Snell’s law in an isoperimetric setting

AIP Conference Proceedings 1260, 126 (2010); https://doi.org/10.1063/1.3479313 1260, 126 © 2010 American Institute of Physics. Snell’s law in an isoperimetric setting Cite as: AIP Conference Proceedings 1260, 126 (2010); https:// doi.org/10.1063/1.3479313 Published Online: 05 August 2010 Antonio Cañete Snell’s law in an isoperimetric setting Antonio Cañete Departamento de Matemática Aplicada I, Universidad de Sevilla Abstract. In this work we focus on the isoperimetric problem in R2endowed with a piecewise constant density. We will see that the boundary of an isoperimetric solution is not a smooth curve in general, since some corners may appear according to a rule analogous to the Snell refraction law from Optics. Keywords: Snell’s law, isoperimetric problem, manifolds with density PACS: 02.40.Ma INTRODUCTION Given a surface M, the isoperimetric problem looks for the least-perimeter set in M enclosing a prescribed quantity of area. A priori, the existence of such a set is not assured, and it will be called isoperimetric region if it exists. In the literature we can find several papers classifying the isoperimetric regions for different surfaces ([9], [5], [2]). In the last years, this problem has been studied considering a density function on the plane, which is just a positive function that weights the area and perimeter functionals (see [1], [4], [8]). More precisely, if R2is endowed with a density f:R2→R+, the area and the perimeter of a set Ω⊂R2will be given by A(Ω) = ZΩ f da,P(Ω) = Z∂Ω f dx,(1) where da and dx are the area and perimeter elements [8]. Observe that when f=1, we obtain the standard Euclidean definitions of area and perimeter, but in general we will have that the area and the perimeter of a set Ωwill depend on the values of the density on the points of Ωand ∂Ω. We remark that this new setting, apart from being a generalization of the classical isoperimetric problem, corresponds to a change of the measure in R2(see [7] for further details). In particular, in this work we will focus on piecewise constant densities defined on R2, summarizing some interesting results we have obtained in [3]. We note that this kind of densities will always have a set of discontinuities, and up to [3], the isoperimetric problem in a discontinuous density setting had not been treated in literature. Also observe that, due to definitions (1), each different density will yield a different isoperimetric problem. Example 1 Fix µ∈R,µ>1, and consider the function in R2defined by taking value one in the lower half-plane {x2≤0}, and taking value µin the upper half-plane {x2>0} (see Figure 1). This is a piecewise constant positive function, and so it gives a piecewise 126 CREDIT LINE (BELOW) TO BE INSERTED ON THE FIRST PAGE OF EACH PAPER CP1260, XVIII International Fall Workshop on Geometry and Physics edited by M. Asorey, J. F. Cariñena, J. Clemente-Gallardo, and E. Martínez © 2010 American Institute of Physics 978-0-7354-0809-8/10/$30.00 constant density called the half-plane density. For instance, observe that a ball of radius r=1/2has different area and perimeter depending whether it is contained in the lower or in the upper half-plane, from the definitions (1) given above (if it is contained in the lower half-plane, the area is πr2and the perimeter is 2πr, whereas if contained in the upper half-plane, the area equals µπr2and the perimeter is 2µπr). We will describe the isoperimetric regions for this density in Subsection 2.1. FIGURE 1. The half-plane density in R2 Example 2 Let B be the closed unit ball in the plane, and fix λ∈(0,1). We define the ball density by the function in the plane taking value λin B, and taking value one outside B (see Figure 2). As this function is positive and piecewise constant, it yields a piecewise constant density on the plane. We will also show the corresponding isoperimetric regions for this density in Subsection 2.2. FIGURE 2. The ball density in R2 In this setting, an interesting phenomenon arises due to the discontinuity of the density function. In the case that the boundary curve of an isoperimetric region crosses the set of discontinuities, a change of direction will happen and a corner will appear on the crossing point. This behaviour is analogous to the one described in Optics by the Snell refraction law (for instant, see [10]), which explains the change of direction of a ray of ligth when passing through two different media. In that situation, the ray of light looks for the least-time path, which is determined by means of the Snell law expression. In some sense, this fact agrees with the general isoperimetric approach, which tries to minimize the perimeter under an area constraint (that is, in both cases the aim is minimizing the energy). We shall see in Theorem 1.2 that a similiar rule is satisfied in our setting, with the constant values of the density playing the role of the refraction’s coeficients. 