On the regularity of reflector antennas

Transcription

1 Annals of Matematics, 167 (2008), On te regularity of reflector antennas By Luis A. Caffarelli, Cristian E. Gutiérrez, and Qingbo Huang* 1. Introduction By te Snell law of reflection, a ligt ray incident upon a reflective surface will be reflected at an angle equal to te incident angle. Bot angles are measured wit respect to te normal to te surface. If a ligt ray emanates from O in te direction x S n 1, and A is a perfectly reflecting surface, ten te reflected ray as direction: (1.1) x = T (x) =x 2 x, ν ν, were ν is te outer normal to A at te point were te ligt ray its A. Suppose tat we ave a ligt source located at O, and Ω, Ω are two domains in te spere S n 1, f(x) is a positive function for x Ω (input illumination intensity), and g(x ) is a positive function for x Ω (output illumination intensity). If ligt emanates from O wit intensity f(x) for x Ω, te far field reflector antenna problem is to find a perfectly reflecting surface A parametrized by z = ρ(x) x for x Ω, suc tat all reflected rays by A fall in te directions in Ω, and te output illumination received in te direction x is g(x ); tat is, T (Ω) = Ω, were T is given by (1.1). Assuming tere is no loss of energy in te reflection, ten by te law of conservation of energy f(x) dx = g(x ) dx. Ω Ω In addition, and again by conservation of energy, te map T defined by (1.1) is measure-preserving: f(x) dx = g(x ) dx, for all E Ω Borel set, T 1 (E) E *Te first autor was partially supported by NSF grant DMS Te second autor was partially supported by NSF grant DMS Te tird autor was partially supported by NSF grant DMS

2 300 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG and consequently, te Jacobian of T is f(x). It yields te following nonlin- g(t (x)) ear equation on S n 1 (see [GW98]): (1.2) det ( ij u +(u η)e ij ) η n 1 det(e ij ) = f(x) g(t (x)), were u =1/ρ, = covariant derivative, η = u 2 + u 2, and e is te metric 2u on S n 1. Tis very complicated fully nonlinear PDE of Monge-Ampère type received attention from te engineering and numerical points of view because of its applications [Wes83]. From te point of view of te teory of nonlinear PDEs, te study of tis equation began only recently wit te notion of weak solution introduced by Xu-Jia Wang [Wan96] and by L. Caffarelli and V. Oliker [CO94], [Oli02]. Te reflector antenna problem in te case n =3,Ω S+, 2 and Ω S, 2 were S+ 2 and S 2 are te nortern and soutern emisperes respectively, was discussed in [Wan96], [Wan04]. Te existence and uniqueness up to dilations of weak solutions were proved in [Wan96] if f and g are bounded away from 0 and. Regularity of weak solutions was also addressed in [Wan96] and it was proved tat weak solutions are smoot if f, g are smoot and Ω, Ω satisfy certain geometric conditions. Xu-Jia Wang [Wan04] recently discovered tat tis antenna problem is an optimal mass transportation problem on te spere for te cost function c(x, y) = log(1 x y); see also [GO03]. On te oter and, te global reflector antenna problem (i.e., Ω = Ω = S n 1 ) was treated in [CO94], [GW98]. Wen f and g are strictly positive bounded, te existence of weak solutions was establised in [CO94] and te uniqueness up to omotetic transformations was proved in [GW98]. If f, g C 1,1 (S n 1 ), Pengfei Guan and Xu-Jia Wang [GW98] sowed tat weak solutions are C 3,α for any 0 <α<1. Actually, sligtly more general results were discussed in tese references. We mention tat in te case of two reflectors a connection wit mass transportation was found by T. Glimm and V. Oliker [GO04]. It is noted tat te reflector antenna problem is someow analogous to te Monge-Ampère equation, owever, it is more nonlinear in nature and more difficult tan te Monge-Ampère equation. Our purpose in tis paper is to establis some important quantitative and qualitative properties of weak solutions to te global antenna problem, tat is, wen Ω = Ω = S n 1. Tree important results are crucial for te regularity teory of weak solutions to te Monge-Ampère equation: interior gradient estimates, te Alexandrov estimate, and Caffarelli s strict convexity. Our first goal ere is to extend tese fundamental estimates to te setting of te reflector antenna problem. Tis is contained in Teorems In our case tese estimates are muc more complicated to establis tan te coun-

3 ON THE REGULARITY OF REFLECTOR ANTENNAS 301 terpart for convex functions due to te lack of te affine invariance property of te equation (1.2) and te fact tat te geometry of cofocused paraboloids is muc more complicated tan tat of planes. Our second goal is to prove te counterpart of Caffarelli s strict convexity result in tis setting, Teorem 4.2. Finally, te tird goal is to sow tat weak solutions to te global reflector antenna problem are C 1 under te assumption tat input and output illumination intensities are strictly positive bounded. To tis end, in Section 5 we establis some properties of te Legendre transforms of weak solutions and combine tem togeter wit Teorem 4.2 to obtain te desired regularity. 2. Preliminaries Let A be an antenna parametrized by y = ρ(x) x for x S n 1. Trougout tis paper, we assume tat tere exist r 1,r 2 suc tat (2.1) 0 <r 1 ρ(x) r 2, x S n 1. Given m S n 1 and b>0, P (m, b) denotes te paraboloid of revolution in R n wit focus at 0, axis m, and directrix yperplane Π(m, b) of equation m y +2b = 0. Te equation of P (m, b) is given by y = m y +2b. IfP (m,b ) is anoter suc paraboloid, ten P (m, b) P (m,b ) is contained in te bisector of te directrices of bot paraboloids, denoted by Π[(m, b), (m,b )], and tat as equation (m m ) y +2(b b ) = 0; see Figure 1. P (m, b) Π[(m, b), (m,b )] P (m,b ) Figure 1 Lemma 2.1. Let P (e n,a) and P (m, b) be two paraboloids wit m = (m,m n ). Ten te projection onto R n 1 of P (e n,a) P (m, b) is a spere S a,b,m wit equation S a,b,m m x 2 a 2 = 8ab = R 2 a,b,m.

