Given a map $u : \Om \sub \R^n \larrow \R^N$, the $\infty$-Laplacian is the
system \[ \label{1} \De_\infty u \, :=\, \Big(Du \ot Du + |Du|^2 [Du]^\bot \
\ot I \Big) : D^2 u\, = \, 0 \tag{1} \] and arises as the "Euler-Lagrange PDE"
of the supremal functional $E_\infty(u,\Om)= \|Du\|_{L^\infty(\Om)}.$ \eqref{1}
is the model PDE of vector-valued Calculus of Variations in $L^\infty$ and
first appeared in the author's recent work \cite{K1,K2,K3}. Solutions to
\eqref{1} present a natural phase separation with qualitatively different
behaviour on each phase. Moreover, on the interfaces the coefficients of
\eqref{1} are discontinuous. Herein we constuct new explicit smooth solutions
for $n=N=2$ for which the interfaces have triple junctions and nonsmooth
corners. The high complexity of these solutions provides further understanding
of the PDE \eqref{1} and shows there can be no regularity theory of interfaces.Comment: 5 pages, 4 figure

Katzourakis, Nicholas

arXiv

Central Archive at the University of Reading

Explicit 2D Infinity­Harmonic maps whose interfaces have junctions and corners Article Accepted Version Katzourakis, N. (2013) Explicit 2D Infinity­Harmonic maps whose interfaces have junctions and corners. Comptes Rendus Mathematique, 351 (17­18). pp. 677­680. ISSN 1631­073X Available at http://centaur.reading.ac.uk/33367/ It is advisable to refer to the publisher’s version if you intend to cite from the work. Publisher: Elsevier All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs onlineEXPLICIT 2D ∞-HARMONIC MAPS WHOSE INTERFACESHAVE JUNCTIONS AND CORNERSNICHOLAS KATZOURAKISAbstract. Given a map u : Ω ⊆ Rn −→ RN , the ∞-Laplacian is the system(1) ∆∞u :=(Du⊗Du + |Du|2[Du]⊥⊗ I): D2u = 0and arises as the “Euler-Lagrange PDE” of the supremal functional E∞(u,Ω) =‖Du‖L∞(Ω). (1) is the model PDE of vector-valued Calculus of Variations inL∞ and first appeared in the author’s recent work [K1]-[K5]. Solutions to(1) present a natural phase separation with qualitatively different behaviouron each phase. Moreover, on the interfaces the coefficients of (1) are discon-tinuous. Herein we constuct new explicit smooth solutions for n = N = 2for which the interfaces have triple junctions and nonsmooth corners. Thehigh complexity of these solutions provides further understanding of the PDE(1) and limits what might be true in future regularity considerations of theinterfaces.’Etant donne´ une carte u : Ω ⊆ Rn −→ RN , le ∞-Laplacien est le syste`me(1) ∆∞u :=(Du⊗Du + |Du|2[Du]⊥⊗ I): D2u = 0et se pre´sente comme la “E´DP d’Euler-Lagrange” de la fonctionnelle E∞(u,Ω)= ‖Du‖L∞(Ω). (1) est l’ E´DP mode`le du Calcul des Variations a` valeursvectorielles dans L∞ et elle est apparue pour la premie`re fois dans les travauxre´cents de l’ auteur [K1]-[K5]. Les solutions de (1) pre´sentent une se´parationde phase naturelle, avec un comportement qualitativement diffe´rent sur chaquephase. En outre, sur les interfaces les coefficients de (1) sont discontinus. Ici,nous construisons de nouvelles solutions re´gulie`res explicites pour n = N =2 pour lesquelles les interfaces ont des jonctions triples et des coins qui nesont pas lisses. La grande complexite´ de ces solutions permet d’ ame´liorer lacompre´hension de la E´DP (1) et les limites que