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On a class of obstacle problem via Young measure in generalized Sobolev space

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Abstract

This paper deals with the existence and uniqueness of weak solution for a class of obstacle problem of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\displaystyle \int _{\Omega }\mathcal {V}(x,Dw):D(\vartheta -w)\mathrm {~d}x+\displaystyle \int _{\Omega }\left\langle w\vert w\vert ^{p(x)-2},\vartheta - w\right\rangle \mathrm {~d}x\\ &{} \quad \ge \displaystyle \int _{\Omega }\mathcal {U}(x,w)(\vartheta -w)\mathrm {~d}x, \\ \;\\ &{} \vartheta \in \Im _{\Lambda , h}, \end{array}\right. } \end{aligned}$$

where \(\Im _{\Lambda , h}\) is a convex set defined below. By using the Young measure theory and Kinderlehrer and Stampacchia Theorem, we prove the existence and uniqueness result of the considered problem in the framework of generalized Sobolev space.

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Authors and Affiliations

  1. Laboratory LMACS, Faculty of Science and Technics, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco

    Mouad Allalou, Mohamed El Ouaarabi, Hasnae El Hammar & Abderrahmane Raji

  2. Fundamental and Applied Mathematics Laboratory, Faculty of Sciences Aïn Chock, Hassan II University, BP 5366, 20100, Casablanca, Morocco

    Mohamed El Ouaarabi

Authors
  1. Mouad Allalou
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  2. Mohamed El Ouaarabi
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  3. Hasnae El Hammar
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  4. Abderrahmane Raji
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Correspondence to Mohamed El Ouaarabi.

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Communicated by Julio Rossi.

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Allalou, M., El Ouaarabi, M., El Hammar, H. et al. On a class of obstacle problem via Young measure in generalized Sobolev space. Adv. Oper. Theory 9, 48 (2024). https://doi.org/10.1007/s43036-024-00349-2

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