4 302 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG Proof. Since P (e n,a) as focus at 0, it follows tat it as equation x n = 1 4a x 2 a. Te intersection of P (e n,a) and P (m, b) is contained in te yperplane of equation (m e n ) x +2(b a) = 0. Hence te equation of Π[(e n,a), (m, b)] can be written as x n = m x +2 b a. Terefore te points x =(x,x n ) P (e n,a) P (m, b) satisfy te equation 1 4a x 2 a = m x +2 b a, wic simplifies to te spere in R n 1 S a,b,m m ( 2 ( x 2 a 8a(b a) m = +4a 2 ) ) 2 1+ = Ra,b,m 2 1 m. n Since m 2 + m 2 n = 1, a direct simplification yields R 2 a,b,m = 8ab. Definition 2.2 (Supporting paraboloid). We say tat P (m, b) isasup- porting paraboloid to te antenna A at te point y A, or tat P (m, b) supports A at te point y A,ify P (m, b) and A is contained in te interior region limited by te surface described by P (m, b). Definition 2.3 (Admissible antenna). as a supporting paraboloid at eac point. Te antenna A is admissible if it Remark 2.4. We remark tat if P (m, b) is a supporting paraboloid to te antenna A, ten r 1 b r 2. To prove it, assume tat P (m, b) contacts A at ρ(x 0 )x 0 for x 0 S n 1. Obviously, 0 <b ρ(x 0 ) r 2 by (2.1). On te oter and, b ρ( m) r 1 also by (2.1). Definition 2.5 (Reflector map). Given an admissible antenna A parametrized by z = ρ(x) x and y S n 1, te reflector mapping associated wit A is N A (y) ={m S n 1 : P (m, b) supports A at ρ(y) y}. If E S n 1, ten N A (E) = y E N A (y). Obviously, N A is te generalization of te mapping T in (1.1) for nonsmoot antennas. Te set y1 y 2 [N A (y 1 ) N A (y 2 )] as measure 0, and as a consequence, te class of sets E S n 1 for wic N A (E) is Lebesgue measurable is a Borel σ-algebra; see [Wan96, Lemma 1.1]. Te notion of weak solution

5 ON THE REGULARITY OF REFLECTOR ANTENNAS 303 can be introduced troug energy conservation in two ways. Te first one is te natural one and uses N 1 A (E ) fdx = E gdm, troug N 1 A. And te second one uses E fdx = N gdm, troug N A(E) A. For nonnegative functions f, g L 1 (S n 1 ), it is easy to sow using [Wan96, Lemma 1.1] tat tese two ways are equivalent. We will use te second way to define weak solutions. Given g L 1 (S n 1 ) we define te Borel measure μ g,a (E) = g(m) dm. N A(E) Definition 2.6 (Weak solution). Te surface A is a weak solution of te antenna problem if A is admissible and μ g,a (E) = f(x) dx, for eac Borel set E S n 1. By te definition, smoot solutions to (1.2) are weak solutions. If CA is te C-dilation of A wit respect to O, ten N CA = N A. Terefore, any dilation of a weak solution is also a weak solution of te same antenna problem. We make a remark on (2.1). If te input intensity f and te output intensity g are bounded away from 0 and, and A is normalized wit inf s S n 1 ρ(x) = 1, ten tere exists r 0 > 0 suc tat sup x S n 1 ρ(x) r 0, by [GW98]. E 3. Estimates for reflector mapping Trougout tis paper, we assume tat f and g are bounded away from 0 and, and tere exist positive constants in λ, Λ suc tat (3.1) λ E N A (E) Λ E, for all Borel subsets E S n 1. Let A be an admissible antenna and P (m, b 0 ) a paraboloid focused at O suc tat A P (m, b 0 ). Let S A (P (m, b 0 )) be te portion of A cut by P (m, b 0 ) and lying outside P (m, b 0 ), tat is, (3.2) S A (P (m, b 0 )) = {z A: b b 0 suc tat z P (m, b)}. S A (P (m, b 0 )) can be viewed as a level set or cross section of te reflector antenna A. We sall first establis some estimates for te reflector mapping on cross sections of te antenna A Projections of cross sections. We begin wit a geometric lemma concerning te convexity of projections of cross sections of A.

6 304 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG Lemma 3.1. Let A be an admissible antenna and let P (e n,a) be a paraboloid focused at 0 suc tat P (e n,a) A. Ten (a) If x 0,x 1 S A (P (e n,a)), ten tere exists a planar curve C S A (P (e n,a)) joining x 0 and x 1. (b) Let R = S A (P (e n,a)) and R be te projection of R onto R n 1 wic is identified as a yperplane in R n troug O wit te normal e n. Ten R is convex. Proof. Let x 0,x 1 be te projection of x 0,x 1 onto R n 1, and let L be te 2-dimensional plane troug x 0,x 1 and parallel to e n. Consider te planar curve L A tat contains x 0,x 1. We claim tat te lower portion of L A connecting x 0,x 1 lies below P (e n,a). Indeed, let x be on tis lower portion of L A and let P (m, b) be a supporting paraboloid to A at te point x. If m = e n, ten a b and x is below P (e n,a). Now consider te case m e n. Obviously, te points x 0,x 1 are below P (e n,a) and inside P (m, b). Terefore, x 0,x 1 lie below te bisector Π[(e n,a),(m, b)] and ence below te line L Π[(e n,a),(m, b)]. Since L A is a convex curve, it follows tat te lower portion of L Aconnecting x 0 and x 1 lies below L Π[(e n,a), (m, b)] and so does x. It implies tat x is below P (e n,a). Tis proves (a) and as a result part (b) follows. Remark 3.2. Trougout tis section we use te following construction. If P (e n,a) A, R = S A (P (e n,a)), and R is te projection of R onto R n 1 parallel to te directrix yperplane Π(e n,a), ten E will denote te Fritz Jon (n 1)-dimensional ellipsoid of R 1 ; tat is, n 1 E R E; we assume tat E as principal axes λ 1,,λ n 1 in te coordinate directions e 1,,e n Estimates in case te diameter of E is big. For a convex function v(x) on a convex domain Ω, it is well known tat Dv(x) C osc Ω v/dist(x, Ω), for any x Ω, see [Gut01, Lemma 3.2.1]. Tis fact gives rise to an estimate from above of te measure of te image of te norm mapping. Te following teorem extends tis result to te setting of te reflector mapping. Teorem 3.3. Let A be an admissible antenna satisfying (2.1) and let P (e n,a + ) wit >0small be a supporting paraboloid to A. Denote by R = S A (P (e n,a)) te portion of A bounded between P (e n,a+ ) and P (e n,a), and let R and E be defined as in Remark 3.2. Let R 1/2 be te lower portion of R wose projection onto R n 1 1 is 2(n 1) E. (a) Assume d 1 d = diam(e) d 2. If P (m, b) is a supporting paraboloid to A at some Q R 1/2 wit m =(m,m n )=(m 1,,m n 1,m n ), ten m i C/λ i for i =1,,n 1, and m 2 C /d, were C depends only on structural constants, d 1, and d 2.

