Singular perturbations for systems of linear partial differential equations

Livne, A.; Schuss, Z.

doi:10.1090/S0002-9947-1974-0340780-6

Singular perturbations for systems of linear partial differential equations
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by A. Livne and Z. Schuss PDF

Trans. Amer. Math. Soc. 190 (1974), 335-343 Request permission

Abstract:

We consider the system of linear partial differential equations $\varepsilon {A^{ij}}u_{ij}^\varepsilon + {B^i}u_i^\varepsilon + C{u^\varepsilon } = f$ where ${A^{ij}},{B^i}$ are symmetric $m \times m$ matrices and — C is a sufficiently large positive definite matrix. We prove that under suitable conditions ${\left \| {{u^\varepsilon } - u} \right \|_{{L^2}}} \leq c\surd \varepsilon {\left \| f \right \|_{{H^1}}}$ where u is the solution of a suitable boundary value problem for the system ${B^i}{u_i} + Cu = f$.

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Perturbations singulières et prolongements maximaux d’opérateurs positifs

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Regular degeneration and boundary layer for linear differential equations with a small parameter

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