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Some applications of the least squares method to differential equations and related problems

The first part of this paper contains an overview of the least squares method applied to various problems in ordinary and partial differential equations. In particular, we discuss various applications to the homogenization of transport equations, to the characterization of entropy solutions to scalar conservation laws and to the asymptotic behaviour of the action functional obtained through the reaction-diffusion approximation of mean curvature flow. In the last part of the paper we introduce and discuss the related problem of the quasi-potential for scalar conservation laws.