In this paper we consider a steady state phase transition problem with given convection v. We prove, among other things, that the weak solution is locally Lipschitz continuous provided that $\mathbf{v}=D\xi $ and ξ is a harmonic function. Moreover, for continuous casting problem, i.e. when v is constant vector, we show that Lipschitz free boundaries are ${C}^{1}$ regular surfaces.

Classification: 35R35, 35J60, 35R37, 80A22

Keywords: Free boundary, Stefan problem, Phase transition, Convection, Lipschitz regularity, Viscosity solution

@article{AIHPC_2015__32_4_715_0, author = {Karakhanyan, Aram L.}, title = {Optimal regularity for phase transition problems with convection}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {32}, number = {4}, year = {2015}, pages = {715-740}, doi = {10.1016/j.anihpc.2014.03.003}, zbl = {1329.35361}, mrnumber = {3390081}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2015__32_4_715_0} }

Karakhanyan, Aram L. Optimal regularity for phase transition problems with convection. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 715-740. doi : 10.1016/j.anihpc.2014.03.003. http://www.numdam.org/item/AIHPC_2015__32_4_715_0/

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