Viscosity solutions for a polymer crystal growth model

Abstract

We prove existence of a solution for a polymer crystal growth model describing the movement of a front (Γ(t)) evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source δΓ. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with Hölder bounds in space. From this result, we deduce a priori regularity for the front. On the other hand, under this regularity assumption, we prove bounds and regularity estimates for the solution of the heat equation. 1991 Mathematics Subject Classification. 49L25, 35F25, 35A05, 35D05, 35B50, 45G10.We introduce the following set of assumptions, denoted by (A) in the rest of the paper. (A1) κ is a fixed real number (κ is positive in the case of a negative heat source and negative otherwise),ḡ : R N → R is Lipschitz continuous, bounded, and there exist A, B > 0 such that A ≤ḡ(z) ≤ B for all z ∈ R .(A2) v 0 : R N → R is Lipschitz continuous and bounded.

Regularity of expanding front and its application to solidification/melting in undercooled liquid/superheated solid

“…It improves the estimate for the length of trajectory from Su and Burger's l C O.l˛/ to its optimal l C O.l 1C2ˇ/ . Similar estimate has also been established in [8] with Hölder exponent2 under a different setting. The most significant application of this new estimate is that it allows us to construct efficiently comparison particle motion paths to show that .s/ near x .s/ is sandwiched between two C 1C˛=2 graphs touching at x .s/, for each s 2 .0; t/; technically, writing p D x .s/ and n D n .s/, we can prove the following inclusions: where C D C .n; c;˛/ is a constant depending only on n; c and˛.…”

“…Their methods of proof, based on an interior sphere property of the attainable sets, are totally different from ours, and they also require somewhat stronger condition on expanding speed and initial shape. Recently, similar regularity results have also been obtained by Cardaliaguet, Ley and Mointeiliet [8] where in addition to the Hölder continuity of the expanding speed, they needed stronger condition jv.x; t/ v.y; t/j 6 C jy xj.1 C j ln jx yjj/:…”

Regularity results for eikonal-type equations with nonsmooth coefficients

“…An essential step of the analysis is the C 1,α -regularity of the extremal trajectories of (2), see Theorem 4.1. For time-dependent and isotropic Hamiltonians (H = a(t, x)|p|), such a regularity property-interesting in its own right-has already been observed in [2] for N = 2, and [6] for general N . However, the unexpected connection between Theorem 4.1 and the semiconcavity of the solution of ( 1) is, to our best knowledge, entirely new.…”