# American Institute of Mathematical Sciences

December  2017, 37(12): 6165-6181. doi: 10.3934/dcds.2017266

## $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem

 1 Normandie Univ, UNIHAVRE, LMAH, 76600 Le Havre, France 2 Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK 3 Universitá degli Studi della Campania "Luigi Vanvitelli", Scuola Politecnica e delle Scienze di Base, Dipartimento di Matematica e Fisica, Viale Lincoln, 5, 81100 Caserta, Italy

‡ Corresponding author

† N.K. has been partially financially supported by the EPSRC grant EP/N017412/1

Received  December 2016 Revised  July 2017 Published  August 2017

For
 $\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$
and
 $u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$
, consider the system
 $\label{1}\mathrm{A}_∞ u\, :=\,\Big(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^\bot \mathrm{H}_{PP}\Big)(\text{D} u):\text{D}^2u\, =\,0. \tag{1}$
We construct
 $\mathcal{D}$
-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our
 $\mathcal{D}$
-solutions are
 $W^{1,∞}$
-submersions and are obtained without any convexity hypotheses for
 $\mathrm{H}$
, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions
 $n≠ N$
.
Citation: Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266
##### References:
 [1] H. Abugirda and N. Katzourakis, Existence of 1D vectorial Absolute Minimisers in $L^∞$ under minimal assumptions, Proceedings of the AMS, 145 (2017), 2567-2575. doi: 10.1090/proc/13421. Google Scholar [2] L. Ambrosio and J. Malý, Very weak notions of differentiability, Proceedings of the Royal Society of Edinburgh A, 137 (2007), 447-455. doi: 10.1017/S0308210505001344. Google Scholar [3] G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$, Arkiv für Mat., 6 (1965), 33-53. doi: 10.1007/BF02591326. Google Scholar [4] G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅱ, Arkiv für Mat., 6 (1966), 409-431. doi: 10.1007/BF02590964. Google Scholar [5] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv für Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928. Google Scholar [6] 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. doi: 10.1007/BF02590989. Google Scholar [7] G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅲ, Arkiv für Mat., 7 (1969), 509-512. doi: 10.1007/BF02590888. Google Scholar [8] G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} = 0$, Manuscripta Math, 47 (1984), 133-151. doi: 10.1007/BF01174590. Google Scholar [9] G. Aronsson, Construction of singular solutions to the $p$-harmonic equation and its limit equation for $p=∞$, Manuscripta Math, 56 (1986), 135-158. doi: 10.1007/BF01172152. Google Scholar [10] G. Aronsson, M. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bulletin of the AMS, New Series, 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3. Google Scholar [11] E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. doi: 10.1090/S0002-9947-07-04338-3. Google Scholar [12] E. N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^{∞}$ functionals, Arch. Rational Mech. Analysis, 157 (2001), 255-283. doi: 10.1007/PL00004239. Google Scholar [13] E. N. Barron, R. Jensen and C. Wang, Lower semicontinuity of $L^{∞}$ functionals, Ann. I. H. Poincaré AN, 18 (2001), 495-517. doi: 10.1016/S0294-1449(01)00070-1. Google Scholar [14] A. C. Barroso, G. Croce and A. Ribeiro, Sufficient conditions for existence of solutions to vectorial differential inclusions and applications, Houston J. Math., 39 (2013), 929-967. Google Scholar [15] C. Le Bris and P. L. Lions, Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications, Ann. di Mat. Pura ed Appl., 183 (2004), 97-130. doi: 10.1007/s10231-003-0082-4. Google Scholar [16] L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings, Journal of Differential Equations, 253 (2012), 851-877. doi: 10.1016/j.jde.2012.04.015. Google Scholar [17] C. Castaing, P. R. de Fitte and M. Valadier, Young Measures on Topological spaces with Applications in Control Theory and Probability Theory, Mathematics and Its Applications (Kluwer Academic Publishers, Academic Press), Academic Press, 2004. doi: 10.1007/1-4020-1964-5. Google Scholar [18] M. G. Crandall, A visit with the 1-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, 75-122, Springer Lecture notes 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_3. Google Scholar [19] G. Croce, A differential inclusion: The case of an isotropic set, ESAIM Control Optim. Calc. Var., 11 (2005), 122-138. doi: 10.1051/cocv:2004035. Google Scholar [20] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd edition, Applied Mathematical Sciences, Springer, 2008. Google Scholar [21] B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems, Journal of Functional Analysis, 152 (1998), 404-446. doi: 10.1006/jfan.1997.3172. Google Scholar [22] B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 1999. doi: 10.1007/978-1-4612-1562-2. Google Scholar [23] B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case, Portugaliae Mathematica, 62 (2005), 421-436. Google Scholar [24] B. Dacorogna, G. Pisante and A. M. Ribeiro, On non quasiconvex problems of the calculus of variations, Discrete Contin. Dyn. Syst., 13 (2005), 961-983. doi: 10.3934/dcds.2005.13.961. Google Scholar [25] B. Dacorogna and A. M. Ribeiro, Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 907-921. doi: 10.1017/S0308210500003541. Google Scholar [26] B. Dacorogna and C. Tanteri, On the different convex hulls of sets involving singular values, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1261-1280. doi: 10.1017/S0308210500027311. Google Scholar [27] R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995. Google Scholar [28] L. C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, 2nd edition, 2010. doi: 10.1090/gsm/019. Google Scholar [29] L. C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC press, 1992. Google Scholar [30] L. C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Metric Spaces, De Gruyter, 2012. doi: 10.1515/9783110280517. Google Scholar [31] I. Fonseca and G. Leoni, Modern methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, 2007. Google Scholar [32] R. A. Horn and Ch. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. Google Scholar [33] N. Katzourakis, $L^∞$-variational problems for maps and the Aronsson PDE system, J. Differential Equations, 253 (2012), 2123-2139. doi: 10.1016/j.jde.2012.05.012. Google Scholar [34] N. Katzourakis, $∞$-minimal submanifolds, Proceedings of the AMS, 142 (2014), 2797-2811. doi: 10.1090/S0002-9939-2014-12039-9. Google Scholar [35] N. Katzourakis, On the structure of $∞$-harmonic maps, Communications in PDE, 39 (2014), 2091-2124. doi: 10.1080/03605302.2014.920351. Google Scholar [36] N. Katzourakis, Explicit $2D$ $∞$-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. Google Scholar [37] N. Katzourakis, Optimal $∞$-quasiconformal immersions, ESAIM Control, Opt. and Calc. Var., 21 (2015), 561-582. doi: 10.1051/cocv/2014038. Google Scholar [38] N. Katzourakis, Nonuniqueness in vector-valued calculus of variations in $L^∞$ and some linear elliptic systems, Comm. on Pure and Appl. Anal., 14 (2015), 313-327. doi: 10.3934/cpaa.2015.14.313. Google Scholar [39] N. Katzourakis, An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in $L^∞$, Springer Briefs in Mathematics, 2015. doi: 10.1007/978-3-319-12829-0. Google Scholar [40] N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, Journal of Differential Equations, 263 (2017), 641-686. doi: 10.1016/j.jde.2017.02.048. Google Scholar [41] N. Katzourakis, Absolutely minimising generalised solutions to the equations of vectorial Calculus of Variations in $L^∞$, Calculus of Variations and PDE, 56 (2017), 1-25. doi: 10.1007/s00526-016-1099-z. Google Scholar [42] N. Katzourakis, A new characterisation of $∞$-Harmonic and $p$-Harmonic maps via affine variations in $L^∞$, Electronic Journal of Differential Equations, 2017 (2017), 1-19. Google Scholar [43] N. Katzourakis, Solutions of vectorial Hamilton-Jacobi equations are rank-one Absolute Minimisers in $L^∞$, Advances in Nonlinear Analysis, in press.Google Scholar [44] N. Katzourakis, Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients, preprint, arXiv: 1507.03042.Google Scholar [45] N. Katzourakis and J. Manfredi, Remarks on the validity of the maximum principle for the $∞$-Laplacian, Le Matematiche, 71 (2016), 63-74. Google Scholar [46] N. Katzourakis and T. Pryer, On the numerical approximation of $∞$-Harmonic mappings Nonlinear Differential Equations and Applications, 23 (2016), Art. 51, 23 pp. doi: 10.1007/s00030-016-0415-9. Google Scholar [47] B. Kirchheim, Rigidity and geometry of microstructures, in Issue 16 of Lecture notes, Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, 2003.Google Scholar [48] B. Kirchheim, Deformations with finitely many gradients and stability of convex hulls, Comptes Rendus de l'Académie des Sciences, Séries I, Mathematics, 332 (2001), 289-294. doi: 10.1016/S0764-4442(00)01792-4. Google Scholar [49] S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, (1996), 239-251. Google Scholar [50] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math., 157 (2003), 715-742. doi: 10.4007/annals.2003.157.715. Google Scholar [51] P. Pedregal, Parametrized Measures and Variational Principles Birkhäuser, 1997. doi: 10.1007/978-3-0348-8886-8. Google Scholar [52] G. Pisante, Sufficient conditions for the existence of viscosity solutions for nonconvex Hamiltonians, SIAM J. Math. Anal., 36 (2004), 186-203. doi: 10.1137/S0036141003426902. Google Scholar [53] S. Sheffield and C. K. Smart, Vector-valued optimal Lipschitz extensions, Comm. Pure Appl. Math., 65 (2012), 128-154. doi: 10.1002/cpa.20391. Google Scholar [54] M. Valadier, Young measures, Methods of Nonconvex Analysis, 1446 (1990), 152-188. doi: 10.1007/BFb0084935. Google Scholar

