Snell’s law in an isoperimetric setting

AIP Con e ence P oceedings 1260, 126 (2010); h ps://doi.o g/10.1063/1.3479313 1260, 126 © 2010 Ame ican Ins i u e o Physics. Snell’s law in an isope ime ic se ing Ci e as: AIP Con e ence P oceedings 1260, 126 (2010); h ps:// doi.o g/10.1063/1.3479313 Published Online: 05 Augus 2010 An onio Cañe e Snell’s law in an isope ime ic se ing An onio Cañe e Depa amen o de Ma emá ica Aplicada I, Uni e sidad de Se illa Abs ac . In his wo k we ocus on he isope ime ic p oblem in R2endowed wi h a piecewise cons an densi y. We will see ha he bounda y o an isope ime ic solu ion is no a smoo h cu e in gene al, since some co ne s may appea acco ding o a ule analogous o he Snell e ac ion law om Op ics. Keywo ds: Snell’s law, isope ime ic p oblem, mani olds wi h densi y PACS: 02.40.Ma INTRODUCTION Gi en a su ace M, he isope ime ic p oblem looks o he leas -pe ime e se in M enclosing a p esc ibed quan i y o a ea. A p io i, he exis ence o such a se is no assu ed, and i will be called isope ime ic egion i i exis s. In he li e a u e we can ind se e al pape s classi ying he isope ime ic egions o di e en su aces ([9], [5], [2]). In he las yea s, his p oblem has been s udied conside ing a densi y unc ion on he plane, which is jus a posi i e unc ion ha weigh s he a ea and pe ime e unc ionals (see [1], [4], [8]). Mo e p ecisely, i R2is endowed wi h a densi y :R2→R+, he a ea and he pe ime e o a se Ω⊂R2will be gi en by A(Ω) = ZΩ da,P(Ω) = Z∂Ω dx,(1) whe e da and dx a e he a ea and pe ime e elemen s [8]. Obse e ha when =1, we ob ain he s anda d Euclidean de ini ions o a ea and pe ime e , bu in gene al we will ha e ha he a ea and he pe ime e o a se Ωwill depend on he alues o he densi y on he poin s o Ωand ∂Ω. We ema k ha his new se ing, apa om being a gene aliza ion o he classical isope ime ic p oblem, co esponds o a change o he measu e in R2(see [7] o u he de ails). In pa icula , in his wo k we will ocus on piecewise cons an densi ies de ined on R2, summa izing some in e es ing esul s we ha e ob ained in [3]. We no e ha his kind o densi ies will always ha e a se o discon inui ies, and up o [3], he isope ime ic p ob- lem in a discon inuous densi y se ing had no been ea ed in li e a u e. Also obse e ha , due o de ini ions (1), each di e en densi y will yield a di e en isope ime ic p oblem. Example 1 Fix µ∈R,µ>1, and conside he unc ion in R2de ined by aking alue one in he lowe hal -plane {x2≤0}, and aking alue µin he uppe hal -plane {x2>0} (see Figu e 1). This is a piecewise cons an posi i e unc ion, and so i gi es a piecewise 126 CREDIT LINE (BELOW) TO BE INSERTED ON THE FIRST PAGE OF EACH PAPER CP1260, XVIII In e na ional Fall Wo kshop on Geome y and Physics edi ed by M. Aso ey, J. F. Ca iñena, J. Clemen e-Galla do, and E. Ma ínez © 2010 Ame ican Ins i u e o Physics 978-0-7354-0809-8/10/$30.00 cons an densi y called he hal -plane densi y. Fo ins ance, obse e ha a ball o adius =1/2has di e en a ea and pe ime e depending whe he i is con ained in he lowe o in he uppe hal -plane, om he de ini ions (1) gi en abo e (i i is con ained in he lowe hal -plane, he a ea is π 2and he pe ime e is 2π , whe eas i con ained in he uppe hal -plane, he a ea equals µπ 2and he pe ime e is 2µπ ). We will desc ibe he isope ime ic egions o his densi y in Subsec ion 2.1. FIGURE 1. The hal -plane densi y in R2 Example 2 Le B be he closed uni ball in he plane, and ix λ∈(0,1). We de ine he ball densi y by he