Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

Abstract

We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.Key words: free boundary, Stefan problem, classical solvability, porous medium equation, degenerate parabolic equations. To the memory of Professor B.V.BazaliyThe final publication is available at Springer via http://dx.doi.org/10.1007/s00030-014-0280-3 1 Statement of the problem and the main result Classical solvability of the Stefan problem for uniformly parabolic equations has been well studied -see for example papers [1] -[6] and the references therein. At the same time, as has long been known, the heat transfer model based on uniformly parabolic equations, has some properties which can not be observed in the reality, in particular, the infinite speed of propagation of disturbances. We also know that more accurate model of the heat transfer is the model which is based on degenerate parabolic equations, such as equations of the form where β(u) is a discontinuous function of the form 1

Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting

“…(1.15) As such a function can serve, for example, the bounded solution of the problem ∆d(x) = −1, x ∈ Ω, d(x)| ∂Ω = 0. Note that weighted seminorm (1.16) is equivalent to the usual Hölder seminorm with respect to some Carnot-Caratheodory metric for equation (1.1) (see [26], [25], [15], [22] for the definitions and see [24], [22] for the equivalence).…”

“…Formulate now the main result of the paper. The method of the proving of Theorem 2 consists of reducing of the problem to some nonlinear operator equation and applying the Inverse Function theorem as it was done in [24]. So we formulate a variant of such theorem.…”

“…We will show below, that the free (unknown) boundary S T can be parameterized in terms of its deviation from the given surface Γ T = Γ × [0, T ]. We follow to [28] to give the exact formulation (compare [28], [24]).…”

“…Let us explain for completeness the obtaining of conditions (3.8), (3.10) from conditions (1.1), (1.3) -compare [24]. Denote by (ω y , λ y ) the (ω, λ)-coordinates of the point y and by (ω x , λ x ) the (ω, λ)-coordinates of the point x in N .…”

“…Note that weighted seminorm (1.16) is equivalent to the usual Hölder seminorm with respect to some Carnot-Caratheodory metric for equation (1.1) (see [26], [25], [15], [22] for the definitions and see [24], [22] for the equivalence). Define the space C γ γ/2 (Ω) as the space of functions u(x) with the finite norm…”