7 ON THE REGULARITY OF REFLECTOR ANTENNAS 305 (b) Assume tat d η 0 wit η 0 small. Let ρ 1 (R 1/2 ) be te preimage of R 1/2 on S n 1. Ten N A (ρ 1 (R 1/2 )) {(m,m n ) S n 1 : C /d} and n 1 { } N A (ρ 1 (R 1/2 )) C min d,, λ i i=1 were C depends only on structural constants and η 0. Proof. Suppose tat P (m, b) is a supporting paraboloid to A at some point Q R 1/2. Let τ = m m Rn 1, m τ = m, and write (3.3) m =(m τ τ,m n ). We ave 1 = m 2 = m 2 n + m 2 τ and terefore (3.4) m 2 τ 2(1 m n ). From Lemma 2.1, te points x =(x,x n ) P (e n,a) P (m, b) satisfy te equation S a,b,m x 2 a m τ 2 τ = Ra,b,m 2, wit Ra,b,m 2 = 8ab. Our goal now is to estimate te reflector mapping over te interior lower portion R 1/2 wose projection on R n 1 1 is 2(n 1) E. Recall Remark 2.4 and tat is very small. Let Q denote te projection of Q in te direction e n ; tat is, Q 1 2(n 1) E. We may assume m e n. Obviously, tere exists 0 <ε 0 1 suc tat Q P (e n,a+ ε 0 ) P (m, b); see Figure 2. Let P be te portion of P (m, b) below R and defined over R. Since P (e n,a+ ε 0 ) P (m, b) Π[(e n,a+ ε 0 ), (m, b)], it follows tat P crosses Π[(e n,a+ ε 0 ), (m, b)] and P (e n,a+ ε 0 ). Let S a+ε0,b,m be te spere from Lemma 2.1 obtained projecting Π[(e n,a+ ε 0 ), (m, b)] P (m, b) on R n 1, and let B a+ε0,b,m be te solid ball wose boundary is S a+ε0,b,m. Since Π[(e n,a+ ε 0 ), (m, b)] traverses P (m, b), it follows tat P is below te bisector Π[(e n,a+ ε 0 ), (m, b)] in te region R B a+ε0,b,m, and terefore P is below P (e n,a+ ε 0 ) in te same region. Terefore, P is above (or inside) P (e n,a+ ε 0 )inr \ B a+ε0,b,m. For x =(x,x n ) Pwit x R \B a+ε0,b,m, x must be between P (e n,a) and P (e n,a + ε 0 ). Hence tere exists ε = ε x suc tat 0 ε ε 0 wit

8 306 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG A P (m, b) 0 P (en,a) P (en,a+ ε0) Q P (en,a+ ) Figure 2 x P (e n,a+ ε) P (m, b). Consequently, x S a+ε,b,m and from Lemma 2.1 we ave m τ 2 x 2(a + ε) τ 8(a + ε)b = = Ra+ε,b,m 2 1 m. n On te oter and, x is outside S a+ε0,b,m. It follows tat 8(a + ε 0 )b m τ x 2(a + ε 0 ) τ m τ x 2(a + ε) τ +2(ε 0 ε) m τ 8(a + ε)b mn 8(a + ε 0 )b (1 + C). One ten obtains tat R \ B a+ε0,b,m is contained in a ring wit inner radius R = R a+ε0,b,m and widt CR. Since te inner spere of te ring S a+ε0,b,m passes troug Q 1 2(n 1) E, its tangent at Q 1 traverses 2(n 1) E and te ring.

9 ON THE REGULARITY OF REFLECTOR ANTENNAS 307 Tus, tere exists an ellipsoid E 0 R \ B a+ε0,b,m wose axes are comparable and parallel to tose of E. Moreover, E 0 is contained in a cylinder C wose eigt is CRand wose base is an (n 2)-dimensional ball wit radius CR and center Q. Since diam(c) =CR, one obtains tat d CR (3.5) and terefore 1 mn C /d. As /d is small, m n is close to 1 and R is very large. From (3.4) and (3.5) we obtain te estimate m τ C /d. Let x 0 be te center of E 0 and E C be te center of E. We want to sow tat ( m 1 m ) n E C (3.6) x 2a 0x C, for all x E 0. For simplicity, let C 0 =2(a + ε 0 ) τ be te center of te ring. We claim tat te angle between C 0 E C and te radial direction C 0 Q is very small; tat is, angle( C 0 E C, C 0 Q ) C d/r. In fact, by te law of cosines, we ave tat E C Q 2 = C 0 Q 2 + C 0 E C 2 2 C 0 Q C 0 E C. Witout loss of generality, we may assume tat C 0 E C C 0 Q. If we set C 0 Q = C 0 Q τ r = R 1 τ r, A 1 = E C Q, and C 0 E C = C 0 E C τ E =(R 1 A 2 )τ E, were 0 <A 2 A 1 d, ten Since R is large and R1 R m τ A 2 1 = R 2 1 +(R 1 A 2 ) 2 2R 1 (R 1 A 2 ) τ r τ E. C by (3.5), we get te following A τ r τ E = A2 2 2R 1 (R 1 A 2 ) Cd2 R 2, and te claim is proved. Continuing wit te proof of (3.6), write τ E = k r τ r + k t τ t, were τ t is a unit vector in te tangent plane of te spere S a+ε0,b,m at te point Q ; tat is, τ t τ r, and k t 0. Terefore, we ave For x,x E 0, write τ E τ t = k t = 1 (τ r τ E ) 2 = (1 + τ r τ E )(1 τ r τ E ) 2 Cd2 R 2 C d R. x x = ε 1 CRτ r + ε 2 dτ t + τ, were 1 <ε 1,ε 2 < 1, and τ is perpendicular to bot τ r and τ t. From (3.5) d CR and so

10 308 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG τ E x x ε 1 τ E τ r CR + ε 2 τ E τ t d CR + Cd 2 /R CR. Note tat C 0 E C CR. It follows tat C 0 E C x x CR τ E x x CR 2. Since x x d<r,weave ( m ) E C 2a x x CR 2, and ten by te definition of R we obtain (3.6). We are now ready to prove (a). Since d 1 d d 2, from (3.5) and (3.6), one obtains m x 0x C. Since te ellipsoid E 0 as principal axes Cλ 1,,Cλ n 1 in te coordinate directions e 1,,e n 1, it follows from te last inequality tat te i-t component m i of m must satisfy m i C/λ i. We now prove (b). For m B η0 (0) wit small η 0, let w = M(m )= m 1 1 m 2 E C. It is easy to verify tat te Jacobian of M is close 2a to 1 and tat for m,m 0 B η 0 (0) we ave (3.7) (1 Cη 0 ) m m 0 M(m ) M(m 0) (1 + Cη 0 ) m m 0. We claim tat M is a 1-to-1 mapping from B η0 (0) onto B η0 (w η0 ), were w η = 1 1 η 2 E C for 0 η η 0. In fact, if m = η, ten w w η = η. Itis 2a easy to verify tat w η w η0 Cη 0 (η 0 η). Hence, M(B η0 ) B η0 (w η0 ). On te oter and, given w 0 wit w w η0 = μ<η 0, consider te continuous function f(η) = w w η /η for 0 <η η 0. Obviously, lim η 0 + f(η) = and lim η η0 f(η) =μ/η 0 < 1. Terefore, tere exists 0 <η<η 0 suc tat f(η) = 1, wic implies tat w w η = η and w = M(m ) wit m = w w η. Tus, te claim is proved. From (3.6), if m = (m,m n ) N A (ρ 1 (R 1/2 )), ten w = M(m ) = (w 1,,w n 1 ) is in te dual ellipsoid E of te ellipsoid E given by E = {w : w i C/λ i, 1 i n 1}. Clearly, we ave te following estimate N A (ρ 1 (R 1/2 )) {(m,m n ) S n 1 : C /d and M(m ) E } C {m : M(m ) C /d and M(m ) E } = C M 1 {w : w C /d and w i C/λ i, 1 i n 1} n 1 { } C min d,. λ i i=1 Tis completes te proof of te teorem.

