# 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

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topDominik 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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- [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 Zbl0102.32701
- [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] 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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