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2014 | 34 | 1 | 89-103
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On the solvability of Dirichlet problem for the weighted p-Laplacian

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In this paper we are concerned with the existence and uniqueness of the weak solution for the weighted p-Laplacian. The purpose of this paper is to discuss in some depth the problem of solvability of Dirichlet problem, therefore all proofs are contained in some detail. The main result of the work is the existence and uniqueness of the weak solution for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the solution belongs to the Sobolev space $W₀^{1,p}(Ω)$.
Twórcy
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
Bibliografia
  • [1] J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936) 396-414. doi: 10.1090/S0002-9947-1936-1501880-4
  • [2] P.A. Cojuhari, Generalized Hardy type inequalities and some applications to spectral theory, Operator theory, operator algebras and related topics, Theta Found., Bucharest (1997) 79-99.
  • [3] P. Cojuhari and A. Gheondea, Closed embeddings of Hilbert spaces. J. Math. Anal. Appl. 369 (2010) 60-75. doi: 10.1016/j.jmaa.2010.02.027
  • [4] P. Cojuhari and, A. Gheondea, Closely embedded Kreĭn spaces and applications to Dirac operators, J. Math. Anal. Appl. 376 (2) (2011) 540-550. doi: 10.1016/j.jmaa.2010.10.059
  • [5] J.R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1961) 436-446. doi: 10.1090/S0002-9947-1967-0217574-1
  • [6] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265-292. doi: 10.1090/S0002-9947-1947-0021241-4
  • [7] A. Kufner, Weighted Sobolev Spaces (Teubner-Texte zur Mathematik, Band 31, 1980).
  • [8] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961) 29-43. doi: 10.1090/S0002-9947-1961-0133024-2
  • [9] V. Smulian, Sur la dérivabilité de la norme dans l'espace de Banach, Dokl. Akad. Nauk SSSR 27 (1940) 643-648.
  • [10] E. Szlachtowska, On weak solutions of Dirichlet problem for weighted p-Laplacian, Opuscula Math. 32 (4) (2012) 775-781. doi: 10.7494/OpMath.2012.32.4.775
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1156
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