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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.

  • AMS Subject Headings

35K45, 35K61, 35K65

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COPYRIGHT: © Global Science Press

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@Article{ATA-30-318, author = {Y. Akdim and C. Allalou}, 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 AU - Y. Akdim & C. Allalou 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 and C. Allalou. (2014). 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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