Weak solution to a phase-field transmission problem in a concentrated capacity

Singular limit of a transmission problem for the parabolic phase-field model

“…The ε-limit procedure is now the same as before, except for the α ε i -parts which require a new estimate. First of all, we recall a result of [15], adapting the compatibility assumption (19) to the Yosida approximate graphs α ε i .…”

“…As in our former paper [15], we suppose that D(α 1 ) = D(α 2 ) =: D 1,2 , where, by definition, D(α i ) = {x ∈ Ê: α i (x) = ∅} is the domain of α i , and that there exists a fixed constant C 1 such that…”

“…We omit the checkings which are very similar to the corresponding ones in [15]. We just point out that, thanks to the density of V ∞ in V , we are allowed to take the test functions v for the limit equations in the whole space V .…”

“…Now, the proof is analogous to that of a similar result of [15], to which we refer again for more details.…”

The Penrose-Fife phase-field modelwith coupled dynamic boundary conditions

“…'s, the walls of the the container have a significant effect on the phase transition process. This may include the case of two phase changing substances in contact, where one occupies a small layer surrounding the other one and this layer is approximated by a surface in the mathematical model (concentrated capacity phenomenon, see, e.g., [37], [38] for more details).…”