127 We remark that, in this piecewise constant density setting, this last property constitutes the main difference with respect to the classical isoperimetric problem (without considering a density), where the isoperimetric boundaries are always smooth curves [6]. 1. PROPERTIES OF THE ISOPERIMETRIC BOUNDARIES Let {Ω1,...,Ωk}be a family of closed sets partitioning R2, such that the interiors ˚ Ω1,..., ˚ Ωkare disjoint sets, and call Γ= k [ i=1 ∂Ωi. Consider a piecewise constant density f:R2→R+, that is, a function defined in the plane as f(p) = fi,p∈˚ Ωi, min{fi:p∈∂Ωi},p∈Γ, where f1,..., fkare positive real numbers. Note that Γis the set of discontinuities of f. Let Ebe an isoperimetric region in the plane with density f, and call Σ=∂Eits boundary. In this setting, Σsatisfies an interesting property: the geodesic curvature of Σ−Γis constant (when considering the Euclidean metric), and so it will be composed of arcs of a circle (all of them with the same radius), or of line segments. We point out that this kind of density functions defined on the plane will not affect the value of the geodesic curvature, since their derivatives vanish in each set ˚ Ωi(the definition of the geodesic curvature for general densities can be found in [8, §. 3]). On the other hand, we remark that some pieces of Γmay bound our isoperimetric region (that is, Σmay contain pieces of Γ). The main property in this setting is the following. It may happen that the isoperimetric boundary Σcrosses transversally Γ, passing through regions of the plane with different values of density. If this is the case, a similar rule to the Snell law must be satisfied, as we will see in Theorem 1.2. In order to prove this fact, we need the first variations of area and perimeter (see [3, Prop. 2.11 and eq. (11)]). Following Proposition 1.1 gives these expressions in a more general way, for piecewise regular densities. These densities are piecewise defined in each set ˚ Ωiby means of a smooth positive function fi. It is clear that when fiare positive real numbers, we will have a piecewise constant density of our family. A detailed proof of this Proposition can be found in [3]. Proposition 1.1 Let f be a piecewise regular density in R2, and Γits set of discontinuities. Consider E ⊂R2and call Σ=∂E its boundary. Assume Σ∩Γ6=/0, and consider p ∈Σ∩Γwith p ∈∂Ωi∩∂Ωj. For a smooth one-parameter variation {Φt:R2→R2}t≥0with compact support in a neighborhood of p, set A(t) = A(Φt(E)) and P(t) = P(Φt(E)). Then, if we denote by X the associated vector field of the variation, and by νΣthe inward unit normal vector to Σ, we have that the first variation of 128 area and perimeter of E are given by A0(0) = −ZΣ f uda,(2) and P0(0) = ZΣ h∇ψ,νΣif udx −ZΣ H f udx +fihX,νΣii(p) + fjX,νΣj(p),(3) where u =:hX,νΣi, f =eψ, H is the geodesic curvature of Σand Σi=Σ∩Ωi. When above Proposition 1.1 is applied for a piecewise constant density, we obtained the following consequence, which constitutes the main result of our work: Theorem 1.2 Let f be a piecewise constant density in R2, with Γthe set of discontinuities. Let E be an isoperimetric region in the plane with density f , and Σ=∂E. Assume that Σcrosses transversally Γat a point p ∈∂Ωi∩∂Ωj. Then ficosαi=fjcosαj,(4) where αi,αjare the angles at p between Σand Γ(see Figure 3). FIGURE 3. Snell’s law in our isoperimetric setting Proof. We give an sketch of the proof of (4). Consider a variation of Ewith compact support contained in a neighborhood of ppreserving the area enclosed, that is, A0(0) = 0, with X(p)tangent to Γ(intuitively, you can construct such variation deforming Efar away from pin order to balance the area change, see [3, Prop. 2.13] or [8, §. 3] for further details). As Eis an isoperimetric region, we have that its boundary Σis a stationary set, and so P0(0) = 0. Observe that the first term in (3) vanishes since ∇ψ=0 (recall that f is piecewise constant, and so ψis). Moreover, taking into account that the geodesic curvature Hof Σis constant, the second term in (3) also vanishes due to the areapreserving condition (2). Hence we have fihX,νΣii(p)+ fjX,νΣj(p) = 0, 129 which easily yields Snell’s law (4) by applying the standard definition of the scalar product, taking into account that cosαi=cos(−X(p),νΣi(p)),cosαj=cos(X(p),νΣj(p)), from Figure 3. Remark 1.3 We note that Theorem 1.2 follows easily from Proposition 1.1, since the density is piecewise constant. In a more general case, when considering piecewise regular densities, this Theorem also holds (even in Rn), although the proof is more elaborate. For interested readers, it can be found in [3, Prop. 2.13]. 