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Volume 30, Issue 3
Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

Y. Akdim & C. Allalou

Anal. Theory Appl., 30 (2014), pp. 318-343.

Published online: 2014-10

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  • Abstract

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion $\beta(u)-div(a(x,Du)+F(u))\ni f$ in $\Omega $, where $f\in L^{1}(\Omega )$. A vector field $a(\cdot,\cdot)$ is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general $L^1$-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.

  • Keywords

Weighted Sobolev spaces, Hardy inequality, Truncations, maximal monotone graph, degenerated elliptic operators.

  • AMS Subject Headings

35K45, 35K61, 35K65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-30-318, author = {}, title = {Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems}, journal = {Analysis in Theory and Applications}, year = {2014}, volume = {30}, number = {3}, pages = {318--343}, abstract = {

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion $\beta(u)-div(a(x,Du)+F(u))\ni f$ in $\Omega $, where $f\in L^{1}(\Omega )$. A vector field $a(\cdot,\cdot)$ is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general $L^1$-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2014.v30.n3.8}, url = {http://global-sci.org/intro/article_detail/ata/4496.html} }
TY - JOUR T1 - Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems JO - Analysis in Theory and Applications VL - 3 SP - 318 EP - 343 PY - 2014 DA - 2014/10 SN - 30 DO - http://doi.org/10.4208/ata.2014.v30.n3.8 UR - https://global-sci.org/intro/article_detail/ata/4496.html KW - Weighted Sobolev spaces, Hardy inequality, Truncations, maximal monotone graph, degenerated elliptic operators. AB -

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion $\beta(u)-div(a(x,Du)+F(u))\ni f$ in $\Omega $, where $f\in L^{1}(\Omega )$. A vector field $a(\cdot,\cdot)$ is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general $L^1$-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.

Y. Akdim & C. Allalou. (1970). Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems. Analysis in Theory and Applications. 30 (3). 318-343. doi:10.4208/ata.2014.v30.n3.8
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