pourrait eˆtre vrai dans desconside´rations de re´gularite´ futures des interfaces.1. IntroductionLet u : Ω ⊆ Rn −→ RN be a smooth map. In this note we are interested inconstructions of solutions to the∞-Laplace PDE system, which in index form reads(1.1) DiuαDjuβ D2ijuβ + |Du|2[Du]⊥αβD2iiuβ = 0.Here Diuα is the i-partial derivative of the α-component of u, [Du(x)]⊥ is theorthogonal projection on the nullspace of Du(x)> which is the transpose of thegradient matrix Du(x) : Rn −→ RN and | · | is the Euclidean norm on RN×n, i.e.|Du| = (DiuαDiuα) 12 . The summation convention is tacitly employed for indices2010 Mathematics Subject Classification. Primary 35J47, 35J62, 53C24; Secondary 49J99.Key words and phrases. ∞-Laplacian, Vector-valued Calculus of Variations in L∞, Interfaces,Phase separation.12 NICHOLAS KATZOURAKIS1 ≤ i, j ≤ n and 1 ≤ α, β ≤ N . In compact vector notation, we write (1.1) as(1.2) ∆∞u :=(Du⊗Du+ |Du|2[Du]⊥⊗ I): D2u = 0.(1.2) arises as the “Euler-Lagrange PDE system” in vector-valued Calculus of Vari-ations in the space L∞ for the model supremal functional(1.3) E∞(u,Ω) :=∥∥Du∥∥L∞(Ω)which we interpret as ess supΩ |Du|. (1.2) has first been derived by the author in[K1] and has been subsequently studied together with (1.3) in [K2, K3]. (1.2) is aquasilinear degenerate elliptic system in non-divergence form (with discontinuouscoefficients) which can be derived in the limit of the p-Laplace system ∆pu =Div(|Du|p−2Du) = 0 as p→∞. The special case of the scalar ∞-Laplacian reads∆∞u = DiuDjuD2iju = 0 and has a long history. In this case the coefficient|Du|2[Du]⊥ of (1.2) vanishes identically. The scalar ∆∞ was first derived in thelimit of the ∆p as p → ∞ and studied in the ’60s by Aronsson [A3, A4]. It hasbeen extensively studied ever since (see e.g. [C] and references therein).The motivation to study L∞ variational problems stems from their frequentappearance in applications (see e.g. [B]) because minimising maximum values fur-nishes more realistic models when compared to minimisation of averages with inte-gral functionals. The associated PDE systems are also very challenging since theyare nonlinear, in nondivergence form and with discontinuous coefficients and cannot be studied by classical techniques. Moreover, certain geometric problems areinherently connected to L∞. In the vector case N ≥ 2 our motivation comes fromthe problem of optimisation of quasiconformal deformations of Geometric Analysis(see [CR] and [K4]). For N = 1, the motivation is the optimisation of Lipschitzextensions (see [A3, C] and also [SS] of a recent vector-valued extension).A basic difficulty arising already in the scalar case is that DiuDjuD2iju = 0is degenerate elliptic and in non-divergence form and generally does not have dis-tributional, weak, strong or classical solutions. In [A6, A7] Aronsson demostrated“singular solutions” (see also [K]), which later were rigorously interpreted as vis-cosity solutions ([CIL]). In the vector case of N ≥ 2, “singular solutions” of (1.2)still appear (see [K1]). A further difficulty associated to (1.2) which is a genuinelyvectorial phenomenon and does not appear when N = 1 is that [Du]⊥ may bediscontinuous even for C∞ solutions. Such an example on R2 was given in [K1] andis u(x, y) = eix − eiy. This