3-PHASE BOUNDARY MOTIONS UNDER CONSTANT VELOCITIES .1. THE VANISHING SURFACE-TENSION LIMIT

In this paper, we deal with the dynamics of material interfaces such a s solid-liquid, grain or antiphase boundaries. We concentrate on the s ituation in which these internal surfaces separate three regions in th e material with different physical attributes (e.g. grain boundaries i n a polycrystal). The basic two-dimensional model proposes that the mo tion of an interface Gamma(ij) between regions i and j (i, j = 1, 2, 3 , i not equal j) is governed by the equation (0.1) V-ij = mu(ij)(f(ij) k(ij) + F-ij). Here V-ij, k(ij), mu(ij) and f(ij) denote, respectively , the normal velocity, the curvature, the mobility and the surface ten sion of the interface and the numbers F-ij stand for the (constant) di fference in bulk energies. At the point where the three phases coexist , local equilibrium requires that (0.2) the curves meet at prescribed angles. In case the material constants f(ij) are small, f(ij) = <(epsi lon f)over cap (ij)> and epsilon much less than 1, previous analyses b ased on the parabolic nature of the equations (0.1) do not provide goo d qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with f(ij) = 0. It turns out that this problem, (0.1) with f(ij) = 0, admits infinitel y many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution-'the vanishing surface tension (VST) solution'-is selected by letting epsilon --> 0. Indeed, a formal analy sis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with epsilon = 0. As we show, this weak solution is unique and is therefore expected to coincide wit h the VST solution. To support this statement, we present a perturbati on analysis and a construction of self-similar solutions; a rigorous c onvergence result is established in the case of symmetric configuratio ns. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VS T solution differs from results proposed previously.