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

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  • 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 $Du(x) \in \mathcal{K} \subseteq R^{n\times n}$, $x\in \Omega$.
    Mathematics Subject Classification: Primary: 30C70, 30C75; Secondary: 35J47.


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  • [1]

    L. V. Ahlfors, On quasiconformal mappings, J. Anal. Math., 3 (1954), 1-58; J. Anal. Math., 3 (1954), 207-208.


    L. V. Ahlfors, Quasiconformal deformations and mappings in $R^n$, J. Anal. Math., 30 (1976), 74-97.


    S. N. Armstrong, M. G. Crandall, V. Julin and C. K. Smart, Convexity criteria and uniqueness of absolutely minimizing functions, Archive for Rational Mechanics and Analysis, 200 (2011), 405-443.doi: 10.1007/s00205-010-0348-0.


    S. N. Armstrong and C. K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differential Equations, 37 (2010), 381-384.doi: 10.1007/s00526-009-0267-9.


    G. Aronsson, Minimization problems for the functional su$p_x F(x, f(x), f'(x))$, Arkiv für Mat., 6 (1965), 33-53.


    G. Aronsson, Minimization problems for the functional su$p_x F(x,f(x), f'(x))$ II, Arkiv für Mat., 6 (1966), 409-431.


    G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv für Mat., 6 (1967), 551-561.


    G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy}=0$, Arkiv für Mat., 7 (1968), 395-425.


    G. Aronsson, Minimization problems for the functional su$p_x F(x, f(x), f'(x))$ III, Arkiv für Mat., 8 (1969), 509-512.


    K. Astala, T. Iwaniec and G. J. Martin, Deformations of annuli with smallest mean distortion, Arch. Ration. Mech. Anal., 195 (2010), 899-921.doi: 10.1007/s00205-009-0231-z.


    K. Astala, T. Iwaniec, G. J. Martin and J. Onninen, Optimal mappings of finite distortion, Proc. London Math. Soc., 91 (2005), 655-702.doi: 10.1112/S0024611505015376.


    G. Barles and Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Diff. Equations, 26 (2001), 2323-2337.doi: 10.1081/PDE-100107824.


    N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^\infty$ functionals, Arch. Rational Mech. Anal., 157 (2001), 255-283.doi: 10.1007/PL00004239.


    L. Bers, Quasiconformal mappings and Teichmuüller's theorem, in Analytic Functions, Princeton Univ. Press, Princeton, NJ (1960), 89-119.


    L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings, J. Differential Equations, 253 (2012), 851-877.doi: 10.1016/j.jde.2012.04.015.


    T. Champion and L. De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex Anal., 14 (2007), 515-541.


    T. Champion and L. De Pascale, $\Gamma$-convergence and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var., 10 (2004), 14-27.doi: 10.1051/cocv:2003036.


    M. G. Crandall, A visit with the $\infty$-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, Springer Lecture notes in Mathematics 1927, CIME, Cetraro Italy, 2005.


    M. G. Crandall, G. Gunnarsson and P. Y. Wang, Uniqueness of $\infty$-harmonic Functions and the Eikonal Equation, Comm. Partial Differential Equations, 32 (2007), 1587-1615.doi: 10.1080/03605300601088807.


    B. Dacorogna and P. Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and Their Applications, Vol. 37, Birkhäuser, 1999.doi: 10.1007/978-1-4612-1562-2.


    R. Gariepy, C. Wang and Y. Yu, Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers, Communications in PDE, 31 (2006), 1027-1046.doi: 10.1080/03605300600636788.


    F. W. Gehring, Quasiconformal mappings in Euclidean spaces, in Handbook of complex analysis: geometric function theory, Vol. 2, Elsevier, Amsterdam, 2005, 1-29.doi: 10.1016/S1874-5709(05)80005-8.


    E. Gusti, Direct Methods in the Calculus of Variations, River Edge, New Jersey London Singapore, World Scientific Publishing, 2003.doi: 10.1142/9789812795557.


    R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74.doi: 10.1007/BF00386368.


    R. Jensen, C. Wang and Y. Yu, Uniqueness and nonuniqueness of viscosity solutions to Aronsson's equation, Arch. Rational Mech. Anal., 190 (2008), 347-370.doi: 10.1007/s00205-007-0093-1.


    P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717.doi: 10.1137/S0036141000372179.


    N. Katzourakis, Maximum principles for vectorial approximate minimizers of nonconvex functionals, Calculus of Variations and PDE, 46 (2013), 505-522.doi: 10.1007/s00526-012-0491-6.


    N. Katzourakis, $L^{\infty}$ variational problems for maps and the Aronsson PDE system, J. Differential Equations, 253 (2012), 2123-2139.doi: 10.1016/j.jde.2012.05.012.


    N. Katzourakis, $\infty$-minimal submanifolds, Proc. Amer. Math. Soc.,142 (2014), 2797-–2811.doi: 10.1090/S0002-9939-2014-12039-9.


    N. Katzourakis, The subelliptic $\infty$-Laplace system on Carnot-Carathéodory spaces, Adv. Nonlinear Analysis., 2 (2013), 213-233.


    N. Katzourakis, Explicit $2D$ $\infty$-Harmonic Maps whose Interfaces have Junctions and Corners, Comptes Rendus Acad. Sci. Paris Ser.I, 351 (2013), 677-680.doi: 10.1016/j.crma.2013.07.028.


    N. Katzourakis, On the structure of $\infty$-harmonic maps, Communications in PDE 39 (2014), 1–-34.


    N. Katzourakis, Optimal $\infty$-quasiconformal maps, Control, Optimization and Calculus of Variations, to appear.


    S. Müller and V. Sverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math., 157 (2003), 715-742.doi: 10.4007/annals.2003.157.715.


    S. Sheffield and C. K. Smart, Vector valued optimal Lipschitz extensions, Comm. Pure Appl. Math., 65 (2012), 128-154.doi: 10.1002/cpa.20391.


    S. Strebel, Extremal quasiconformal mappings, Results Math., 10 (1986), 168-210.doi: 10.1007/BF03322374.


    O. Teichmüler, Extremale quasikonforme Abbildungen und quadratische differentiale, Abhandlungen der Preussischen Akademie der Wissenschaften, Math-naturw. Klasse, No. 22, 1939.


    J. Väisälä, Lectures on $n$-dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin, 1971.


    Y. Yu, $L^\infty$ variational problems and Aronsson equations, Arch. Rational Mech. Anal., 182 (2006), 153-180.doi: 10.1007/s00205-006-0424-7.

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