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Asymptotic series and the method of vanishing viscosity. (English) Zbl 0573.35034
We consider a second-order quasilinear elliptic partial differential equation in a bounded region \(\Omega\) of \({\mathbb{R}}^ N\) with zero data on the boundary \(\partial \Omega\). The second order term is Laplacian multiplied by a small coefficient \(\epsilon >0\). When \(\epsilon =0\) the equation degenerates to a first-order Hamilton-Jacobi equation. The solution \(u^{\epsilon}\) for \(\epsilon >0\) is expressed as an asymptotic series in powers of \(\epsilon\), with coefficients depending on the solution \(u^ 0\) for \(\epsilon =0\). This expansion is valid in subregions of \(\Omega\) where \(u^ 0\) coincides with a smooth classical solution obtained by the method of characteristics. The proof uses in several ways the maximum principle for elliptic differential equations.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35L50 Initial-boundary value problems for first-order hyperbolic systems
49J20 Existence theories for optimal control problems involving partial differential equations
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