Study of a non-classical singular perturbation problem. (Etude d’un problème de perturbation singulière non classique.) (French) Zbl 0902.35011

Benkiran, Taoufiq

Boll. Unione Mat. Ital., VII. Ser., A 11, No. 1, 93-103 (1997).

Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^n\) with a smooth boundary \(\Gamma\). This paper deals with the behavior of the solution \(u_\varepsilon\) of the perturbed problem \[ (\varepsilon A+B)u= f_\varepsilon\quad\text{in }\Omega,\quad B_1u= g_1\quad\text{and}\quad B_2u= g_2\quad\text{on }\Gamma,\tag{P} \] where \(\varepsilon\) is a small parameter which tends to zero. The limit problem is \[ Bu= f\quad\text{in }\Omega,\quad B_1u= g_1\quad\text{on }\Gamma.\tag{L} \] Here \(A\) and \(B\) are linear elliptic operators of order 4 and 2 respectively, \(B_1\) and \(B_2\) are boundary operators of order 2 and 3 respectively.
In the first section, it is assumed that \(f_\varepsilon\) and \(f\) are smooth enough, (P) and (L) have a unique solution, \(f_\varepsilon\) converges to \(f\) in \(H^s(\Omega)\), \(s>1/2\). The author shows that \(u_\varepsilon\) converges to \(u\) in a suitable Sobolev space. The proof is based on a proper convergence theorem due to F. Stummel [Math. Ann., II. Ser. 190, 45-92 (1970; Zbl 0203.45301)] and D. Huet [Asymptotic Anal. 2, 5-19 (1989; Zbl 0704.35002)].
The second section is devoted to simple one-dimensional examples, where \(f\) is not smooth enough or \(f_\varepsilon\) does not converge to \(f\) in \(H^s(\Omega)\), \(s>1/2\).

Reviewer: D.Huet (Nancy)

MSC:

35B25	Singular perturbations in context of PDEs
35J40	Boundary value problems for higher-order elliptic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

proper convergence

Citations:

Zbl 0203.45301; Zbl 0704.35002