11 ON THE REGULARITY OF REFLECTOR ANTENNAS 309 A fundamental estimate for convex functions is te Alexandrov geometric inequality wic asserts tat if u(x) is a convex function in a bounded convex domain Ω R n suc tat u C(Ω) and u =0on Ω, ten for x 0 Ω u(x 0 ) n C dist(x 0, Ω) diam(ω) n 1 Du(Ω) ; see [Gut01, Lemma 1.4.2]. We extend tis result to te setting of te reflector mapping in te following teorem. Teorem 3.4. Let A be an admissible antenna satisfying (2.1) and let P (e n,a + ) wit >0small be a supporting paraboloid to A. Denote by R = S A (P (e n,a)) te portion of A bounded between P (e n,a+ ) and P (e n,a), and let R and E be defined as in Remark 3.2. Assume tat E as center E C and principal axes λ 1,,λ n 1 in te coordinate directions e 1,,e n 1. Denote by ρ 1 (R) te preimage of R on S n 1. (a) Assume tat d 1 d = diam(e) d 2. Given δ>0 and z =(z 1,,z n 1 ) R suc tat z =(z,z n ) R P (e n,a+ ) wit K δλ 1 z 1 K, were K = sup x R x 1, ten tere exists ε 0, independent of δ and z, suc tat F = {m S n 1 : ε 0 /d, 0 m 1 ε 0, m i ε 0,i=2,,n 1} N A (ρ 1 (R)). δλ 1 λ i In oter words, if m F, ten P (m, b) is a supporting paraboloid to A at some point on R for some b>0. (b) Assume tat /d C 0. Let B be te linear transformation given by B(y 1,,y n 1 )=(λ 1 y 1,,λ n 1 y n 1 ) suc tat E E C = BB 1, were B 1 is te unit ball. Given δ>0 and z =(z,z n ) R P (e n,a+ ) wit z = E C +(θ δ)by, y =1,E C + θby R, and 1 n 1 θ 1, ten tere exist a small ε 0 > 0, independent of δ and z, and n 1 ortonormal vectors e 1,,e n 1 in Rn 1 suc tat C {w R n 1 : w ε 0 /d, Bw E } N A (ρ 1 (R)), { n 1 were E = i=1 w i e i : ε 0 3δ w 1 0, ( ) } n 1 ε0 2 i=2 (w i )2 3 a cylinder wit circular base B ε0/3 and eigt ε 0 3δ. is z 2(a + ) Proof. Let z R P (e n,a+ ). We ave 1 e n = const z z by Remark 2.4 and (2.1). If m S n 1 and m e n ε 0 wit ε 0 small, ten

12 310 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG S m, S m C m, C m ΔR z Figure 3: Teorem 3.4 ( 2b z 1 m z ) const and so z P (m, b). Recall tat E C is te center z of E and /d C 0, and set { F = m S n 1 : ( ε 0 /d, sup m + 1 m ) n E C } z x ε 0. x R 2a In order to prove (a) and (b), we first sow tat (3.8) F N A (ρ 1 (R)). To prove tis, we will first claim tat for m F, te portion of P (m, b) tat contains z and is over R is below P (e n,a), and second we will sow tat tis implies tat P (m, b 0 ) (peraps wit b 0 different from b) is a supporting paraboloid to te wole antenna A at a point on R. To sow te first claim, since z is below P (e n,a), it suffices to prove tat (3.9) R S m, were S m = S a,b,m is te spere from Lemma 2.1 wic is te projection of te intersection of P (e n,a) and te bisector Π[(e n,a), (m, b)]. As in te proof of Teorem 3.3, S m as equation S m x 2 a m τ 2 τ = 8ab = Rm 2 = R 2, were m τ = m and m = m τ τ. In order to prove (3.9) we now sow tat (3.10) z is inside S m and dist(z,s m ) CR, and next construct a cylinder C defined by (3.11) so tat R C S m. Indeed, since z P (e n,a+ ) P (m, b), z must be on te spere S m, =

13 ON THE REGULARITY OF REFLECTOR ANTENNAS 311 S a+,b,m, te projection of te intersection of P (e n,a + ) and te bisector Π[(e n,a+ ), (m, b)]. Similarly to S m, S m, as equation S m, m τ x 2(a + ) τ 2 = 8(a + )b = R 2 m,. We claim tat b a Cm τ + Cε 0. In fact, if z = ρ(y)y for some y S n 1, ten and consequently ρ(y) = 2(a + ) 1 e n y = 2b 1 m y, b a a + = (e n m) y 1 e n y. Terefore b a = a + 1 e n y (e n m) y 1 2 ρ(y) e n m Cm τ, and te claim follows. Let C m and C m, be te centers of S m and S m, respectively. By te law of cosines 1+τ C m, z C m, z =1 C m, z C m, z C m, O C m, O Oz 2 2 C m, z C m, O C R 2, were O is te origin in R n 1. Tis means tat te angle between τ and C m, z is less tan C/R. We now estimate C m z to locate te position of z inside S m. Again by using te inner product, we ave C m z 2 = C m, z 2 + C m, C m 2 2 C m, z C m, C m ( = R m, 2 m ) [ 2 τ +2R m, 2 m ] τ C m, z 1 C m, z ( τ) ( R m, 2 m ) 2 τ + CR R C R 2 ( R m, 2 m ) 2 τ + C. Since R m, 2 mτ 1 m n R, it follows tat C m z ( R m, 2 m τ ) + C R.

