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Weak solution for obstacle problem with variable growth. (English) Zbl 1064.46022

Summary: In this paper, we introduce the weighted spaces \(L^{p(x)}(\Omega,\omega)\) and \(W^{k,p(x)}(\Omega,\omega)\). After discussing the properties of these spaces, we obtain the existence and uniqueness of weak solutions for obstacle problem with variable growth in the setting of these spaces.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

[1] Adams, R., Sobolev Spaces (1975), Academic Press: Academic Press London · Zbl 0314.46030
[2] J. Chabrowski, The Dirichlet Problem with \(L^2\)-Boundary Data for Elliptic Linear Equations, Lecture Notes in Mathematics, vol. 1482, Springer, Berlin, 1991.; J. Chabrowski, The Dirichlet Problem with \(L^2\)-Boundary Data for Elliptic Linear Equations, Lecture Notes in Mathematics, vol. 1482, Springer, Berlin, 1991. · Zbl 0734.35024
[3] Fan, X.; Dun, Z., A class of De Giorgi type and Hölder continuity of minimizers of variational functionals with \(m(x)\)-growth conditions, Nonlinear Anal., 36, 295-318 (1999) · Zbl 0927.46022
[4] Fan, X.; Dun, Z., The quasi-minimizer of integral functionals with \(m(x)\) growth conditions, Nonlinear Anal., 39, 807-816 (2000) · Zbl 0943.49029
[5] Fu Yongqiang, The existence of solutions for elliptic systems with nonuniform growth, Studia Math., 151, 227-246 (2002) · Zbl 1007.35023
[6] Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs (1993), Oxford University Press: Oxford University Press Oxford
[7] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[8] Kováčik, O.; Rákosnik, J., On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\), Czechoslovak Math. J., 41, 592-618 (1991) · Zbl 0784.46029
[9] Kovalevsky, A.; Nicolisi, F., Solvability of Dirichlet problem for a class of degenerate nonlinear high-order equations with \(L^1\)-data, Nonlinear Anal., 47, 435-446 (2001) · Zbl 1042.35528
[10] Kufner, A., Weighted Sobolev Spaces (1985), Wiley: Wiley New York · Zbl 0567.46009
[11] Maz’ya, V., Sobolev Spaces (1985), Springer: Springer Berlin · Zbl 0727.46017
[12] E. Stredulinsky, Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1074, Springer, Berlin, 1984.; E. Stredulinsky, Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1074, Springer, Berlin, 1984. · Zbl 0541.35001
[13] Zhikov, V., On weight Sobolev space, Mat. Sb., 189, 7-8, 1139-1170 (1998) · Zbl 0919.46026
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