník, J.: On spaces ${L}^{\phantom{\rule{0.166667em}{0ex}}p\left(x\right)}$ and ${W}^{1,p\left(x\right)}$ . Czechoslov. Math. J. 41(116), 592–618 (1991) [28] 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) [29] Marcellini, P.: Regularity and existance of solutions of elliptic equations with $p,q$ -growth conditions. J. Differ. Equ. 50(1), 1–30 (1991) · Zbl 0724.35043 · doi:10.1016/0022-0396(91)90158-6 [30] Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin Heidelberg New York (1983) [31] Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3, 200–212 (1931) [32] Pick, L., Ružička, 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) [33] 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) [34] Rudin, W.: Functional Analysis, TMH Edition, 14th Reprint. Tata McGraw-Hill, New Delhi (1990) [35] Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin Heidelberg New York (2000) [36] Samko, S.: Denseness of ${C}_{0}^{\infty }\left({\mathrm{I}\phantom{\rule{-2.0pt}{0ex}}\mathrm{R}}^{n}\right)$ in the generalized Sobolev spaces ${W}^{\phantom{\rule{0.166667em}{0ex}}m,p\left(x\right)}\left({\mathrm{I}\phantom{\rule{-2.0pt}{0ex}}\mathrm{R}}^{n}\right)$ (Russian). Dokl. Ross. Acad Nauk 369(4), 451–454 (1999); translation in Dokl. Math. 60, 382–385 (1999) [37] Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45(3), 1021–1050 (2001) [38] Sharapudinov, I.I.: The topology of the space ${ℒ}^{p\left(t\right)}\left(\left[0,1\right]\right)$ . Mat. Zametki 26(4), 613–632 (1979) [39] 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) [40] 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) [41] Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3(2), 249–269 (1995)