Research Online Research Online On the classical solvability of near field reflector problems On the classical solvability of near field reflector problems

Abstract In this paper we prove the existence of classical solutions to near field reflector problems, both for a point light source and for a parrallel light source, with planar recievers. These problems involve monge-Ampere type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditioins by Li, Liu and Nguyen. Abstract. In this paper we prove the existence of classical solutions to near ﬁeld reﬂector problems, both for a point light source and for a parallel light source, with planar receivers. These problems involve Monge-Amp`ere type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditions by Li, Liu and Nguyen.


1.
Introduction. The near field reflector problem, with a point light source, can be described as follows: there is a light source at the origin O, a reflecting surface Γ which is a radial graph over a domain U ⊂ S n , where S n is the unit sphere in R n+1 , and a bounded smooth receiver Σ to be illuminated. Letf ∈ L 1 (U ) be the illumination on U , i.e. the intensity of incident rays, and letĝ ∈ L 1 (Σ) be a nonnegative function satisfying (1) We are concerned with the existence of reflector Γ such that the light emitting from O with intensityf is reflected off to the receiver Σ and the intensity of reflected light on Σ is equal toĝ. We always assume that the reflection system is ideal, namely there is no loss of energy in reflection, (1). We represent the reflector Γ in the polar coordinate system as Γ = {Xρ(X) : X ∈ U } (2) with a positive function ρ. Assuming that U lies in the northern hemisphere S n + := S n ∩ {x n+1 > 0}, we project U to Ω ⊂ {x n+1 = 0} so that x = (x 1 , · · · , x n ) ∈ Ω The corresponding boundary condition is where T = T u is the reflection mapping given by Another special case of reflector problems is the far field case, which is related to the reflector antenna design problem [27] and has been extensively studied. It can be regarded as the limit of the above problem with Σ = {dX : X ∈ V }, d → ∞, where V is a domain in S n [9]. The existence and interior regularity for weak solutions were first established in [27] in dimension two, which can be extended to higher dimensions by the a priori estimates in [5]. By a duality, the far field case can be formulated as an optimal transportation problem [28]. Global regularity and the existence of classical solutions then follows from [22]. Mathematically, one may also consider the case when the reflector is a closed surface without boundary. In this case the existence of weak solutions was proved in [2], and the regularity was proved in [5] if f, g ∈ C ∞ and f, g are pinched by two positive constants.
In the near field reflector problem, weak solutions were introduced and obtained in [9,10], and criteria for local interior regularity for general targets were found in [9]. Along the lines of [28], in [14] the near field reflector problem was formulated as a nonlinear optimization problem, to which the regularity results in optimal transportation theory cannot be applied directly. In this paper we first establish the global C 2 estimate for solutions of (3)- (6) and then obtain the existence of classical solutions.
Our essential estimate is the following. Let Ω, Ω * be bounded C 4 domains in R n and Ω B 1 (0). Suppose that Ω is uniformly convex, and Ω * is uniformly Y * -convex with respect to Ω × I for any bounded interval I ⊂ (δ, ∞) for some fixed δ > 0. Let u > δ, ∈ C 4 (Ω) be an elliptic solution of (3)- (6). Then we have the a priori estimate where C depends on f, g, Ω, Ω * and inf Ω u.

CLASSICAL SOLVABILITY OF NEAR FIELD REFLECTORS 3
The notion of Y * -convexity, which is adapted from [19,20], is defined in Section 2.2. In particular if 0 ∈ Ω * and Ω * is convex in the usual sense then Ω * is uniformly Y * -convex with respect to Ω× I for any domain Ω B 1 (0) and interval I (0, ∞). We point out that the boundary condition (5)-(6) is related, but not equivalent, to the boundary condition of prescribing the image of the gradient mapping, where Ω * is a domain in R n . The boundary problem (8) has been extensively studied; see for example [1,3,23,24] and references therein. As a consequence of Theorem 1.1, we have the following existence result for classical solutions. The proof is based on a degree theory recently developed in [11] for second order elliptic operators with nonlinear oblique boundary conditions. Theorem 1.2. Suppose in addition to the hypotheses in Theorem 1.1 that the balance condition (1) is satisfied. Then, there exists a solution ρ ∈ C 3 (Ω) of the near field reflector problem satisfying ρ ≤ 1/δ.
