Structural stability for nonlinear elliptic problems of the $p(x)-$ and $p(u)$-laplacian kind

Boris Andreianov (1) , Mostafa Bendahmane (2) , Stanislas Ouaro (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 CI²MA - Centro de Investigación en Ingeniería Matemática [Concepción]
3 LAME - LAME Ouagadougou, Burkina-Faso

This work consists of two parts. In the first one, we prove the structural stability (i.e., the continuous dependence on the coefficients) of solutions of the elliptic problems under the form $b(u_n)-\div \mathfrak{a}_n(x,\Grad u_n)=f_n$ in a bounded domain $\Om$ of $\R^N$ with homogeneous Dirichlet boundary data on $\ptl\Om$. Here $b$ is a non-decreasing function on $\R$, and $\Bigl(\mathfrak{a}_n(x,\xi)\Bigr)_n$ is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent $p_n(x)$, $1

Mots clés: p(x)-laplacian; Leray-Lions operator; variable exponent; thermo-rheological fluids; well-posedness; continuous dependence; convergence of minimizers; Young measures

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Mostafa Bendahmane, Stanislas Ouaro. Structural stability for nonlinear elliptic problems of the $p(x)-$ and $p(u)$-laplacian kind. 2009. ⟨hal-00363284v2⟩