map is ∞-Harmonic in a neibourhood of the origin butthe projection [Du]⊥ is discontinuous on the diagonal.In general,∞-Harmonic maps present a phase separation, which is better under-stood when n = 2. For every C2 map u : Ω ⊆ R2 −→ RN solving ∆∞u = 0, there isa partition of Ω to the sets Ω2,Ω1,S of (2.1) below and u has 2- and 1- dimensionalbehaviour on Ω2 and Ω1 respectively (for details see [K3]). Also, [Du]⊥ is discon-tinuous on S. However, no information was provided on the possible structure ofthese interfaces. For the example eix − eiy, the interface S is a straight line.Herein, following [K1], we construct explicit examples of smooth solutions to(1.2) on the plane, whose interfaces have surprisingly complicated structure, pre-senting multiple junctions and corners. In particular, these examples show thatthere can be no regularity theory of interfaces, and the study of the system (1.2)itself is complicated even for smooth solutions. Moreover, these examples relate toquestions posed in [SS] for the interfaces of solutions to a different “∞-Laplacian”EXPLICIT 2D ∞-HARMONIC MAPS WHOSE INTERFACES HAVE JUNCTIONS AND CORNERS 3which arises when using the nonsmooth operator norm on RN×n instead of the Eu-clidean norm. The more complicated ∞-Laplacian of [SS] relates to vector-valuedLipschitz extensions rather than to Calculus of Variations in L∞.2. Constructions of 2-Dimensional ∞-Harmonic mappings.Let u : R2 −→ R2 be a map in C1(R2)2. We setΩ2 :={rk(Du) = 2}, Ω1 := int{rk(Du) ≤ 1}, S := ∂Ω2,(2.1)where “rk” denotes rank and “int” topological interior. We call Ω2 the 2-D phaseof u, Ω1 the 1-D phase of u and S the interface of u. Evidently, R2 = Ω2 ∪Ω1 ∪S.On Ω2 u is local diffeomorphism and on Ω1 “essentially scalar”.Proposition 2.1. Let u : R2 −→ R2 be a map given by(2.2) u(x, y) :=∫ xyeiK(t)dtwhere eia = (cos a, sin a)> and K ∈ C1(R) with supR |K| < pi2 . Then,(a) If K ≡ 0 on (−∞, 0] and K ′ > 0 on (0,∞), then ∆∞u = 0, u is affine onΩ1 and Ω2, Ω1, S are as in Figure 1, i.e.Ω1 = {x, y < 0}, S = ∂Ω1 ∪ {x = y ≥ 0}, Ω2 = R2 \ (Ω1 ∪ S).(2.3)(b) If K ≡ 0 on [−1,+1] and K ′ > 0 on (−∞,−1) ∪ (1,∞), then ∆∞u = 0, u isaffine on Ω1 and Ω2, Ω1, S are as in Figure 2, i.e.Ω1 = {−1 < x, y < 1}, S = ∂Ω1 ∪ {x = y, |y| ≥ 1}, Ω2 = R2 \ (Ω1 ∪ S).(2.4)Figure 1 Figure 2Example 2.2. For (a), an explicit K is K(t) = 1 − (t2 + 1)−1 for t > 0 andK(t) = 0 for t ≤ 0. For (b), an explicit K is K(t) = 1− ((t− 1)2 + 1)−1 for t > 1,K(t) = 0 for t ∈ [−1, 1] and K(t) = ((t+ 1)2 + 1)−1 − 1 for t < −1 (Fig. 3, 4).Figure 3 Figure 4Proof of Proposition 2.1. We begin with a little greater generality, in orderto obtain formulas needed later in Proposition 2.3. Fix two planar curves f, g ∈C2(R)2 which satisfy |f ′|2 = |g′|2 ≡ 1 and set v(x, y) := f(x)+g(y). Then, we haveDv(x, y) =(f ′(x), g′(y)) ∈ R2×2 and also D2xxv(x, y) = f ′′(x), D2yyv(x, y) = g′′(y),D2xyv = D2yxv = 0. Since |f ′| = |g′| ≡ 1, the rark of Dv is determined by the angle4 NICHOLAS KATZOURAKISof f ′, g′. Hence, rk(Dv(x, y)) = 2 if and only if f ′(x) is not colinear to g′(y) andrk(Dv(x, y)) = 1 otherwise. We recall from [K1] that a direct calculation gives(2.5) ∆∞v(x, y) = 2[(f ′(x), g′(y))]⊥(f ′′(x) + g′′(y)).We observe that [(f ′(x), g′(y))]⊥ = I − f ′(x)⊗ f ′(x) when f ′(x) = ±g′(y) and(2.6) [(f ′(x), g′(y))]⊥ = 0 ⇔ rk(Dv(x, y)) = 2 ⇔ f ′(x) 6= ±g′(y).We now choose f(t) :=∫ t0eiK(s)ds and g(t) := −f(t) forK ∈ C1(R) with supR |K| <pi/2. Then, u of (2.2) can be written as u(x, y) = f(x)− f(y) and also Du(x, y) =(f ′(x),−f ′(y)) ∈ R2×2. In view of (2.5), we deduce(2.7) ∆∞u(x, y) = 2[(f ′(x),−f ′(y))]⊥(f ′′(x)− f ′′(y)).Since |f ′| ≡ 1, for the angle of the 2 partials Dxu = f ′ and Dyu = −f ′ we have(2.8) cos(∠(f ′(x),−f ′(y))) = −f ′α(x)f ′α(y) = − cos (K(x)−K(y)).Since supR |K| < pi/2, we have∣∣K(x)−K(y)∣∣ < pi and as a result(2.9) [(f ′(x),−f ′(y))]⊥ = 0 ⇔ rk(Du(x, y)) = 2 ⇔ K(x) 6= K(y).(a) We now show that u is a solution on each quadrant separately.On {x, y > 0} we have K(x) 6= K(y) if and only if x 6= y, since K is strictlyincreasing on (0,∞). For x 6= y, (2.9) and (2.7) give ∆∞u(x, y) = 0. On the otherhand, for x = y, (2.7) readily gives ∆∞u(x, x) = 0.On {x, y ≤ 0}, we have K(x) = K(y) = 0 since K ≡ 0 on (−∞, 0]. Moreover,K ′ ≡ 0 on (−∞, 0] because K ∈ C1(R). By recalling that f ′(t) = eiK(t), by (2.2)we have u(x, y) = ei0(x− y) = e1(x− y) and Du(x, y) = (e1,−e1) = e1 ⊗ (e1 − e2)and also D2u ≡ 0. Hence, ∆∞u(x, y) = 0.On {x ≤ 0, y > 0}, we have K(x) = 0 and 0 < K(y) < pi/2 because K ≡ 0 on(−∞, 0] and 0 < K < pi/2 on (0,∞). Hence, K(x) 6= K(y) and by (2.9), (2.7) wehave ∆∞u(x, y) = 0.On {y ≤ 0, x > 0}, we have K(y) = 0 and 0 < K(x) < pi/2 and hence K(x) 6=K(y). By (2.9) and (2.7) we again deduce ∆∞u(x, y) = 0.We conclude (a) by observing that rk(Du) = 1 on {x = y} ∪ {x, y ≤ 0} andrk(Du) = 2 otherwise. Hence, (2.3) follows too.(b) On {−1 ≤ x, y ≤ 1}, we have K(x) = K(y) = K ′(x) = K ′(y) = 0. Henceu(x, y) = e1(x− y), Du(x, y) = e1 ⊗ (e1 − e2) and D2u ≡ 0. Thus, ∆∞u(x, y) = 0.On {x, y > 1}, we have K(x) 6= K(y) if and only if x 6= y, since K is strictlyincreasing on (1,∞). By (2.7) we evidently have ∆∞u(x, x) = 0 and for x 6= y by(2.9) and (2.7) we again deduce ∆∞u(x, y) = 0.On {y > 1,−1 ≤ x ≤ 1}, we have K(x) = 0 < K(y) < pi/2 and by (2.9) and(2.7) we again have ∆∞u(x, y) = 0. By arguing in the same way in the remainingsubsets of R2, (b) follows together with (2.4). The following result shows that Proposition 2.1 covers all possible qualitativebehaviours of 2-D ∞-Harmonic maps in separated variables:Proposition 2.3. Let u : R2 −→ R2 be a map of the form u(x, y) = f(x) + g(y)which satisfies ∆∞u = 0, where f, g are unit speed curves in C2(R)2. Then,(a) If Ω1 6= ∅, then u is affine on (connected components of) Ω1.(b) If S is a C1 graph near a certain point, then near that point either u is affineon S or S is part of the diagonals {x = ±y} of R2.EXPLICIT 2D ∞-HARMONIC MAPS WHOSE INTERFACES HAVE JUNCTIONS AND CORNERS 5Proof of Proposition 2.3. By (2.1), Ω2 is open and {rk(Du) ≤ 1} is closed andequals Ω1 ∪ S = Ω1. Since ∆∞u = 0 and |f ′|2 = |g′|2 ≡ 1, by (2.5), (2.6) we have(2.10) Ω1 ={(x, y) ∈ R2 ∣∣ f ′(x) = ±g′(y), f ′′(x) + g′′(y) // f ′(x), g′(y)}.Hence, there is a λ : Ω1 −→ R such that f ′′(x) + g′′(y) = λ(x, y)f ′(x) and alsof ′(x) = ±g′(y). Thus, we have λ(x, y) = λ(x, y)|f ′(x)|2 = (λ(x, y)f ′α(x))f ′α(x) =(f ′′α(x) + g′′α(y))f ′α(x) = f′′α(x)f′α(x) + g′′α(y)(± g′α(y)) = 0. Hence, (2.10) becomes(2.11) Ω1 ={(x, y) ∈ R2 ∣∣ f ′(x) = ±g′(y), f ′′(x) = −g′′(y)}.