In the last part of this paper, we introduce another model of reflector problems with a parallel light source. Consider the situation when light is emitted from a bounded domain Ω ⊂ R n → R n+1 along direction e n+1 , where R n is identified with R n+1 × {0}, and e n+1 = (0, · · · , 0, 1). We assume that the reflector Γ is represented as a graph over Ω and the light is reflected back to a bounded domain Ω * ⊂ R n so that the prescribed intensities f, g on Ω, Ω * are realized, respectively. This problem has many applications, for example, in the design of reflectors for lamps.
Similarly as above we show that for uniformly Y, Y * -convex domains Ω, Ω * ⊂ R n with smooth distributions C −1 ≤ f, g ≤ C supported on Ω, Ω * , respectively, there exists a reflector Γ such that light emitted with intensity f from Ω is reflected to Ω * and the intensity g is realized provided that A detailed description of this model and a formulation of corresponding theorems are contained in Section 50. Moreover the essential features in this case also occur in refractor problems for parallel beams, as introduced in the work of Gutiérrez and Tournier [6] and Oliker, Rubinstein and Wolanski [16,17] so we also conclude classical existence in these cases by applying our general estimates and existence procedure.
This paper is organized as follows. In Section 2 we first derive equation (3) by considering general prescribed Jacobian equations, and then introduce some preliminary notations and results. In Section 3 we prove that the boundary condition (5)-(6) is oblique in the context of general prescribed Jacobian equations and estimate the obliqueness under the hypotheses of uniform Y and Y * -convexity of Ω and Ω * , by following the argument in [20]. Specializing to the Monge-Ampère equation (3),Theorem 1.1 then follows from [25]. In Section 4 we prove Theorem 1.2 by using the degree theory for oblique boundary value problems developed in [11]. In Section 5 we present the reflector problem with a parallel light source and state the main theorem for this problem as well as the extensions to refractor problems.
Finally we point out that the existence of an infinite number of classical solutions follows from Theorem 1.2 and moreover we can correspondingly refine our domain convexity conditions . In a sequel paper we consider the existence of solutions with value prescribed at a fixed point. For this we use the method proposed in [20] which necessitates more complicated estimates.

Preliminaries.
2.1. Derivation of equation. Suppose that the ray X ∈ U is reflected off at Xρ(X) ∈ Γ in directionŶ and reaches Y ∈ Σ. Let γ be the unit normal of Γ at Xρ(X). By calculations one has then from the reflection law, the reflection direction iŝ where a := |Dρ| 2 − (ρ + Dρ · x) 2 and b := |Dρ| 2 + ρ 2 − (Dρ · x) 2 . Let d = |Y − Xρ| be the length of the reflected ray. Then, where T : U → Σ is the reflection mapping. Since Y ∈ {x n+1 = 0}, from (12) we have 0 = x n+1 ρ +ŷ n+1 d and from (11) Therefore, d = − b a ρ, and by (11)-(12) again, we obtain Regarding T as a mapping from Ω ⊂ R n to Ω * ⊂ R n , we then have Let u = ρ −1 . The reflection mapping T in (14) can be written as This is a special case of considering a general mapping Y from Ω × R × R n into R n [20]. Denoting points in Ω × R × R n by (x, z, p), we see that from (15) Y (x, z, p) = 2p The reflector equation is a special case of a prescribed Jacobian equation Since CLASSICAL SOLVABILITY OF NEAR FIELD REFLECTORS 5 we then obtain, when det Y p = 0, the following Monge-Ampère type equation for elliptic solutions u, that is D 2 u > A(·, u, Du), where the matrix function A and scalar function B are given by For Y = Y (x, z, p) in (16), it is easy to check that that is A = 0, and Using the formula det [I + ξ ⊗ η] = 1 + ξ · η for any vector ξ, η ∈ R n , we have Combining (19), (20), (21) and (23), we then obtain equation (3) for elliptic solutions u.
