Nonconvex Variational Problems with General Singular Perturbations

We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of T-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in L1 as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem. Introduction. Let /: [0, oo) x IR —> R satisfy (HI) f(p,u)EC2, (H2) fp(0,u) = 0 VueR, (H3) there exists fci > 0 such that f(p, u) > kyp2 Vp E [0, oo), Vu E R, (H4) fpp(p, u)>0 Vp G [0, co), Vu E R, ,„.. f(Q, u) > 0 Vu E R, with equality if and only if u = a or b (-oo < ( ] a < b < oo), and /uu(0, a) ^ 0, fuu(0, b) ± 0, (H6) /(0, u) —> oo as u —► ±oo. Let fi be a bounded, open subset of R2 with Lipschitz continuous boundary. Define a sequence of functionals, Ie, by Ie(u)= f /(e|Vu|,u)dz, Jn where Vu = (du(xy,x2)/dxy,du(xy,x2)/dx2) and e > 0. For a fixed c E (a, b), consider the regular calculus of variations problem: (Pe) Minimize I£ on A = {u E W1'1^): fQudx = c,I£(u) < oo}. Here, W1'1(Sl) denotes the Sobolev space of functions mapping fi into R with integrable generalized derivative. In this paper we shall study the following problem: If ue is a solution of (P£) and ue —» u as £ —> 0, then what variational problem does u satisfy? By our convexity (H4) and growth assumptions (H3) on /, J£ is weakly lower semicontinuous and Received by the editors January 25, 1987 and, in revised form, September 20, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 34D15; Secondary 82A25. Research partially supported by National Science Foundation Grant DMS 8600710 under the Center for Applied Mathematics, Purdue University and National Science Foundation Grant DMS 8701448. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 393 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use