SOME RECENT RESULTS ON THE EQUATION OF PRESCRIBED GAUSS CURVATURE

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1 215 SOME RECENT RESULTS ON THE EQUATION OF PRESCRIBED GAUSS CURVATURE John I.E. Urbas In this article we discuss some recently established results concerning convex solutions u E c 2 (Q) of the equation of prescribed Gauss curvature (1) K(x) (1 + loul2> (n+2)/2 Here Q is a domain in lrn, Du and o 2 u denote the gradient and the Hessian of the function u, and K(x) denotes the Gauss curvature of the graph of u at (x,u(x)), which we shall assume is positive in n We start with a necessary condition for the existence of a convex c 2 (Q) solution of (1). If u is such a solution, then the gradient mapping Du : n -+ lrn is one to one with Jacobian 2 det D u, so by integrating (1) we obtain K= w n where w is the measure of the unit ball in lrn. Thus the condition n (2) is necessary for the existence of a convex solution 2 U E C (n) of (1).

2 216 The first prob1em we consider is the Dirichlet problem for (1), which was recently studied by P.L. Lions [9], [10], Trudinger and Urbas [12] and Ivochkina [5]. The following theorem was proved in [12]. THEOREM 1: Let n be a c 1 ' 1 uniformly convex domain in lrn 3 ~ E c 1 ' 1 <n> and K E c 1 ' 1 {n) a positive function such that (3) w n and (4) K(x) ~ ~ dist(x,an> for some positive constant ~. Then the classical Dirichlet problem (5) det D2u = K(x) (1 + jduj 2) (n+2)/2 in n U = ~ On an 1 has a unique convex solution u Theorem 1 can be obtained from the results of Lions [9], [10] as in [12], or directly from the results of Caffarelli, Nirenberg and Spruck [2], Krylov [6], [7], f8] and Ivochkina [5] on the existence of globally smooth solutions of the Dirichlet problem for equations of Monge-Ampere type, by using the interior second derivative estimate established in [13]. The existence of a convex solution u E c 2 <n> n c 0 (Q) of (5) was proved under the hypotheses of Theorem 1 by Lions [9], [10], and the case ~ = 0 was also proved by Gilbarg and Trudinger [4]. The Dirichlet problem for convex generalized solutions of (5) was studied by Bakelman [1], who proved a generalized version of Theorem 1. Additional references to this work are given in [1]. The condition (4) causes the equation (1) to become degenerate near an 1 Which precludes US from Obtaining globally SmOOth SOlUtiOnS Of (5). However, a partial result on the global regularity of convex solutions of (5) is given in [12]. Specifically, if an c 2 ' 1 I~= 0 and

3 217 (modulo convex functions), 'chen the convex The existence of globally smoo th convex solutions of (5) was :recen tly established by Ivochkina [5]. Her hypotheses,3.re different to the ones of 'fheorem 1; in particular, K is assumed 'co be bounded away from zero in :\6 and a restriction on the size of l 1 2 ;:\6 is necessary. The sharpness of the condition (4) for the classical solvability of the Dirichlet p:o:-oblem (5) for arbitrary smooth boundary data, at least in terms of power functions, is shown in [12] using a barrier _a:cgument. Related to 'chis is the follmving global Holdei: es i:imate which is proved in [15], and v-rhich yields nonexistence results for 'che Dirichle-t problem (5). THEOREI~ 2: Let n be a,l.bounded domain. in and u E em a convex solution of (1), where K.c;atisfies (6) for some cons1':ants )1 > o and S E [ 0, l). Then (7) I I I I (l-sl/2n u (x) - u (y), :;; C x - y, 1uhere c depends on n, )1 ~ s and n. This result is an extension of the global oscillation estima te proved in [14], and is proved by a careful application of the barrier technique us<=d 'chere. Although we cannot generally satisfy the boundary condition in (5) in the classical sense if (4) is not satisfied, it is possible to satisfy it in a certain optlmal or generalized sense. This lftras proved Bakelman [l] for generalized solutions. In [15] we have established the following result for smooth solutions.