Remark 1. Noting that U lies in the northern hemisphere, we have x n+1 > 0, which implies (4) and shows also that T is well defined. By computing the Jacobian determinant of T = T ρ in (14), Karakhanyan and Wang obtained the equation for ρ in [9] det which is equivalent to equation (3).

Domain convexity.
We introduce some domain convexity notions adapted from [19,20]. Let us suppose that the mapping Y is defined and C 1 in an open set U 0 ⊂ R n ×R×R n , with Y p = 0 in U 0 . In our reflector problem, Y is equal to (16) and We first define the appropriate convexity notions for the target domain Ω * as it is already used in the formulation of Theorem 1.1. Namely, By pulling back from P(x, z) to Ω * and using the local invertibility of Y with respect to p, we may express these notions in terms of boundary data for C 2 domains Ω * , and mappings Y ∈ C 2 (U 0 ), as done in [22]. Accordingly we have that the target domain Ω * is Y * -convex (uniformly Y * -convex) with respect to Ω × I, if it is connected and for all (x, z) ∈ Ω × I, y = Y (x, z, p) ∈ ∂Ω * , unit outer normal γ * and unit tangent vector τ , (for some constant δ * In optimal transportation, Y is independent of z and the Y * -convexity condition agrees with the corresponding c * -convexity of the target Ω * introduced in [15,22]. As in [22] the key role of the uniform convexity condition in global estimates is in barrier constructions arising from a further formulation in terms of defining functions. Using the distance function as in [22], we obtain that Ω * is uniformly Y *convex with respect to Ω × I, if it is connected and there exists a defining function φ * ∈ C 2 (Ω * ) satisfying φ * = 0, Dφ * = 0 on ∂Ω * together with Again using the local invertibility of Y with respect to p, it follows that (27) holds more generally for y ∈ N * ∩ Ω * for some neighbourhood N * of ∂Ω * , for a further constant κ * 0 > 0. Note that by dividing φ * by an appropriate constant, we can assume κ * 0 = 1. For optimal generality in our obliqueness estimate, we will employ the analogue of this characterisation of uniform convexity for our definitions for the initial domain Ω.
if it is connected and there exists a defining function φ ∈ C 2 (Ω) satisfying φ = 0, Dφ = 0 on ∂Ω together with, for all x ∈ N ∩ Ω, z ∈ I, y = Y (x, z, p) ∈ Ω * , for some neighbourhood N of ∂Ω, where A is the matrix in (20) generated by the mapping Y .
Note that for our reflector problem A ≡ 0 and uniform Y -convexity is equivalent to the usual uniform convexity. These convexity notions can be equivalently expressed in terms of boundary data, corresponding to (26), when the mapping Y is globally invertible with respect to p for each (x, z) ∈ Ω × I, which is the case for our parallel beam examples in Section 6, (and more generally when Y arises from a generating function as in [21]). It then follows that Ω is uniformly Y -convex with respect to Ω * × I, if it is connected and for all x ∈ ∂Ω, z ∈ I, Y (x, z, p) ∈ Ω * , unit outer normal γ and unit tangent vector τ , for some constant δ 0 > 0. Conversely we note that (28) always implies (29).
2.3. Gradient estimates. By writing the constraint (4) in the form, we immediately infer a bound where d 0 = dist(Ω, ∂B 1 ) from which follows a Harnack inequality with C depending on d 0 , which is a special case of Lemma 4.2 in [9]. More generally, if we only assume Ω ⊂ B 1 , as in [9], then we obtain corresponding estimates under the condition |x · y| ≤ (1 − d 0 )|y|.