14 312 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG On te oter and ( ΔR R R m, 2 m ) τ =2 m τ 8ab + = 2 1+m n 1 mn 8(a + )b 8b 1 mn a + + a 1 mn [2 1+m n 2 ] b. a C Since b a Cε 0 and m n is close to 1, we obtain tat ΔR = 1 mn CR. Terefore R ( C m z R R m, 2 m ) τ C R =ΔR C R CR. So (3.10) is proved. Write C m z = C m z τ r, i.e., τ r is te radial direction at z. We obtain tat te cylinder (3.11) C = {z + k r τ r + k t τ t : R/2 k r CR, k t CR, τ t τ r, τ t =1} is contained inside te spere S m for an appropriate coice of C. We next prove tat if m F, ten R C wic will complete te proof of (3.9). Write C m E C = C m E C τ E. Ten te angle between τ r and τ E is small. Indeed, by te law of cosines E C z 2 1 τ r τ E 2 C m z C m E C Cd2 R 2. Write τ r = k E τ E + k τ suc tat k 0, τ τ E, and τ = 1. Terefore, 1 2 k E 1 and 0 k C d R. Since m F and R = C/, z x τ t z x d ε 0 CR for x R, and we ave te following estimate z x τ r = k E z x z x τ E + k z x τ k E k E 2a 1 m n ε 0 C m E C ( ) E C 2a m 1 m n + C d C m E C R d + C d2 R Cε 0 R+ Cε 2 0R= Cε 0 R,

15 ON THE REGULARITY OF REFLECTOR ANTENNAS 313 for all x R, and terefore R C and so (3.9) follows. As a result, te portion of P (m, b) over R passing troug z is strictly below (or outside) P (e n,a). Furtermore, te portion of P (e n,a) over R is strictly contained in P (m, b). Geometrically, if we drag P (m, b) downward (i.e., aving b increase), ten we can get a supporting paraboloid P (m, b 0 ). In fact, if x = ρ(y)y R wit y = 1, ten x P (e n,a + ε) for some 0 ε 1 and 1 e n y = 2(a+ε)/ρ(y) const. Tus, x P (m, b x ) were 2b x ρ(y)(1 my) const. Let b 0 = sup{b x : x R}. We want to prove tat P (m, b 0 ) is a supporting paraboloid to A at a point in te interior of R. Coose z k Rsuc tat b zk b 0. Witout loss of generality, assume tat z k z 0. Let z k = ρ(y k )y k, y k = 1, and z 0 = ρ(y 0 )y 0, y 0 = 1. By taking te limit as k in te equation ρ(y k )(1 my k )=2b zk, we get tat z 0 P (m, b 0 ). On te oter and, every x Rmust be on some P (m, b x ) wic lies inside P (m, b 0 ) since b x b 0. Hence, R is inside P (m, b 0 ) and touces P (m, b 0 )atz 0. It follows tat P (m, b 0 ) is a supporting paraboloid to R and R is strictly inside P (m, b 0 ). To sow tat P (m, b 0 ) is a supporting paraboloid to A, it is enoug to sow tat A\R is also contained inside P (m, b 0 ). Indeed, suppose by contradiction tat tere exists x A\R lying outside P (m, b 0 ). Ten x, z 0 lie on or outside P (m, b 0 ). By Lemma 3.1, tere exists a curve C on A connecting z 0 and x and lying on or outside P (m, b 0 ). Ten C must cross te boundary of R wic is strictly contained inside P (m, b 0 ), a contradiction. Tus, te proof of (3.8) is complete. Now to te proof of Part (a). Given x R, write z x = x z = ε 1 λ 1 e 1 + n 1 i=2 ε iλ i e i, were 2 ε 1 δ and 2 ε i 2 for i =2,,n 1. If m F and since d 1 d d 2, ten one obtains ( ) n 1 1 z x mn E C m Cε 2 2a 0 ε 1 m 1 λ 1 ε i m i λ i i=2 Cε δ ε 0 δλ 1 λ 1 +2(n 2)ε 0 = Cε 0. Coosing ε 0 in F sufficiently small we get tat F F (F is now defined wit Cε 0 instead of ε 0 ) and (a) follows from (3.8). To prove (b), and as in te proof of Teorem 3.3, we consider te mapping w = M(m )=m 1 1 m 2 E C. Let 2a Proj F = {m : m n suc tat (m,m n ) F } be te projection on R n 1. Since ε 0 is small, it follows from (3.7) tat

16 314 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG (3.12) We claim tat {m : M(m ) ε 0 2 d, {m : m ε 0 /d, Proj F. sup [ M(m )] z x ε 0 } x R sup [ M(m )] z x ε 0 } x R {m : BM(m ) E } {m : sup [ M(m )] z x ε 0 }. x R Let R = B 1 (R E C ). Obviously, B 1 R B 1. By te assumptions, n 1 E C + θby R and ence θy R. Let z (θ δ)y = B 1 (z E C ), and let y R be suc tat dist(z, R )= y z. Let e 1 = y z and coose {e i }n 1 i=2 suc tat {e i }n 1 i=1 e 1 is normal to R at y and y z is a set of ortonormal vectors. Clearly, n 1 R {z + u i e i : 2 u 1 δ, u i 2, i=2,,n 1}. i=1 Terefore, for Bw E, it is easy to verify sup ( w) z x = sup [ Bw] (u z ) ε 0, x R u R and te claim follows. Terefore from (3.12) we get {m : M(m ) ε 0 2 d, BM(m ) E } Proj F. Since te Jacobian of M is close to one, te conclusion in part (b) follows from (3.8) Estimates in case te diameter of E is small. Teorem 3.3 and Teorem 3.4 extend te gradient estimate and Alexandrov estimate for convex functions to reflector antennas in te case /d η 0. Tese two teorems are sufficient for te discussion of strict reflector antennas in Section 4. However, to get complete extension of te estimates, we also need to prove te following teorem addressing te case /d η 0. Teorem 3.5. Let A be an admissible antenna satisfying (2.1) and (3.1), and let P (e n,a + ) be a supporting paraboloid to A for small >0. Let R = S A (P (e n,a)) and let R and E be defined as in Remark 3.2, let E C be te center of E, and d = diam(e). Assume tat d η 0 > 0. (a) Tere exists C>0suc tat C 1 d λ i Cd, for i =1,,n 1, and η 0 d C.

17 ON THE REGULARITY OF REFLECTOR ANTENNAS 315 (b) Let B be te linear transformation given by B(y 1,,y n 1 )=(λ 1 y 1,,λ n 1 y n 1 ) and be suc tat E E C = BB 1. Given δ > 0 and z = (z,z n ) R P (e n,a+ ) suc tat z = E C +(1 δ)by wit 1 n 1 y 1, and E C + By R, tere exists a small ε 0 > 0 suc tat { 1 Cε n 1 0 min, 1 } ( ) n 1 NA (ρ 1 (R)), δ were ρ 1 (R) is te preimage of R on S n 1. (c) Let R 1/2 be te lower portion of R wose projection onto R n 1 is 1 2(n 1) E and ρ 1 (R 1/2 ) be its preimage on S n 1. Ten N A (ρ 1 (R 1/2 )) {(m,m n ) S n 1 : C }, were C depends only on te structural constants and η 0. Proof. We first prove part (a). Let z =(z,z n ) R P (e n,a+ ). We remark tat to prove (3.10), it suffices to assume tat m e n ε 0 wit ε 0 small, and terefore under tis assumption one can conclude as in Teorem 3.4 tat te cylinder C = {z + k r τ r + k t τ t : R/2 k r CR, k t CR, τ t τ r, τ t =1} is contained strictly inside te spere S m, were te symbols ave te same meaning as in tat teorem. If on te oter and ε 0 /d, ten d ε 0 CR were R is te radius of S m and R = C/ and consequently R B d (z ) C. Terefore, if min{ε 0,ε 0 /d}, ten R S m. Using te tecnique of dragging te paraboloid as in te proof of Teorem 3.4, we ten obtain tat m N A (ρ 1 (R)); tat is, {m S n 1 : min{ε 0,ε 0 /d}} N A (ρ 1 (R)). Let D = ρ 1 (R) and P (e n,a) D be te restriction of P (e n,a) over D, i.e., te portion of P (e n,a) contained in A. Obviously, D C P (e n,a) D C R. By (3.1), we obtain (3.13) [min{ε 0,ε 0 /d}] n 1 C D C R Cλ 1 λ n 1. We claim tat if is small enoug, ten /d 1. Oterwise, if /d > 1, ten from (3.13), ε n 1 0 Cλ 1 λ n 1. Tis implies tat λ i Cε n 1 0 for i =1,,n 1, and terefore d Cε n 1 <η 0 for small, a contradiction. 0 ( ) n 1 Tus, te claim is proved. Terefore from (3.13), ε 0 Cλ 1 λ n 1, d

