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

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