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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.

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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