(a): If Ω1 6= ∅, for any (x0, y0) ∈ Ω1, there is an r > 0 such that (x0 − r, x0 +r) × (y0 − r, y0 + r) ⊆ Ω1. Hence, for y = y0 and x ∈ (x0 − r, x0 + r), we havef ′(x) = ±g′(y0) and hence f ′′(x) = 0. Similarly, g′′ = 0 on (y0 − r, y0 + r) andhence u is affine on connected components of Ω1.(b): If{(x, a(x)) : |x− x0| < r} ⊆ S for some r > 0 and a ∈ C1(x0 − r, x0 + r),we have f ′(x) = ±g′(a(x)) and by differentiating we get f ′′(x) = ±g′′(a(x))a′(x).Recall that we also have f ′′(x) = −g′′(a(x)). By these two we deduce (a′(x) ±1)g′′(a(x)) = 0. As a result, either a′ = ±1 near x0, or g′′ = 0 near a(x0). Theconclusion follows. Acknowledgement. I thank F. Fanelli for his help in the preparation of this note.I am grateful to the anonymous referee for the careful reading of the manuscript.ReferencesA3. G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv fu¨r Mat. 6 (1967),551 - 561.A4. G. Aronsson, On the partial differential equation u2xuxx + 2uxuyuxy + u2yuyy = 0, Arkiv fu¨rMat. 7 (1968), 395 - 425.A5. G. Aronsson, Minimization problems for the functional supxF (x, f(x), f ′(x)) III, Arkiv fu¨rMat. (1969), 509 - 512.A6. G. Aronsson, On Certain Singular Solutions of the Partial Differential Equation u2xuxx +2uxuyuxy + u2yuyy = 0, Manuscripta Math. 47 (1984), no 1-3, 133 - 151.A7. G. Aronsson, Construction of Singular Solutions to the p-Harmonic Equation and its LimitEquation for p =∞, Manuscripta Math. 56 (1986), 135 - 158.B. N. Barron, Viscosity Solutions and Analysis in L∞, Nonlinear analysis, differential equationsand control (Montreal QC, 1998), Kluwer Acad. Publ. Dordrecht, 1999, 1 - 60.CR. L. Capogna, A. Raich, An Aronsson type approach to extremal quasiconformal mappings, J.Differential Equations, Volume 253, Issue 3, 1 August 2012, Pages 851 - 877.C. M. G. Crandall, A visit with the ∞-Laplacian, in Calculus of Variations and Non-LinearPDE, Springer Lecture notes in Mathematics 1927, CIME, Cetraro Italy 2005.CIL. M. G. Crandall, H. Ishii, P.-L. Lions, User’s Guide to Viscosity Solutions of 2nd OrderPartial Differential Equations, Bulletin of the AMS, Vol. 27, Nr 1, Pages 1 - 67, 1992.K. N. Katzourakis, Explicit Singular Viscosity Solutions of the Aronsson Equation, C. R. Acad.Sci. Paris, Ser. I 349, No. 21 - 22, 1173 - 1176 (2011).K1. N. Katzourakis, L∞ Variational Problems for Maps and the Aronsson PDE System, J. Dif-ferential Equations, Volume 253, Issue 7, 1 October 2012, Pages 2123 - 2139.K2. N. Katzourakis, ∞-Minimal Submanifolds, Proceedings of the AMS, to appear.K3. N. Katzourakis, The Subelliptic ∞-Laplace System on Carnot-Carathe´odory Spaces, Adv.Nonlinear Analysis. Vol. 2, Issue 2, 213 - 233, 2013.K4. N. Katzourakis, On the Structure of ∞-Harmonic Maps, preprint, 2012.K5. N. Katzourakis, Optimal ∞-Quasiconformal Immersions, preprint, 2012.SS. S. Sheffield, C.K. Smart, Vector Valued Optimal Lipschitz Extensions, Comm. Pure Appl.Math., Vol. 65, Issue 1, January 2012, 128 - 154.6 NICHOLAS KATZOURAKISBCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009, Bilbao, Spain AND Department of Mathematics and Statistics, Whiteknights,PO Box 220, Reading RG6 6AX, Berkshire, UK.E-mail address: n.katzourakis@reading.ac.uk

Explicit 2D Infinity-Harmonic maps whose interfaces have junctions and corners

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