unc ion in he plane aking alue λin B, and aking alue one ou side B (see Figu e 2). As his unc ion is posi i e and piecewise cons an , i yields a piecewise cons an densi y on he plane. We will also show he co esponding isope ime ic egions o his densi y in Subsec ion 2.2. FIGURE 2. The ball densi y in R2 In his se ing, an in e es ing phenomenon a ises due o he discon inui y o he densi y unc ion. In he case ha he bounda y cu e o an isope ime ic egion c osses he se o discon inui ies, a change o di ec ion will happen and a co ne will appea on he c ossing poin . This beha iou is analogous o he one desc ibed in Op ics by he Snell e ac ion law ( o ins an , see [10]), which explains he change o di ec ion o a ay o lig h when passing h ough wo di e en media. In ha si ua ion, he ay o ligh looks o he leas - ime pa h, which is de e mined by means o he Snell law exp ession. In some sense, his ac ag ees wi h he gene al isope ime ic app oach, which ies o minimize he pe ime e unde an a ea cons ain ( ha is, in bo h cases he aim is minimizing he ene gy). We shall see in Theo em 1.2 ha a similia ule is sa is ied in ou se ing, wi h he cons an alues o he densi y playing he ole o he e ac ion’s coe icien s. 127 We ema k ha , in his piecewise cons an densi y se ing, his las p ope y cons i- u es he main di e ence wi h espec o he classical isope ime ic p oblem (wi hou conside ing a densi y), whe e he isope ime ic bounda ies a e always smoo h cu es [6]. 1. PROPERTIES OF THE ISOPERIMETRIC BOUNDARIES Le {Ω1,...,Ωk}be a amily o closed se s pa i ioning R2, such ha he in e io s ˚ Ω1,..., ˚ Ωka e disjoin se s, and call Γ= k [ i=1 ∂Ωi. Conside a piecewise cons an densi y :R2→R+, ha is, a unc ion de ined in he plane as (p) =  i,p∈˚ Ωi, min{ i:p∈∂Ωi},p∈Γ, whe e 1,..., ka e posi i e eal numbe s. No e ha Γis he se o discon inui ies o . Le Ebe an isope ime ic egion in he plane wi h densi y , and call Σ=∂Ei s bounda y. In his se ing, Σsa is ies an in e es ing p ope y: he geodesic cu a u e o Σ−Γis cons an (when conside ing he Euclidean me ic), and so i will be composed o a cs o a ci cle (all o hem wi h he same adius), o o line segmen s. We poin ou ha his kind o densi y unc ions de ined on he plane will no a ec he alue o he geodesic cu a u e, since hei de i a i es anish in each se ˚ Ωi( he de ini ion o he geodesic cu a u e o gene al densi ies can be ound in [8, §. 3]). On he o he hand, we ema k ha some pieces o Γmay bound ou isope ime ic egion ( ha is, Σmay con ain pieces o Γ). The main p ope y in his se ing is he ollowing. I may happen ha he isope ime ic bounda y Σc osses ans e sally Γ, passing h ough egions o he plane wi h di e en alues o densi y. I his is he case, a simila ule o he Snell law mus be sa is ied, as we will see in Theo em 1.2. In o de o p o e his ac , we need he i s a ia ions o a ea and pe ime e (see [3, P op. 2.11 and eq. (11)]). Following P oposi ion 1.1 gi es hese exp essions in a mo e gene al way, o piecewise egula densi ies. These densi ies a e piecewise de ined in each se ˚ Ωiby means o a smoo h posi i e unc ion i. I is clea ha when ia e posi i e eal numbe s, we will ha e a piecewise cons an densi y o ou amily. A de ailed p oo o his P oposi ion can be ound in [3]. P oposi ion 1.1 Le be a piecewise egula densi y in R2, and Γi s se o dis- con inui ies. Conside E ⊂R2and call Σ=∂E i s bounda y. Assume Σ∩Γ6=/0, and conside p ∈Σ∩Γwi h p ∈∂Ωi∩∂Ωj. Fo a smoo h one-pa ame e a ia ion {Φ :R2→R2} ≥0wi h compac suppo in