Advances in Differential Equations

Uniqueness result for nonlinear anisotropic elliptic equations

Rosaria Di Nardo, Filomena Feo, and Olivier Guibé

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Abstract

We consider here a class of anisotropic elliptic equations, in a bounded domain $\Omega$ with Lipschitz continuous boundary $\partial \Omega$, of the type $$ -\sum_{i=1}^{N}\partial_{x_{i}}\big(a_{i}(x,u) |\partial_{x_{i}}u|^{p_{i}-2}\partial_{x_{i}} u\big) =f- {\rm div} g $$ with Dirichlet boundary conditions. Using the framework of renormalized solutions we prove the uniqueness of the solution under a very local Lipschitz condition on the coefficients $a_{i}(x,s)$ with respect to $s$ and with $f$ belonging to $L^1(\Omega)$.

Article information

Source
Adv. Differential Equations Volume 18, Number 5/6 (2013), 433-458.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1363266253

Mathematical Reviews number (MathSciNet)
MR3086461

Zentralblatt MATH identifier
1272.35092

Subjects
Primary: 35K55: Nonlinear parabolic equations 35K20: Initial-boundary value problems for second-order parabolic equations 35R05: Partial differential equations with discontinuous coefficients or data 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness

Citation

Di Nardo, Rosaria; Feo, Filomena; Guibé, Olivier. Uniqueness result for nonlinear anisotropic elliptic equations. Adv. Differential Equations 18 (2013), no. 5/6, 433--458. https://projecteuclid.org/euclid.ade/1363266253.


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