18 316 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG and so ( ) n 1 (ε 0 η0) 2 n 1 ε n 1 0 d 2 C λ 1 λ n 1 d n 1 C λ i d C, for i =1,,n 1. Tis completes te proof of part (a). Now prove part (b). By part (a), η 0 /d C. By Teorem 3.4 (b) C {w R n 1 : w ε 0 η 0, Bw E } N A (ρ 1 (R)), were E is a cylinder wit circular base B ε0/3 and eigt ε 0. By part (a), 3δ Bw Cd w. Terefore C B 1 (B C0ε 0η 0d E ) C {w R n 1 : Bw C 0 ε 0 η 0 d, Bw E } N A (ρ 1 (R)). Since /C d /η 0, it is easy to verify tat N A (ρ 1 (R)) C { min C dn 1 0 ε 0 η 0 d, ε 0 3δ C(ε 0 η 0 ) n 1 min }[ min { 1, 1 δ } ( ) n 1. { C 0 ε 0 η 0 d, ε }] n Tis completes te proof of part (b). To prove part (c), let z = (z,z n ) R 1/2 and P (m, b) be a supporting paraboloid at z. As in te proof of (3.5) in Teorem 3.3, tere exists an ellipsoid E 0 R wose axes are comparable and parallel to tose of E suc tat E 0 is contained in a cylinder C wose eigt is CR and wose base is an (n 2)-dimensional ball wit radius CR and center z, were R = C/. By part (a), it follows tat B σ0d C for some small σ 0. Terefore, σ 0 d CR = C/. Since d C, we obtain tat 1 mn C. Te proof of te teorem is finised. 4. Strict antennas In tis section, we use te estimates establised in Section 3 to sow tat a reflector antenna satisfying (2.1) and (3.1) must be a strict reflector antenna. Definition 4.1 (Strict antenna). An admissible antenna A is a strict antenna if every supporting paraboloid of A touces A at only one point. Te following result is concerned wit strict antenna. Teorem 4.2. If A is an admissible antenna satisfying (2.1) and (3.1), ten A is a strict antenna, and consequently, te map N A is injective.

19 ON THE REGULARITY OF REFLECTOR ANTENNAS 317 Proof. Let P (e n,a 1 ) be a supporting paraboloid to A. We need to sow tat P (e n,a 1 ) Ais a single point set. By Lemma 3.1 (b), te projection Δ on R n 1 of P (e n,a 1 ) A is a convex set. Suppose by contradiction tat Δ contains at least two points. Ten diam(δ) = constant > 0. For sufficiently small, let R be te portion of A cut by P (e n,a 1 ), R 0 te portion of A cut by P (e n,a 1 ) and relabel a = a 1, a + = a 1. We claim tat R converges to R 0 in te Hausdorff metric as 0. Indeed, suppose by contradiction tat tere exist δ 0 > 0 and z R suc tat dist(z, R 0 ) δ 0. By compactness, passing troug a subsequence z z 0 R 0 and z z 0 δ 0, we obtain a contradiction. Let R be te projection of R on R n 1. Ten by te claim, R Δin te Hausdorff metric as 0. Let E be te Jon ellipsoid for te set R and let λ 1 () be te longest axis of E. Ten λ 1 () C diam(δ) and tere exists z Δ suc tat K δ λ 1 () (z ) 1 K, were K = sup z R z 1. Notice tat δ 0as 0. We now apply Teorems 3.3 and 3.4 to get a contradiction. Let ˆR and ( ˆR ) 1/2 be te lower portions of R defined over R and 1 2(n 1) E, respectively, and let D and (D ) 1/2 be te preimages on S n 1 of ˆR and ( ˆR ) 1/2, respectively. We want to sow tat D R. Given y = ρ(x) x Awit x S n 1, let P (e, b) be a supporting paraboloid to A at y. Let n be te inner normal of P (e, b) and A at y. Ten by Snell s law, n ( x) =n e and n ( x) const > 0. It follows tat D ˆR. Similarly, D P (e n,a) D, were P (e n,a) D is te restriction of P (e n,a) on D. Obviously, D C P (e n,a) D C R C ˆR C D. Tis proves tat D R. Since (D ) 1/2 ( ˆR ) 1/2, to sow tat (4.1) (D ) 1/2 1 2(n 1) E, it suffices to prove tat ( ˆR ) 1/2 is a Lipscitz grap. For y = ρ(x) x ( ˆR ) 1/2, let P (m, b) be a supporting paraboloid to A at y, and n be te inner normal to P (m, b) and ( ˆR ) 1/2 at y. By Teorem 3.3(a), e n m C. Since P (m, b) is smoot, m n const > 0. Tis implies tat e n n const > 0. Terefore ( ˆR ) 1/2 is a Lipscitz grap and so (4.1) olds. From Teorem 3.3(a) and (3.1) we get C E N A ((D ) 1/2 ) min { C C } n 1, λ 1 diam(e ) i=2 min { C C },. λ i diam(e )

