×

On a singular perturbation problem. I: Elasticity. (Research announcement). (Sur un problème de perturbation singulière. I: Elasticité.) (French) Zbl 1047.34512

The author considers the singular perturbation problem \(\varepsilon^2(u_{\varepsilon}^{(4)}- u_{\varepsilon}^{(2)})- u_{\varepsilon}^{(2)}+ u_{\varepsilon}=f\), \(0<x<\pi\) subject to some boundary conditions at \(x=0\) and at \(x=\pi\). This problem arises in elasticity theory. The paper deals with the convergence of \(u_{\varepsilon}\) when \(f\) is in the Sobolev space \(H^{\beta}((0,\pi))\), \(\beta>\frac12\). It is shown that the limit function solves the problem \(-u''+u=f\), \(u(0)=u(\pi)=0\).

MSC:

34E15 Singular perturbations for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
74B05 Classical linear elasticity
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: EuDML