Next we need to use the boundary condition (5) to strengthen (4), in order to estimate |detY p | in (20), thereby controlling the right hand side of (3). First writing y = Y (x, z, p) in (16), we can estimate, for (x, z, p) ∈ U 0 , x ∈ Ω and a further constant C depending on d 0 . Consequently setting d * = sup y∈Ω * |y|, we obtain from (5), (6) and (32), Also we note here that the mapping Y is globally invertible with respect to p in U 0 and we have an explicit formula for the inverse, which gives us also an explicit expression for the gradient Du in terms of u and T u.

Convex targets.
For our reflector problem we clearly have that the sets P(x, z) are bounded, independently of the target domain Ω * . Let us now suppose that Ω * is convex and contains the origin. We claim that Ω * is uniformly Y * -convex with respect to any Ω B 1 (0) and I (0, ∞). To see this we fix a point p 0 = P (x, z, y 0 ) ∈ ∂P(x, z) and a support hyperplane H 0 to Ω * at y 0 , given by {α · y = 1} for some vector α ∈ R n . Then for y ∈ H 0 , p = P (x, y, z), we have 2α · p |p| 2 − (z − x · p) 2 = 1 and hence there exists a supporting enclosing ellipsoid to P(x, z) at p 0 , with equation, (37) If 0 ∈ ∂Ω * , then we clearly have that Ω * is Y * -convex and uniformly Y * -convex if Ω * is uniformly convex. We remark that in general, if the origin is outside Ω * , and Ω * is uniformly convex then Ω * will be uniformly Y * -convex if z is sufficiently small, that is sup I is sufficiently small which is equivalent to the reflector being sufficiently high above the target hyperplane, {y n+1 = 0}. This may be shown in a similar way by considering a supporting enclosing sphere to Ω * instead of the hyperplane H 0 .
3. Obliqueness. Recall that a boundary condition of the form G(·, u, Du) = 0 on ∂Ω for a second order partial differential equation in a domain Ω is called oblique (or degenerate oblique), with respect to u ∈ C 1 (Ω) if where c 0 is a positive constant and ν is the unit outer normal to ∂Ω.
Let Ω * Y (U 0 ) be a C 2 domain in R n and φ * be a C 2 defining function for Ω * satisfying φ * = 0, |∇φ * | = 0 on ∂Ω * and φ * < 0 in Ω * . The condition T u (Ω) = Y (·, u, Du)(Ω) = Ω * implies the boundary condition, The main estimate in this section is the following Proof. The proof essentials have already been given in [20]. For completeness and the convenience of readers, we provide the detailed proof here. The boundary condition Y u (Ω) = Ω * implies that By differentiation we have for any unit tangential vector τ on ∂Ω, and where ν is the outer normal to ∂Ω, whence for some χ. At this point we observe that χ > 0 on ∂Ω, since |∇φ * | = 0 on ∂Ω and det DY = 0. Denote w ij = D ij u − A ij (·, u, Du). Since det Y p = 0, from (18) one has Letting {w ij } denote the inverse matrix of {w ij }, we then have Combining (40) and (47), we denote which subsequently indicates that on ∂Ω. However, from (46) we also see that Eliminating χ from (48)-(50), we have Let x 0 ∈ ∂Ω be a point where β · ν| ∂Ω has its minimum value. We may make a rotation of coordinates so that e 1 , · · · , e n−1 are tangential to ∂Ω at x 0 and ν(x 0 ) = e n . Then (42) and (43) become We now consider a function for a sufficiently large constant K, where ν is extended C 2 smoothly inside Ω and G is given by (40) with the defining function φ * chosen so that for Y (x, u, Du) ∈ N * ∩ Ω * for some neighbourhood N * of ∂Ω * , in accordance with (27) and the uniform Y * -convexity of Ω * . By appropriate modification of G as in [22], we can then