20 318 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG On te oter and, from Teorem 3.4(a) and (3.1) we ave E R C N A(D ) { C min Terefore, ε n 1 0 min { δ λ 1, ε 0 δ λ 1, } n 1 ε 0 diam(e ) i=2 min } { C min, diam(e ) λ 1 { } ε 0 ε 0,. λ i diam(e ) }. diam(e ) Since λ 1 diam(e ) const, we obtain for any sufficiently small >0 { ε n 1 0 min, } C, δ wic gives a contradiction. Remark 4.3. We notice tat if A k = {xρ k (x) :x S n 1 } is a sequence of admissible antennas satisfying (2.1), ten A k converges to te antenna A = {xρ(x) :x S n 1 } in te Hausdorff metric if and only if ρ k converges to ρ uniformly on S n 1. Lemma 4.4. Let A j = {xρ j (x) :x S n 1 }, j 1, be admissible antennas satisfying (2.1). Assume tat ρ j converges to ρ uniformly on S n 1. Ten (a) lim sup j N Aj (K) N A (K), for any compact set K S n 1 ; (b) lim inf j N Aj (O) N A (O), for any open set O S n 1. Proof. Part (a) is easy to prove by definition. For completeness, we prove part (b). Let K O be compact and E = x1 x 2 [N A (x 1 ) N A (x 2 )]. Ten E = 0. Let m 0 N A (K) \ E. Tere exist x 0 K and a > 0 suc tat P (m 0,a) is a supporting paraboloid to A at x 0 ρ(x 0 ). To finis te proof, it suffices to sow tat m 0 N Aj (O) for sufficiently large j. Since m 0 / E, lim 0 diam(s A (P (m 0,a ))) = 0. By te continuity of te mapping y y, lim 0 diam(d ) = 0, were D = ρ 1 (S A (P (m 0,a ))) is te radial projection on S n 1 of S A (P (m 0,a )). Coose D O. Let ε>0 and coose j 0 large. If j j 0, ten for x S n 1 \ D, a ρ j (x) (1 + ε)ρ(x) (1 + ε) 1 m 0 x, and a ρ j (x 0 ) (1 ε)ρ(x 0 )=(1 ε). 1 m 0 x 0 Let b 0 = (1 + ε)(a ) > 0 and coose ε small enoug suc tat δ = (1 ε)a b 0 > 0. Ten A j is inside P (m 0,b 0 ) along directions in S n 1 \ D

21 ON THE REGULARITY OF REFLECTOR ANTENNAS 319 and A j is outside P (m 0,b 0 + δ) in te direction x 0. For x D, since a ρ(x)(1 m 0 x) a, 2b x ρ j (x)(1 m 0 x) > 0 for j j 0. Tis means tat xρ j (x) P (m 0,b x ). Let b = sup{b x : x D and xρ j (x) P (m 0,b x )}. Witout loss of generality, assume tat b = b x1 were x 1 D. Obviously, b b 0 + δ. Terefore, P (m 0,b) is a supporting paraboloid to A j at x 1 ρ j (x 1 ) O. Tis completes te proof of te lemma. Corollary 4.5. Te class of admissible antennas satisfying (2.1) and (3.1) is compact wit respect to te Hausdorff metric. Proof. By Remark 4.3 and Lemma 4.4, to prove te corollary it suffices to estimate uniformly te Lipscitz constant of te radial function defining te antennas. Let A be an antenna parametrized by ρ(x) suc tat (2.1) and (3.1) old. Let x 0, x 1 S n 1 and x 0 x 1 ε 0. Let P (m 0,a 0 ) be a supporting paraboloid of A at x 0 ρ(x 0 ). Obviously 2a 0 m 0 (x 1 x 0 ) ρ(x 1 ) ρ(x 0 ). 1 m 0 x 0 1 m 0 x 1 Since 1 m 0 x 0 =2a 0 /ρ(x 0 ) const, ten 1 m 0 x 1 const, if ε 0 is small. We conclude tat ρ(x 1 ) ρ(x 0 ) C x 0 x 1. Te corollary is proved. As a corollary of Teorem 4.2 and Corollary 4.5, we ave te following result on te diameter of sections. Corollary 4.6. Let A be an admissible antenna satisfying (2.1) and (3.1). Ten tere exists an increasing function σ() depending only on r 1, r 2, λ, Λ,and n wit lim 0 + σ() =0suc tat diam(s A (P (m, b ))) σ() for any supporting paraboloid P (m, b) of A. 5. Legendre transform Our purpose in tis section is to discuss some properties of te Legendre transform (see Definition 5.1) of weak solutions to te reflector antenna problem, a notion introduced in [GW98]. Definition 5.1 (Legendre transform). Given an admissible antenna A = {xρ(x) :x S n 1 }, te Legendre transform of A, denoted by A, is defined by A = {mρ (m) :m S n 1 }, were ρ (m) = inf x S n 1,x m 1 ρ(x)(1 m x) = 1 sup x S n 1[ρ(x)(1 m x)]. Lemma 5.2. If A is an admissible antenna satisfying (2.1), ten its Legendre transform A 1 is also an admissible antenna and satisfies 2r 2

22 320 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG ρ (m) 1 2r 1 for any m S n 1. Moreover, if m 0 N A (x 0 ), ten x 0 N A (m 0 ). Proof. Te proof is similar to tat of Lemma 4.1 in [GW98]. For m 0 S n 1, let 2a = sup x S n 1 ρ(x)(1 m 0 x) and assume tat tis supremum is attained at x 0 S n 1. Ten P (m 0,a) is a supporting paraboloid to A at x 0, and ρ (m 0 )= 1 2a. By Remark 2.4, one concludes tat 1 ρ (m 0 ) 1 2r 2 2r 1 and N A (S n 1 )=S n 1. Let m 0 N A (x 0 ). We now prove x 0 N A (m 0 ). Denote by P (m 0,a) te supporting paraboloid to A at x 0 ρ(x 0 ) wit te axis direction m 0. Ten ρ(x)(1 m 0 x) attains te maximum 2a at x 0 and ρ 1 (m 0 )= ρ(x 0 )(1 m 0 x 0 ). Obviously, ρ 1 (m) ρ(x 0 )(1 m x 0 ) for m Sn 1 1. Terefore, P (x 0, 2ρ(x ) 0) is a supporting paraboloid to A at m 0 ρ (m 0 ), and x 0 N A (m 0 ). Now, given m 0 S n 1, tere exists x 0 S n 1 suc tat m 0 N A (x 0 ). Hence, x 0 N A (m 0 ) and A is an admissible antenna. Lemma 5.3. Let A k = {xρ k (x) :x S n 1 }, k 1, be a sequence of admissible antennas satisfying (2.1). Assume tat A k converges to te antenna A = {xρ(x) :x S n 1 } under te Hausdorff metric as k. Ten A k also converges to A under te Hausdorff metric. Proof. By Lemma 5.2, 1/(2r 2 ) ρ k 1/(2r 1). So tere exists η 0 > 0 suc tat ρ k (m) = inf 1 1 m x η 0. We obtain tat ρ k (x)(1 m x) ρ k (m) ρ (m) 1 sup 1 η 0 ρ k (x) 1 ρ(x). 1 m x η 0 It follows from Remark 4.3 tat ρ k converges to ρ uniformly on S n 1. Te lemma is proved. We now establis te following important lemma about te Legendre transform of antennas in te setting of weak solutions. Lemma 5.4. Let A = {xρ(x) :x S n 1 } be an admissible antenna suc tat (2.1) and (3.1) old. Ten A satisfies (5.1) Λ 1 E N A (E ) λ 1 E, for all Borel subsets E S n 1. Proof. Let E = N 1 A (E ) = {x S n 1 : m E suc tat m N A (x)}. We first prove