2. SOME PARTICULAR EXAMPLES As particular situations, in this Section we will describe the isoperimetric regions for the piecewise constant densities described in Examples 1 and 2 in the Introduction, the half-plane density, and the ball density with λ∈(0,1). We shall check that the isoperimetric regions satisfy the Snell law (as stated in Theorem 1.2) when the boundaries cross transversally the set of discontinuities of the density, and that different kinds of isoperimetric regions may appear for a given density, depending on the prescribed quantity of area. We shall omit the proofs of Theorems 2.1 and 2.4, which can be found in [3, §. 3]. 2.1. The half-plane density Consider R2endowed with the half-plane density from Example 1, taking value one in the lower half-plane {x2≤0}and value µ>1 in the upper half-plane {x2>0}. First, we point out that the existence of isoperimetric regions for this density is assured for any value of the area (essentially because any minimizing sequence is convergent, see [3, Th. 3.2]). The key result in this case is that any isoperimetric solution must be contained in the lower half-plane [3, Prop. 3.1]. Hence, from the standard classical isoperimetric problem, we deduce the following consequence. Theorem 2.1 ([3, Th. 3.2]) The isoperimetric region of area v in the plane for the halfplane density with µ>1is a round ball contained in {x2≤0}. Remark 2.2 Observe that the boundary of any isoperimetric region for this density does not cross the corresponding set of discontinuities {x2=0}, and so the Snell law cannot be applied in this case. Remark 2.3 We note that the case 0<µ<1in the half-plane density is analogous to the above one, with the isoperimetric regions consisting of round balls contained in the upper half-plane, where the density takes its minimum value. 130 FIGURE 4. Isoperimetric regions in R2for the half-plane density 2.2. The ball density We now focus on the isoperimetric problem in R2when considering the density defined in the Example 2, the ball density with λ∈(0,1). Recall that this density takes value λin the closed unit ball B, and value one outside B, being the corresponding set of discontinuities Γequal to ∂B. As for the previous density, the existence of isoperimetric solutions is guaranteed (since any minimizing sequence is convergent [3, Th. 3.18]). In this case, after classifying the different isoperimetric candidates by using the properties from Section 1, and discarding non-possible solutions, we finally have the following result. Theorem 2.4 ([3, Th. 3.23]) The isoperimetric region of area v in the plane for the ball density with λ∈(0,1)is (see Figure 5): i) a ball of type a), entirely contained in B, if v ≤λπ; ii) a set of type b), bounded by a piece of Γand an arc of circle outside B, if λπ ≤v≤v1; iii) a set of type b) or c), if v1<v<v2; iv) a ball of type c), crossing orthogonally ∂B, if v ≥v2. where v1, v2are certain real numbers depending on λ, v1,v2>λπ. FIGURE 5. Isoperimetric regions in R2for the ball density, λ∈(0,1) 131 Remark 2.5 We point out that Snell’s law (4) must be satisfied in the two vertices of sets of type b), and therefore we have some isoperimetric boundaries which are not smooth curves. On the other hand, notice that orthogonal crossing with the set of discontinuities Γis allowed by Snell’s law (4), and it actually occurs for sets of type c) below. Remark 2.6 Regarding Theorem 2.4, we conjecture that v1=v2, and so the possibility iii) from the statement will not occur in any case, due to some numerical computations we have done. Remark 2.7 The description of the isoperimetric regions in the plane for others particular piecewise constant densities can be found in [3, §. 3], as well as some related open questions. ACKNOWLEDGMENTS This work has been partially supported by the MCyT research project MTM2007-61919. REFERENCES 1. C. BORELL, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207–216. 2. A. CAÑETE, The isoperimetric problem in surfaces of revolution, Proc. of the XV International Workshop on Geometry and Physics 11, Publ. RSME (2007), 258–263. 3. A. CAÑETE, M. MIRANDA AND D. VITTONE, Some isoperimetric problems in planes with density, J. Geom. Anal. 20 (2009), 243–290. 4. C. CARROLL, A. JACOB, C. QUINN AND R. WALTERS, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc. 78 (2008), 177–197. 5. H. HOWARDS, M. HUTCHINGS AND F. MORGAN, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), 430–439. 6. F. MORGAN, Geometric measure theory. A beginner’s guide. Fourth edition, Elsevier/Academic Press, Amsterdam (2009). 7. F. MORGAN, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853–858. 8. C. ROSALES, A. CAÑETE, V. BAYLE AND F. MORGAN, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differential Equations 31 (2008), 27–46. 9. E. SCHMIDT, Über eine neue Methode zur Behandlung einer Klasse isoperimetrischer Aufgaben im Grossen, Math. Z. 47 (1942), 489–642. 10. J. W. SHIRLEY, An early experimental determination of Snell’s law, Amer. J. Phys. 19 (1951), 507– 508. 132