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In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting Euler-Lagrange equation will be studied by means of the analytical methods of set-valued analysis.

Continuous dependence for solution classes of Euler-Lagrange equations generated by linear growth energies