4 218 THEOREM 3: Let n be a c 1 ' 1 unifoy'rrlly convex domain in IRn, cp E c 1 1 and K E c 1 ' 1 <m n r~p(rl), p > n, a positive function satisfying (3). Then there is a unique convex funetion u E c 2 (nj n L 00 (Q) such -that u solves (1) in Q, (8) lim sup u(x) ~ cp(y) x-+y for all y E ds-2, and if v E c 2 (Q) n L 00 (QJ is another convex solution of (1), and lim SUp v(x) ~ <fl(y) for all y E ()Q ~ -!;hen V ~ U in Q x-+y The func tion u is therefore the supremum of the convex subsolu-ti.ons of (l) which lie below on ()Q, and the proof of the theorem shows that u is also the infimum of the convex supersolutions of (l) which lie above on ()Q To prove Theorem 3 ;~e solve approximating Dirichlet problems with boundary values and obtain a sequence of (Q) convex functions converging in (Q) to a convex generalized solution u of (l), 'ltjhich satisfies (8) and t_he final conclusion of the theorem" To deduce the regularity of u we first use some measure theory to obtain information about the behaviour of u near :m, and then use a s tandard method of Pogorelov [11] and Cheng and Yau [3]. where while if K u e CO,(l-8)/2n(~,...O,l(r, U E ~,, n BE/ 2 and satisfies (6) in Q n B (x 0 ), then n BE/2(x0)) If K satisfies (4) in Q n BE The final theorem we men tion summarizes the results we have proved in [14], [15] in the case u = (9) f Q K=tu n THEOREM 4: Let n be a uniformly convex domain in IR.n a:ttd K E c 1 ' 1 (Q) n LP(Q), p > n, a positive function satisfying (9). there is a convex solution u e c 2 (Q) such solutions differ by a constant. Then of the equation (1), and any -~;o

5 219 To prove Theorem 4, we first obtain a generalized solution u of (1), which is done by solving approximating Dirichlet problems and passing to a limit with the help of an interior oscillation estimate, for example, Theorem 2 applied to smooth compactly contained subdomains of Q. The regularity proof is similar to that in 'rheorem 3, and the uniqueness assertion follows from a comparison principle. If K satisfies (4) near a point X E (\Q 9 then 0 lim u (x) x-+:ic 0 oo while if K sa'cisfies (6) near x 0 E (lq, and near then u is Holder continuous there wi th exponent (l-i3)/2n Finally, "''e mention that in [15], these resul ts have been extended to l'ylonge-ampere equations of the form det f(x,u,du), under suitable hypotheses on f. REFERENCES [l] I. Bakelman, 'The Dirichlet problem for the n-dirnensional Monge- Ampere equations and related problems in the theory of quasilinear equations', Institute N.azionale di Alta Matematica, Roma (1982), [2] L. Ca.ffarelli, L. Nirenberg, and J. Spruck, 'The Dirichlet problem for nonlinear second order elliptic equations, I. Monge-Arr~ere equa tion', Corron. Pu1~e Appl. Math. :2.1_ (1984), [3] s.-y. Cheng and s.-t. Yau, 'On the regularity of the Monge-Ampere equation det ( () 2 u;ax. ax.) l J (1977), F(x,u) ', Comrfl. Pure.4ppl. Math. 29 [4] D. Gilbarg and N.S. Trudinger, Elliptic pcu~tial differential

6 220 equat1:ons of second order, 2nd edition, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, [5] l'lm. Ivochkina, 'Classical solvability of the Dirichlet problem for equa tion', Zap. Naucn. Sem. Lening.t'ad~ Otdel. Mat. Inst. StekLov (LOMI) (1983), [ 6] N. V. Krylov, 'Boundedly inhomogen.eous elliptic and parabolic equations', Izvest ia Akad. Nauk. SSSR (1982)' [7] N.V. Krylov, 'Boundedly inhomogeneous elliptic and parabolic equations in a domain', Izvestia Akad. Nauk. SSSR fl (1983), [8] N. V. Krylov, 'On degenera te elliptic equat~ions', Mat. Sb..:!:20. (1983)' [9] P.L. Lions, 'Sur les equations de Mange-Ampere I", ManuscYM~pta Nlai;h. ~1 (1983), [10] p,l Lions' I Sur les equations de l!'longe-a.'llpere II'' Ax cho Rat. Mech. AnaL (to appear), [ll] A.V. Pogorelov, The MinkouYski ;mdtidimensionaz problem, Wiley, New York, [12] N.S. Trudinger and J.I.E. Urbas, 'The Dirichlet problem for t.he equation of prescribed Gauss curvaoture', Bull. Austral. Math.Soc. 28 (1983), [13] N.S. Trudinger and J.I.E. Urbas, 'On second derivative estimates for equations of l!'longe-ampere type', BuZ L. Aust1~aL Math. Soc. [14] J.I.E. Urbas, 'The equation of prescribed Gauss curvature ;~ithout boundary conditions', J. Diff. Geom. ( to appear). [15 J.I.E. Urbas, 'Elliptic equations of Monge-Ampere type', Thesis, Australian National University, Department of Mathematics Faculty of Science Australian National University