科研之友
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摘要
Given a connected Lipschitz domain Omega we let Lambda(Omega) be the set of functions in W-2,W-2 (Omega) with u = 0 on partial derivative Omega and whose gradient (in the sense of trace) satisfies del u (x) . eta(x) = 1, where eta(x) is the inward pointing unit normal to partial derivative Omega at x. The functional I-is an element of(u) = 1/2 integral(Omega) is an element of(-1)| 1 -|del u|(2) |(2) + is an element of |V(2)u|(2) dz, minimised over Lambda(Omega), serves as a model in connection with problems in liquid crystals and thin film blisters. It is also the most natural higher order generalisation of the Modica and Mortola functional. In [16] Jabin, Otto and Perthame characterised a class of functions which includes all limits of sequences u(n) is an element of Lambda(Omega) with I-is an element of n (u(n)) -> 0 as is an element of(n) -> 0. A corollary to their work is that if there exists such a sequence (u(n)) for a bounded domain Omega, then Omega must be a ball and (up to change of sign) u := lim(n ->infinity) u(n) is equal dist(., partial derivative Omega). We prove a quantitative generalisation of this corollary for the class of bounded convex sets. Namely we show that there exists a positive constant yi such that, if Omega is a convex set of diameter 2 and u is an element of Lambda(Omega) with I-is an element of(u) = beta then |B-1 (x)Delta Omega| <= c beta(gamma 1) for some x and
integral(Omega)|del u(z) + z-x'|z-x||(2) dz <= c beta(gamma 1)
xi A corollary of this result is that there exists a positive constant gamma(2) < gamma(1) such that if Omega is convex with diameter 2 and C-2 boundary with curvature bounded by is an element of(-1/2), then for any minimiser v of I-is an element of over Lambda(Omega) we have
||v - zeta|| w(1,2)(Omega) <= c(is an element of + inf(y) |Omega Delta B-1(y)|)gamma(2), where zeta(z) = dist(z, partial derivative Omega). Neither of the constants gamma(1) or gamma(2) are optimal.