23 ON THE REGULARITY OF REFLECTOR ANTENNAS 321 Claim 1. E N A (E) and N A (E) \ E =0. It is easy to see by definition tat E N A (E). Let M = {x S n 1 : A is not differentiable at xρ(x)}. We ave M = 0. We claim tat N A (E) \ E N A (M), and ten from (3.1) we obtain tat N A (E) \ E = 0. To prove te claim, given y E, let m N A (y) \ E. By definition of E, N A (y) E, and terefore tere is m 0 N A (y) E, m 0 m. Terefore, A as at least two supporting paraboloids and ence two supporting yperplanes at yρ(y), and so is not differentiable at yρ(y), wic proves te claim. Claim 2. E N A (E ) and N A (E ) \ E =0. Given x E, by definition of E tere exists m E suc tat m N A (x). By Lemma 5.2, x N A (m), and ence E N A (E ). Let y N A (E ) \ E. Ten y N A (m 0 ) for some m 0 E. By Claim 1, tere exists x 0 E suc tat m 0 N A (x 0 ). Since y x 0, by Teorem 4.2, N A (y) N A (x 0 )=. If m 1 N A (y), ten m 1 m 0. By Lemma 5.2, y N A (m 1 ). Terefore, y N A (m 0 ) N A (m 1 ). From Lemma 5.2, tis implies tat m 0, m 1 N A (y), were A is te Legendre transform of A. So, A is not differentiable at yρ (y) were ρ is te radial function of A.We conclude tat N A (E )\E is a subset of te set were A is not differentiable and wic as measure zero. Tis proves Claim 2. To finis te proof, from Claims 1 and 2 we get E = N A (E) and N A (E ) = E, and so (5.1) follows from (3.1). 6. C 1 regularity We are now ready to prove te C 1 regularity for weak solutions of te antenna problem. First sow te following lemma. Lemma 6.1. If A = {xρ(x) :x S n 1 } is an admissible antenna satisfying (2.1) and (3.1), ten N A is a omeomorpism from S n 1 onto S n 1 wit N 1 A = N A. Moreover, {N A} is equicontinuous, i.e., tere exists an increasing continuous function σ wit lim 0 + σ() =0suc tat N A (x) N A (y) σ( x y ), were σ depends only on r 1, r 2, λ, Λ,and n. Proof. We first prove tat N A is single-valued. Oterwise, if m 1,m 2 N A (x 0 ) wit m 1 m 2, ten by Lemma 5.2, x 0 N A (m 1 ) N A (m 2 ). On te oter and, by Lemmas 5.2 and 5.4, A satisfies (2.1) and (3.1) wit different constants, and so by applying Teorem 4.2 to A we get N A (m 1 ) N A (m 2 )=, a contradiction. In ligt of Teorem 4.2, and since N A (S n 1 )=S n 1, see te proof of Lemma 5.2, we obtain tat N A is bijective. If m = N A (x), ten x = N A (m). Tis proves N 1 A = N A.

24 322 L. A. CAFFARELLI, C. E. GUTIÉRREZ, AND Q. HUANG It remains to sow tat {N A } is equicontinuous. Assume by contradiction tat tere exist A k, x k, y k, and ε 0 > 0 suc tat x k y k 0 and N Ak (x k ) N Ak (y k ) ε 0. By compactness and Corollary 4.5, replaced by a subsequence if necessary, we may assume tat x k z 0, y k z 0, m k = N Ak (x k ) m 0, m k = N A k (y k ) m 0, and A k A 0. It is easy to verify tat m 0 N A0 (z 0 ) and m 0 N A 0 (z 0 ). Since N A0 is single-valued, m 0 = m 0 wic contradicts te assumption m 0 m 0 ε 0. We tus prove te lemma. Teorem 6.2. If A = {xρ(x) : x S n 1 } is an admissible antenna satisfying (2.1) and (3.1), ten A and A are C 1, wit C 1 modulus of continuity depending only on r 1, r 2, λ, Λ,and n. Proof. If P (m, b) is a supporting paraboloid to A at xρ(x), ten by te Snell law A as a supporting yperplane at xρ(x) wit te inward normal m x. By Lemma 6.1, tis field of inward normals for A is continuous. m x It remains to sow tat A is differentiable and ence as only one supporting yperplane at eac point. Let Y =(y 1,,y n ) Aand assume tat {z n = y n } is te equation of a supporting yperplane Π 0 to A at Y.ForX A near Y and witout loss of generality we can write X Y = x 1 e 1 + x n e n wit x 1, x n > 0. By te continuity of te inward normals mentioned before, tere exists a supporting yperplane at X wit te equation ν(x) (z X) =0, were te inward normal ν(x) =(ν 1 (X),,ν n (X)) is close to ν(y )=e n. From te convexity, ν(x) (Y X) 0, and so x n ν 1(X) ν n (X) x 1 εx 1, i.e, dist(x, Π 0 ) ε X Y. Terefore, Π 0 is te tangent plane to A at Y and te only supporting yperplane at Y. Since te field of inward normals of tangent planes to A is continuous, one concludes tat A is of class C 1. Te proof is complete. Corollary 6.3. If A is a weak solution in te sense of Definition 2.6 of te reflector antenna problem wit input illumination intensity f(x) and output illumination intensity g(m) were 0 <λ f(x) Λ and λ g(m) Λ on S n 1, ten A is a C 1 antenna. University of Texas at Austin, Austin, TX address: caffarel@mat.utexas.edu Temple University, Piladelpia, PA address: gutierre@temple.edu Wrigt State University, Dayton, OH address: quang@mat.wrigt.edu References [CO94] L. A. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics, preprint, 1994.

25 ON THE REGULARITY OF REFLECTOR ANTENNAS 323 [GO03] T. Glimm and V. Oliker, Optical design of single reflector systems and te Monge- Kantorovic mass transfer problem, J. Mat. Sci. 117 (2003), [GO04], Optical design of two-reflector systems, te Monge-Kantorovic mass transfer problem and Fermat s principle, Indiana Univ. Mat. J. 53 (2004), [Gut01] C. E. Gutiérrez, Te Monge Ampère Equation, Birkäuser, Boston, MA, [GW98] Pengfei Guan and Xu-Jia Wang, On a Monge Ampère equation arising from optics, J. Diff. Geom. 48 (1998), [Oli02] V. Oliker, On te geometry of convex reflectors, Banac Center Publications 57 (2002), [Wan96] Xu-Jia Wang, On te design of a reflector antenna, Inverse Problems 12 (1996), [Wan04], On te design of a reflector antenna II, Calc. Var. Partial Diff. Eq. 20 (2004), [Wes83] B. S. Wescott, Saped Reflector Antenna Design, Researc Studies Press, Letcwort, UK, (Received Marc 25, 2005)