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Stabilité et convergence dans les problèmes de perturbation singulière. (Stability and convergence in singular perturbation problems). (French) Zbl 0604.47006

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1986, Conf. No. 6, 9 p. (1986).
Let \(\epsilon\) be a positive small parameter, \(H_{\epsilon}\), \(F_{\epsilon}\) be normed spaces, and \(A_{\epsilon}\) be a linear operator from \(H_{\epsilon}\) to \(F_{\epsilon}\). The family \(A_{\epsilon}\) is said to be inversely stable, if there exists a positive constant C such that (1) \(\| u\|_{H_{\epsilon}}\leq C\| A_{\epsilon}u\|_{F_{\epsilon}}\) for all \(u\in H_{\epsilon}\), and all \(\epsilon\) sufficiently small. If \(A_{\epsilon}\) is bijective, we can consider the solution \(u_{\epsilon}\) of the equation (2) \(A_{\epsilon}u_{\epsilon}=f_{\epsilon}\) where \(f_{\epsilon}\) is given in \(F_{\epsilon}\). In many cases, the inverse stability of \(A_{\epsilon}\) implies the convergence of \(u_{\epsilon}\) to the solution u of an equation of the form (3) \(Au=f\), where A is a linear operator from a normed space H to a normed space F.
In the first part of the conference, I build a proper approximation of the usual Sobolev space \(H^{s_ 1}\) by spaces \({\mathcal H}^ s_{\epsilon}\), \(s=(s_ 1,s_ 2)\in {\mathbb{R}}^ 2\). Then, I show how the à-priori estimates of the form (1), obtained by L. S. Frank and W. D. Wendt, in \({\mathcal H}^ s_{\epsilon}\)-spaces [Ann. Mat. Pura Appl. 119, 41-113 (1979; Zbl 0468.35011), Commun. Partial Differ. Equations 7, 469-535 (1982; Zbl 0501.35007), J. Anal. Math. 43, 88-135 (1984; Zbl 0572.35040)], imply the convergence of the solution \(u_{\epsilon}\in {\mathcal H}^ s_{\epsilon}(\Omega)\) of equation (2) to the solution \(u\in H^{s_ 1}(\Omega)\) of equation (3).
In the second part of the conference, I consider operators \(A_{\epsilon}\) which satisfy an à-priori estimate of the form (4) \(\| u\|_{H_{\epsilon}}\leq C\{\| A_{\epsilon}u\|_{F_{\epsilon}}+\| u\|_{E_{\epsilon}}\}\) where \(E_{\epsilon}\) is a suitable normed space. I investigate sufficient conditions under which (4) implies the inverse stability of \(A_{\epsilon}\). Applications to differential problems in \(L_ p\) spaces, related to B. Najman’s results [Glas. Mat., III. Ser.] are given.

MSC:

47A55 Perturbation theory of linear operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47A50 Equations and inequalities involving linear operators, with vector unknowns
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