# American Institute of Mathematical Sciences

January  2015, 14(1): 313-327. doi: 10.3934/cpaa.2015.14.313

## Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems

 1 Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, United Kingdom

Received  March 2014 Revised  April 2014 Published  September 2014

For a Hamiltonian $H \in C^2(R^{N \times n})$ and a map $u:\Omega \subseteq R^n \to R^N$, we consider the supremal functional \begin{eqnarray} E_\infty (u,\Omega) := \|H(Du)\|_{L^\infty(\Omega)} . \end{eqnarray} The Euler-Lagrange" PDE associated to (1) is the quasilinear system \begin{eqnarray} A_\infty u := (H_P \otimes H_P + H[H_P]^\bot H_{PP})(Du):D^2 u = 0. \end{eqnarray} (1) and (2) are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [28]. Herein we show that the Dirichlet problem for (2) admits for all $n = N \geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ scalar uniqueness theory of Jensen [24] has no counterpart when $N \geq 2$. The key idea in the proofs is to recast (2) as a first order differential inclusion $Du(x) \in \mathcal{K} \subseteq R^{n\times n}$, $x\in \Omega$.
Citation: Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure & Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313
##### References:
 [1] L. V. Ahlfors, On quasiconformal mappings,, \emph{J. Anal. Math.}, 3 (1954), 1. Google Scholar [2] L. V. Ahlfors, Quasiconformal deformations and mappings in $R^n$,, \emph{J. Anal. Math.}, 30 (1976), 74. Google Scholar [3] S. N. Armstrong, M. G. Crandall, V. Julin and C. K. Smart, Convexity criteria and uniqueness of absolutely minimizing functions,, \emph{Archive for Rational Mechanics and Analysis}, 200 (2011), 405. doi: 10.1007/s00205-010-0348-0. Google Scholar [4] S. N. Armstrong and C. K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 381. doi: 10.1007/s00526-009-0267-9. Google Scholar [5] G. Aronsson, Minimization problems for the functional su$p_x F(x, f(x), f'(x))$,, \emph{Arkiv f\, 6 (1965), 33. Google Scholar [6] G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x), f'(x))$ II,, \emph{Arkiv f\, 6 (1966), 409. Google Scholar [7] G. Aronsson, Extension of functions satisfying Lipschitz conditions,, \emph{Arkiv f\, 6 (1967), 551. Google Scholar [8] G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy}=0$,, \emph{Arkiv f\, 7 (1968), 395. Google Scholar [9] G. Aronsson, Minimization problems for the functional su$p_x F(x, f(x), f'(x))$ III,, \emph{Arkiv f\, 8 (1969), 509. Google Scholar [10] K. Astala, T. Iwaniec and G. J. Martin, Deformations of annuli with smallest mean distortion,, \emph{Arch. Ration. Mech. Anal.}, 195 (2010), 899. doi: 10.1007/s00205-009-0231-z. Google Scholar [11] K. Astala, T. Iwaniec, G. J. Martin and J. Onninen, Optimal mappings of finite distortion,, \emph{Proc. London Math. Soc.}, 91 (2005), 655. doi: 10.1112/S0024611505015376. Google Scholar [12] G. Barles and Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, \emph{Comm. Partial Diff. Equations}, 26 (2001), 2323. doi: 10.1081/PDE-100107824. Google Scholar [13] N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals,, \emph{Arch. Rational Mech. Anal.}, 157 (2001), 255. doi: 10.1007/PL00004239. Google Scholar [14] L. Bers, Quasiconformal mappings and Teichmuüller's theorem,, in \emph{Analytic Functions}, (1960), 89. Google Scholar [15] L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings,, \emph{J. Differential Equations}, 253 (2012), 851. doi: 10.1016/j.jde.2012.04.015. Google Scholar [16] T. Champion and L. De Pascale, Principles of comparison with distance functions for absolute minimizers,, \emph{J. Convex Anal.}, 14 (2007), 515. Google Scholar [17] T. Champion and L. De Pascale, $\Gamma$-convergence and absolute minimizers for supremal functionals,, \emph{ESAIM Control Optim. Calc. Var.