a neighbo hood o p, se A( ) = A(Φ (E)) and P( ) = P(Φ (E)). Then, i we deno e by X he associa ed ec o ield o he a ia- ion, and by νΣ he inwa d uni no mal ec o o Σ, we ha e ha he i s a ia ion o 128 a ea and pe ime e o E a e gi en by A0(0) = −ZΣ uda,(2) and P0(0) = ZΣ h∇ψ,νΣi udx −ZΣ H udx + ihX,νΣii(p) + jX,νΣj(p),(3) whe e u =:hX,νΣi, =eψ, H is he geodesic cu a u e o Σand Σi=Σ∩Ωi. When abo e P oposi ion 1.1 is applied o a piecewise cons an densi y, we ob ained he ollowing consequence, which cons i u es he main esul o ou wo k: Theo em 1.2 Le be a piecewise cons an densi y in R2, wi h Γ he se o discon inui- ies. Le E be an isope ime ic egion in he plane wi h densi y , and Σ=∂E. Assume ha Σc osses ans e sally Γa a poin p ∈∂Ωi∩∂Ωj. Then icosαi= jcosαj,(4) whe e αi,αja e he angles a p be ween Σand Γ(see Figu e 3). FIGURE 3. Snell’s law in ou isope ime ic se ing P oo . We gi e an ske ch o he p oo o (4). Conside a a ia ion o Ewi h compac suppo con ained in a neighbo hood o pp ese ing he a ea enclosed, ha is, A0(0) = 0, wi h X(p) angen o Γ(in ui i ely, you can cons uc such a ia ion de o ming E a away om pin o de o balance he a ea change, see [3, P op. 2.13] o [8, §. 3] o u he de ails). As Eis an isope ime ic egion, we ha e ha i s bounda y Σis a s a iona y se , and so P0(0) = 0. Obse e ha he i s e m in (3) anishes since ∇ψ=0 ( ecall ha is piecewise cons an , and so ψis). Mo eo e , aking in o accoun ha he geodesic cu a u e Ho Σis cons an , he second e m in (3) also anishes due o he a ea- p ese ing condi ion (2). Hence we ha e ihX,νΣii(p)+ jX,νΣj(p) = 0, 129 which easily yields Snell’s law (4) by applying he s anda d de ini ion o he scala p oduc , aking in o accoun ha cosαi=cos(−X(p),νΣi(p)),cosαj=cos(X(p),νΣj(p)), om Figu e 3. Rema k 1.3 We no e ha Theo em 1.2 ollows easily om P oposi ion 1.1, since he densi y is piecewise cons an . In a mo e gene al case, when conside ing piecewise egula densi ies, his Theo em also holds (e en in Rn), al hough he p oo is mo e elabo a e. Fo in e es ed eade s, i can be ound in [3, P op. 2.13]. 2. SOME PARTICULAR EXAMPLES As pa icula si ua ions, in his Sec ion we will desc ibe he isope ime ic egions o he piecewise cons an densi ies desc ibed in Examples 1 and 2 in he In oduc ion, he hal -plane densi y, and he ball densi y wi h λ∈(0,1). We shall check ha he isope i- me ic egions sa is y he Snell law (as s a ed in Theo em 1.2) when he bounda ies c oss ans e sally he se o discon inui ies o he densi y, and ha di e en kinds o isope i- me ic egions may appea o a gi en densi y, depending on he p esc ibed quan i y o a ea. We shall omi he p oo s o Theo ems 2.1 and 2.4, which can be ound in [3, §. 3]. 2.1. The hal -plane densi y Conside R2endowed wi h he hal -plane densi y om Example 1, aking alue one in he lowe hal -plane {x2≤0}and alue µ>1 in he uppe hal -plane {x2>0}. Fi s , we poin ou ha he exis ence o isope ime ic egions o his densi y is assu ed o any alue o he a ea (essen ially because any minimizing sequence is con e gen , see [3, Th. 3.2]). The key esul in his case is ha any isope ime ic solu ion mus be con ained in he lowe hal -plane [3, P op. 3.1]. Hence, om he s anda d classical isope ime ic p oblem, we deduce he ollowing consequence. Theo em 2.1 ([3, Th. 3.2]) The isope ime ic egion o a ea in he plane o he hal - plane densi y wi h µ>1is a ound ball con ained in {x2≤0}. Rema k 2.2 Obse e ha he bounda y o any isope ime ic egion o his densi y does no c oss he co