assume (55) holds throughout Ω. Since v| ∂Ω has a minimum at x 0 , D α v(x 0 ) = 0 for α = 1, · · · , n − 1, which can be written as We claim at the moment that D n v(x 0 ) ≤ C for some constant C. This can be rewritten as By (40) and (48), Combining (58) and (59), we obtain Multiplying (60) by φ * r ∂ pi Y r and sum over i from 1 to n, we have by (27), (29) and noticing that ν i = δ n i at x 0 . From (56), (57) and (61), we assert that for positive constants τ 0 and C. Hence if β · ν ≤ τ 0 /2C, we have the lower bound To complete the estimation of β · ν, by (51) it remains to obtain a lower bound of w ij ν i ν j at x 0 . For this we use the technique introduced in [20,26], which avoids invoking a dual problem as in [22,24], and which is not available in our generality. At x 0 , D α v = 0 for α = 1, · · · , n − 1 and the assertion D n v ≤ C indicate that It then suffices to have a lower bound for w ij ν i D j v at x 0 . From (47) we have (64) Therefore, by (50) and (61) for a constant c 0 > 0. Combining (51), (62) and (66), the desired obliqueness estimate (39) is thus derived G p · ν ≥ c 0 (67) on ∂Ω for a different positive constant c 0 depending only on domains Ω, Ω * and the claim that D n v(x 0 ) ≤ C.
In the rest, it remains to prove this assertion. By differentiating equation (3), we obtain, for r = 1, · · · , n, where h denotes the inhomogeneous term. Introducing the linearised operator L, we need to compute Lv for v given by (54).
Then we obtain Lv = w ij {F pr D ij u r + F prps D ir uD js u + F zpr D j uD ir u + F zpr D i uD jr u By choosing K in (70) sufficiently large, from (55) we can then ensure that near ∂Ω. Substituting (73) into (72), it follows that where C is a constant depending on h, Ω, Ω * , u C 1 (Ω) and K, see [22]. A suitable barrier is now provided by the uniform Y -convexity of Ω, which provides a defining function φ of Ω satisfying (28) in a fixed neighbourhood of ∂Ω. From this, we infer by the standard barrier argument (which entails further modifying φ and fixing a small enough neighbourhood of ∂Ω, [4]) that where again C is a constant depending on Ω, Ω * , u C 1 (Ω) and h. Since x 0 is a minimum point of v on ∂Ω, we can write where τ ≤ C. This completes the proof of the claim that D n v(x 0 ) ≤ C. Therefore, the proof of the strict obliqueness estimate (39) is finished.
Note that in Theorem 3.1, we need only assume B is defined for x ∈ Ω, z ∈ I and p ∈ P(x, z) so that it applies to (20) when f > 0, ∈ C 1 (Ω), g ∈ C 1 (Ω * ). The The oblique boundary value problem,(1), (2) for Monge-Ampère type equations has been studied in [13,18,22,25]. In particular, the global C 2 estimate in Theorem 1.1 now follows from [25] where the case A = 0 is treated for uniformly convex domains Ω and the uniform convexity condition, (55).
Once the second derivatives are bounded, equations (3) and (25) are effectively uniformly elliptic. This combined with the obliqueness estimate yields global C 2,α estimates, [12]. Moreover, the higher order estimates follow from the theory of linear elliptic equations with oblique boundary conditions [4]. 4. Proof of Theorem 1.2. In this section we prove Theorem 1.2 by the degree theory recently developed in [11] for second order elliptic operators with nonlinear oblique boundary conditions. For the standard Monge-Ampère equation with boundary condition (8), the existence via the method of continuity was given by Urbas in [23]. The situation here is more complicated because of the dependence of Y on u in the boundary condition (5).
