Asymptotic Behavior For Weak Solutions Of A 4-order System Associated With Biharmonic Maps

In this paper, we study a second order energy model appearing in the theories of phase transition.In the study of thin film blisters and smectic type liquid crystals, the model is called a simplized Oseen-Frank energy: Here G(?)Rn is a smooth, bounded and simply connected domain,?>0 is a small parameter. Its minimizer u? in H2(G,Rn) satisfies the following Euler-lagrangeuation We call the H2-weak solution u? a minimal solution,if it is a minimizer of E?(u,G) in Hg2(G,Rn), where g?C?((?)G,Sn-1). In the theories of phase transition, u? is called an order parameter. The paper is concerned with the relation between a biharmonic map and some solution of a 4-order system when a parameter tends to zero. We prove that there exists a subsequence of the minimal solution of the 4-order system, converging locally to a biharmonic map in Cl sense for any ??1, when the degree of the boundary data around the boundary is zero.