esponding se o discon inui ies {x2=0}, and so he Snell law canno be applied in his case. Rema k 2.3 We no e ha he case 0<µ<1in he hal -plane densi y is analogous o he abo e one, wi h he isope ime ic egions consis ing o ound balls con ained in he uppe hal -plane, whe e he densi y akes i s minimum alue. 130 FIGURE 4. Isope ime ic egions in R2 o he hal -plane densi y 2.2. The ball densi y We now ocus on he isope ime ic p oblem in R2when conside ing he densi y de ined in he Example 2, he ball densi y wi h λ∈(0,1). Recall ha his densi y akes alue λin he closed uni ball B, and alue one ou side B, being he co esponding se o discon inui ies Γequal o ∂B. As o he p e ious densi y, he exis ence o isope ime ic solu ions is gua an eed (since any minimizing sequence is con e gen [3, Th. 3.18]). In his case, a e classi ying he di e en isope ime ic candida es by using he p ope ies om Sec ion 1, and disca ding non-possible solu ions, we inally ha e he ollowing esul . Theo em 2.4 ([3, Th. 3.23]) The isope ime ic egion o a ea in he plane o he ball densi y wi h λ∈(0,1)is (see Figu e 5): i) a ball o ype a), en i ely con ained in B, i ≤λπ; ii) a se o ype b), bounded by a piece o Γand an a c o ci cle ou side B, i λπ ≤ ≤ 1; iii) a se o ype b) o c), i 1< < 2; i ) a ball o ype c), c ossing o hogonally ∂B, i ≥ 2. whe e 1, 2a e ce ain eal numbe s depending on λ, 1, 2>λπ. FIGURE 5. Isope ime ic egions in R2 o he ball densi y, λ∈(0,1) 131 Rema k 2.5 We poin ou ha Snell’s law (4) mus be sa is ied in he wo e ices o se s o ype b), and he e o e we ha e some isope ime ic bounda ies which a e no smoo h cu es. On he o he hand, no ice ha o hogonal c ossing wi h he se o discon inui ies Γis allowed by Snell’s law (4), and i ac ually occu s o se s o ype c) below. Rema k 2.6 Rega ding Theo em 2.4, we conjec u e ha 1= 2, and so he possibili y iii) om he s a emen will no occu in any case, due o some nume ical compu a ions we ha e done. Rema k 2.7 The desc ip ion o he isope ime ic egions in he plane o o he s pa i- cula piecewise cons an densi ies can be ound in [3, §. 3], as well as some ela ed open ques ions. ACKNOWLEDGMENTS This wo k has been pa ially suppo ed by he MCyT esea ch p ojec MTM2007-61919. REFERENCES 1. C. BORELL, The B unn-Minkowski inequali y in Gauss space, In en . Ma h. 30 (1975), 207–216. 2. A. CAÑETE, The isope ime ic p oblem in su aces o e olu ion, P oc. o he XV In e na ional Wo kshop on Geome y and Physics 11, Publ. RSME (2007), 258–263. 3. A. CAÑETE, M. MIRANDA AND D. VITTONE, Some isope ime ic p oblems in planes wi h densi y, J. Geom. Anal. 20 (2009), 243–290. 4. C. CARROLL, A. JACOB, C. QUINN AND R. WALTERS, The isope ime ic p oblem on planes wi h densi y, Bull. Aus . Ma h. Soc. 78 (2008), 177–197. 5. H. HOWARDS, M. HUTCHINGS AND F. MORGAN, The isope ime ic p oblem on su aces, Ame . Ma h. Mon hly 106 (1999), 430–439. 6. F. MORGAN, Geome ic measu e heo y. A beginne ’s guide. Fou h edi ion, Else ie /Academic P ess, Ams e dam (2009). 7. F. MORGAN, Mani olds wi h densi y, No ices Ame . Ma h. Soc. 52 (2005), 853–858. 8. C. ROSALES, A. CAÑETE, V. BAYLE AND F. MORGAN, On he isope ime ic p oblem in Euclidean space wi h densi y, Calc. Va . Pa ial Di e en ial Equa ions 31 (2008), 27–46. 9. E. SCHMIDT, Übe eine neue Me hode zu Behandlung eine Klasse isope ime ische Au gaben im G ossen, Ma h. Z. 47 (1942), 489–642. 10. J. W. SHIRLEY, An ea ly expe imen al de e mina ion o Snell’s law, Ame . J. Phys. 19 (1951), 507– 508. 132