First, we adopt the method of domain deformation in [20,22]. By approximation we may assume the domains Ω and Ω * are C 5 smooth. Fix a point x 0 ∈ Ω, by a translation we may assume that x 0 = 0. Define the function for some point y 0 ∈ Ω * . We may also assume that y 0 = 0, thus p 0 = 0 and u 0 has a simple form u 0 (x) = 1 for all x ∈ Ω and T u0 (Ω) Ω * . For r > 0 sufficiently small, one has the image Ω * 0 := T u0 (Ω 0 ) is uniformly Y * -convex with respect to Ω × I, where I is an interval depending on b 0 and r, and the function u 0 is admissible.
By the obliqueness estimate, β · ν > 0 on ∂B r , where ν is the unit outer normal of ∂B r . Let's estimate the coefficient γ, Therefore, for any small > 0, by choosing b 0 > 0 sufficiently large and r > 0 sufficiently small we have |γ| < . This implies that the kernel of (L, B) in (90) is trivial, namely w = 0. Consequently, u 0 is the unique solution of (83) at t = 0.
Note that the intensity functions f, g have positive lower and upper bounds. By integrating equation (83), we obtain uniform bounds for the quantities for two controlled positive constants C 1 and C 2 . From §2.3, it follows that u t and Du t are bounded. In fact, if sup u t > M , by §2.3 u t > cM , which will contradicts the upper bound C 2 when M is sufficiently large. By §2.3 again one has the gradient bound for Du t . Similarly, one can also obtain the lower bound for inf u t > C. Thus we have u t C 1 (Ω0) ≤ C ε . By Theorem 1.1, u t C 4,α (Ω0) ≤ C ε .
Note that there exists a diffeomorphism Φ 0 ∈ C 5 : R n → R n such that Φ 0 (Ω 0 ) = B 1 (0). DefineF for any u ∈ C 4,α (B 1 ). It is straightforward to check that (92) has the same ellipticity and obliqueness as (83) (which is essentially a change of variables), and where C is a uniform positive constant independent of t.
By constructing a family of homotopy problems similar to (83) over each pair (Ω t , Ω * ) and using the degree argument as above, we obtain a solution u ε ∈ C 4,α (Ω) of the boundary value problem T u (Ω) = Ω * for arbitrary small ε > 0. To complete the existence proof we now need to let ε → 0. Write equation (96) in the form of Let {u ε } be the family of solutions of the problems (97). From (82) we see that u ε − u 0 must be zero somewhere in Ω. Hence, from §2.3 sup Ω u ε is bounded independently of ε, so is |Du ε | and u ε . By Theorem 1.1, u ε C 4,α (Ω) is bounded independently of ε. Thus a subsequence of {u ε } converges in C 4,β (Ω) for 0 < β < α to a solution u solving (3)-(5), as required. Note that the C 0 bound of u depends on the initial choice of b 0 in (78), which in turn determines the constant δ in Theorems 1.1 and 1.2. Using the Harnack inequality, (32), we can also find a solution in an interval (δ, Kδ) for some constant K independent of δ for any δ > 0, so in particular there also exist an infinite number of solutions.

Parallel reflector.
Instead of the point light source, in this section we consider another model of reflector problem with parallel light source. We start with the derivation of the equation fulfilled by this model as follows.
(102) Therefore, we obtain the reflection mapping T : x ∈ Ω → y ∈ Ω * given by formula (102). It is easy to see that T is a diffeomorphism onto its image for a smooth positive function u with |Du| < 1; we will assume this in the following content. Write T u (x) = Y (x, u, Du), where Y = Y (x, z, p) is a mapping from U 0 = {(x, z, p) ∈ R n × R × R n : z > 0, |p| < 1} into R n given by From the conservation of energy and (17) By (102) we have the partial derivatives Hence, combining (106)-(107) into (104)-(105) we obtain the equation We remark that (108) is also derived in [8]. Extension of these models to non-flat targets are also considered in [7]. The (uniform) Y and Y * -convexity for domains Ω and Ω * can be defined similarly as in §2.2, see Definitions 2.1, 2.2. Similarly to Theorem 1.2, we have the following classical solvability result.