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

show all references

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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