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In this paper, we focus on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belonging to L1 -data, based on the properties of the renormalized solutions and the generalized monotonicity method in the functional spaces.

In this paper, we study a general class of nonlinear anisotrpic elliptic problems associated with the differential inclusion β(u) − div(a(x, D(u) + F(u)) 3 f in Ω, where f ∈ L∞(Ω). A vector field a(., .) is a Carath´eodory function. Using trunction techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for L∞ − data. [ABSTRACT FROM AUTHOR]