}, 10 (2004), 14. doi: 10.1051/cocv:2003036. Google Scholar [18] M. G. Crandall, A visit with the $\infty$-Laplacian,, in \emph{Calculus of Variations and Non-Linear Partial Differential Equations}, (1927). Google Scholar [19] M. G. Crandall, G. Gunnarsson and P. Y. Wang, Uniqueness of $\infty$-harmonic Functions and the Eikonal Equation,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1587. doi: 10.1080/03605300601088807. Google Scholar [20] B. Dacorogna and P. Marcellini, Implicit partial differential equations,, Progress in Nonlinear Differential Equations and Their Applications, (1999). doi: 10.1007/978-1-4612-1562-2. Google Scholar [21] R. Gariepy, C. Wang and Y. Yu, Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers,, \emph{Communications in PDE}, 31 (2006), 1027. doi: 10.1080/03605300600636788. Google Scholar [22] F. W. Gehring, Quasiconformal mappings in Euclidean spaces,, in \emph{Handbook of complex analysis: geometric function theory}, (2005), 1. doi: 10.1016/S1874-5709(05)80005-8. Google Scholar [23] E. Gusti, Direct Methods in the Calculus of Variations,, River Edge, (2003). doi: 10.1142/9789812795557. Google Scholar [24] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 51. doi: 10.1007/BF00386368. Google Scholar [25] R. Jensen, C. Wang and Y. Yu, Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation,, \emph{Arch. Rational Mech. Anal.}, 190 (2008), 347. doi: 10.1007/s00205-007-0093-1. Google Scholar [26] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179. Google Scholar [27] N. Katzourakis, Maximum principles for vectorial approximate minimizers of nonconvex functionals,, \emph{Calculus of Variations and PDE}, 46 (2013), 505. doi: 10.1007/s00526-012-0491-6. Google Scholar [28] N. Katzourakis, $L^{\infty}$ variational problems for maps and the Aronsson PDE system,, \emph{J. Differential Equations}, 253 (2012), 2123. doi: 10.1016/j.jde.2012.05.012. Google Scholar [29] N. Katzourakis, $\infty$-minimal submanifolds,, \emph{Proc. Amer. Math. Soc.}, (2014). doi: 10.1090/S0002-9939-2014-12039-9. Google Scholar [30] N. Katzourakis, The subelliptic $\infty$-Laplace system on Carnot-Carathéodory spaces,, \emph{Adv. Nonlinear Analysis.}, 2 (2013), 213. Google Scholar [31] N. Katzourakis, Explicit $2D$ $\infty$-Harmonic Maps whose Interfaces have Junctions and Corners,, \emph{Comptes Rendus Acad. Sci. Paris Ser.I}, 351 (2013), 677. doi: 10.1016/j.crma.2013.07.028. Google Scholar [32] N. Katzourakis, On the structure of $\infty$-harmonic maps,, \emph{Communications in PDE} 39 (2014), (2014). Google Scholar [33] N. Katzourakis, Optimal $\infty$-quasiconformal maps,, Control, (). Google Scholar [34] S. Müller and V. Sverák, Convex integration for Lipschitz mappings and counterexamples to regularity,, \emph{Ann. of Math.}, 157 (2003), 715. doi: 10.4007/annals.2003.157.715. Google Scholar [35] S. Sheffield and C. K. Smart, Vector valued optimal Lipschitz extensions,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 128. doi: 10.1002/cpa.20391. Google Scholar [36] S. Strebel, Extremal quasiconformal mappings,, \emph{Results Math.}, 10 (1986), 168. doi: 10.1007/BF03322374. Google Scholar [37] O. Teichmüler, Extremale quasikonforme Abbildungen und quadratische differentiale,, Abhandlungen der Preussischen Akademie der Wissenschaften, (1939). Google Scholar [38] J. Väisälä, Lectures on $n$-dimensional Quasiconformal Mappings,, Lecture Notes in Mathematics, (1971). Google Scholar [39] Y. Yu, $L^\infty$ variational problems and Aronsson equations,, \emph{Arch. Rational Mech. Anal.}, 182 (2006), 153. doi: 10.1007/s00205-006-0424-7. Google Scholar

