Structural stability for variable exponent elliptic problems. I. The $p(x)$-laplacian kind problems.

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

We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form $$b(u_n)-\div mathfrak{a}_n(x,\Grad u_n)=f_n.$$ The equation is set in a bounded domain $\Om$ of $\R^N$ and supplied with the homogeneous Dirichlet boundary condition 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 variable exponent elliptic problems. I. The $p(x)$-laplacian kind problems.. Nonlinear Analysis: Theory, Methods and Applications, 2010, 73 (1), pp.2-24. ⟨10.1016/j.na.2010.02.039⟩. ⟨hal-00363284v3⟩