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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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