On the solvability of Dirichlet problem for the weighted p-Laplacian

Dominik Mielczarek; Jerzy Rydlewski; Ewa Szlachtowska

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2014)

  • Volume: 34, Issue: 1, page 89-103
  • ISSN: 1509-9407

Abstract

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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 W ₀ 1 , p ( Ω ) .

How to cite

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Dominik Mielczarek, Jerzy Rydlewski, and Ewa Szlachtowska. "On the solvability of Dirichlet problem for the weighted p-Laplacian." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 34.1 (2014): 89-103. <http://eudml.org/doc/270397>.

@article{DominikMielczarek2014,
abstract = {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 $W₀^\{1,p\}(Ω)$.},
author = {Dominik Mielczarek, Jerzy Rydlewski, Ewa Szlachtowska},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {weighted p-Laplacian; weak solutions; solvability; semi-inner product spaces; weighted -Laplacian},
language = {eng},
number = {1},
pages = {89-103},
title = {On the solvability of Dirichlet problem for the weighted p-Laplacian},
url = {http://eudml.org/doc/270397},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Dominik Mielczarek
AU - Jerzy Rydlewski
AU - Ewa Szlachtowska
TI - On the solvability of Dirichlet problem for the weighted p-Laplacian
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2014
VL - 34
IS - 1
SP - 89
EP - 103
AB - 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 $W₀^{1,p}(Ω)$.
LA - eng
KW - weighted p-Laplacian; weak solutions; solvability; semi-inner product spaces; weighted -Laplacian
UR - http://eudml.org/doc/270397
ER -

References

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  3. [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 Zbl1198.46022
  4. [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 Zbl1216.46022
  5. [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 Zbl0157.20103
  6. [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. [7] A. Kufner, Weighted Sobolev Spaces (Teubner-Texte zur Mathematik, Band 31, 1980). 
  8. [8] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961) 29-43. doi: 10.1090/S0002-9947-1961-0133024-2 Zbl0102.32701
  9. [9] V. Smulian, Sur la dérivabilité de la norme dans l'espace de Banach, Dokl. Akad. Nauk SSSR 27 (1940) 643-648. Zbl0023.32604
  10. [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 Zbl1263.35084

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