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##### References:
 [1] L. V. Ahlfors, On quasiconformal mappings,, \emph{J. Anal. Math.}, 3 (1954), 1. Google Scholar [2] L. V. Ahlfors, Quasiconformal deformations and mappings in $R^n$,, \emph{J. Anal. Math.}, 30 (1976), 74. Google Scholar [3] S. N. Armstrong, M. G. Crandall, V. Julin and C. K. Smart, Convexity criteria and uniqueness of absolutely minimizing functions,, \emph{Archive for Rational Mechanics and Analysis}, 200 (2011), 405. doi: 10.1007/s00205-010-0348-0. Google Scholar [4] S. N. Armstrong and C. K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 381. doi: 10.1007/s00526-009-0267-9. Google Scholar [5] G. Aronsson, Minimization problems for the functional su$p_x F(x, f(x), f'(x))$,, \emph{Arkiv f\, 6 (1965), 33. Google Scholar [6] G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x), f'(x))$ II,, \emph{Arkiv f\, 6 (1966), 409. Google Scholar [7] G. Aronsson, Extension of functions satisfying Lipschitz conditions,, \emph{Arkiv f\, 6 (1967), 551. Google Scholar [8] G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy}=0$,, \emph{Arkiv f\, 7 (1968), 395. Google Scholar [9] G. Aronsson, Minimization problems for the functional su$p_x F(x, f(x), f'(x))$ III,, \emph{Arkiv f\, 8 (1969), 509. Google Scholar [10] K. Astala, T. Iwaniec and G. J. Martin, Deformations of annuli with smallest mean distortion,, \emph{Arch. Ration. Mech. Anal.}, 195 (2010), 899. doi: 10.1007/s00205-009-0231-z. Google Scholar [11] K. Astala, T. Iwaniec, G. J. Martin and J. Onninen, Optimal mappings of finite distortion,, \emph{Proc. London Math. Soc.}, 91 (2005), 655. doi: 10.1112/S0024611505015376. Google Scholar [12] G. Barles and Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, \emph{Comm. Partial Diff. Equations}, 26 (2001), 2323. doi: 10.1081/PDE-100107824. Google Scholar [13] N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals,, \emph{Arch. Rational Mech. Anal.}, 157 (2001), 255. doi: 10.1007/PL00004239. Google Scholar [14] L. Bers, Quasiconformal mappings and Teichmuüller's theorem,, in \emph{Analytic Functions}, (1960), 89. Google Scholar [15] L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings,, \emph{J. Differential Equations}, 253 (2012), 851. doi: 10.1016/j.jde.2012.04.015. Google Scholar [16] T. Champion and L. De Pascale, Principles of comparison with distance functions for absolute minimizers,, \emph{J. Convex Anal.}, 14 (2007), 515. Google Scholar [17] T. Champion and L. De Pascale, $\Gamma$-convergence and absolute minimizers for supremal functionals,, \emph{ESAIM Control Optim. Calc. Var.}, 10 (2004), 14. doi: 10.1051/cocv:2003036. Google Scholar [18] M. G. Crandall, A visit with the $\infty$-Laplacian,, in \emph{Calculus of Variations and Non-Linear Partial Differential Equations}, (1927). Google Scholar [19] M. G. Crandall, G. Gunnarsson and P. Y. Wang, Uniqueness of $\infty$-harmonic Functions and the Eikonal Equation,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1587. doi: 10.1080/03605300601088807. Google Scholar [20] B. Dacorogna and P. Marcellini, Implicit partial differential equations,, Progress in Nonlinear Differential Equations and Their Applications, (1999). doi: 10.1007/978-1-4612-1562-2. Google Scholar [21] R. Gariepy, C. Wang and Y. Yu, Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers,, \emph{Communications in PDE}, 31 (2006), 1027. doi: 10.1080/03605300600636788. Google Scholar [22] F. W. Gehring, Quasiconformal mappings in Euclidean spaces,, in \emph{Handbook of complex analysis: geometric function theory}, (2005), 1. doi: 10.1016/S1874-5709(05)80005-8. Google Scholar [23] E. Gusti, Direct Methods in the Calculus of Variations,, River Edge, (2003). doi: 10.1142/9789812795557. Google Scholar [24] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 51. doi: 10.1007/BF00386368. Google Scholar [25] R. Jensen, C. Wang and Y. Yu, Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation,, \emph{Arch. Rational Mech. Anal.}, 190 (2008), 347. doi: 10.1007/s00205-007-0093-1. Google Scholar [26] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179. Google Scholar [27] N. Katzourakis, Maximum principles for vectorial approximate minimizers of nonconvex functionals,, \emph{Calculus of Variations and PDE}, 46 (2013), 505. doi: 10.1007/s00526-012-0491-6. Google Scholar [28] N. Katzourakis, $L^{\infty}$ variational problems for maps and the Aronsson PDE system,, \emph{J. Differential Equations}, 253 (2012), 2123. doi: 10.1016/j.jde.2012.05.012. Google Scholar [29] N. Katzourakis, $\infty$-minimal submanifolds,, \emph{Proc. Amer. Math. Soc.}, (2014). doi: 10.1090/S0002-9939-2014-12039-9. Google Scholar [30] N. Katzourakis, The subelliptic $\infty$-Laplace system on Carnot-Carathéodory spaces,, \emph{Adv. Nonlinear Analysis.}, 2 (2013), 213. Google Scholar [31] N. Katzourakis, Explicit $2D$ $\infty$-Harmonic Maps whose Interfaces have Junctions and Corners,, \emph{Comptes Rendus Acad. Sci. Paris Ser.I}, 351 (2013), 677. doi: 10.1016/j.crma.2013.07.028. Google Scholar [32] N. Katzourakis, On the structure of $\infty$-harmonic maps,, \emph{Communications in PDE} 39 (2014), (2014). Google Scholar [33] N. Katzourakis, Optimal $\infty$-quasiconformal maps,, Control, (). Google Scholar [34] S. Müller and V. Sverák, Convex integration for Lipschitz mappings and counterexamples to regularity,, \emph{Ann. of Math.}, 157 (2003), 715. doi: 10.4007/annals.2003.157.715. Google Scholar [35] S. Sheffield and C. K. Smart, Vector valued optimal Lipschitz extensions,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 128. doi: 10.1002/cpa.20391. Google Scholar [36] S. Strebel, Extremal quasiconformal mappings,, \emph{Results Math.}, 10 (1986), 168. doi: 10.1007/BF03322374. Google Scholar [37] O. Teichmüler, Extremale quasikonforme Abbildungen und quadratische differentiale,, Abhandlungen der Preussischen Akademie der Wissenschaften, (1939). Google Scholar [38] J. Väisälä, Lectures on $n$-dimensional Quasiconformal Mappings,, Lecture Notes in Mathematics, (1971). Google Scholar [39] Y. Yu, $L^\infty$ variational problems and Aronsson equations,, \emph{Arch. Rational Mech. Anal.}, 182 (2006), 153. doi: 10.1007/s00205-006-0424-7. Google Scholar
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