show all references

##### References:
 [1] H. Abugirda and N. Katzourakis, Existence of 1D vectorial Absolute Minimisers in $L^∞$ under minimal assumptions, Proceedings of the AMS, 145 (2017), 2567-2575. doi: 10.1090/proc/13421. Google Scholar [2] L. Ambrosio and J. Malý, Very weak notions of differentiability, Proceedings of the Royal Society of Edinburgh A, 137 (2007), 447-455. doi: 10.1017/S0308210505001344. Google Scholar [3] G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$, Arkiv für Mat., 6 (1965), 33-53. doi: 10.1007/BF02591326. Google Scholar [4] G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅱ, Arkiv für Mat., 6 (1966), 409-431. doi: 10.1007/BF02590964. Google Scholar [5] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv für Mat., 6 (1967), 551-561. doi: 10.1007/BF02591928. Google Scholar [6] 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. doi: 10.1007/BF02590989. Google Scholar [7] G. Aronsson, Minimization problems for the functional $sup_x \mathcal{F}(x, f(x), f'(x))$ Ⅲ, Arkiv für Mat., 7 (1969), 509-512. doi: 10.1007/BF02590888. Google Scholar [8] G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} = 0$, Manuscripta Math, 47 (1984), 133-151. doi: 10.1007/BF01174590. Google Scholar [9] G. Aronsson, Construction of singular solutions to the $p$-harmonic equation and its limit equation for $p=∞$, Manuscripta Math, 56 (1986), 135-158. doi: 10.1007/BF01172152. Google Scholar [10] G. Aronsson, M. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bulletin of the AMS, New Series, 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3. Google Scholar [11] E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. doi: 10.1090/S0002-9947-07-04338-3. Google Scholar [12] E. N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of $L^{∞}$ functionals, Arch. Rational Mech. Analysis, 157 (2001), 255-283. doi: 10.1007/PL00004239. Google Scholar [13] E. N. Barron, R. Jensen and C. Wang, Lower semicontinuity of $L^{∞}$ functionals, Ann. I. H. Poincaré AN, 18 (2001), 495-517. doi: 10.1016/S0294-1449(01)00070-1. Google Scholar [14] A. C. Barroso, G. Croce and A. Ribeiro, Sufficient conditions for existence of solutions to vectorial differential inclusions and applications, Houston J. Math., 39 (2013), 929-967. Google Scholar [15] C. Le Bris and P. L. Lions, Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications, Ann. di Mat. Pura ed Appl., 183 (2004), 97-130. doi: 10.1007/s10231-003-0082-4. Google Scholar [16] L. Capogna and A. Raich, An Aronsson type approach to extremal quasiconformal mappings, Journal of Differential Equations, 253 (2012), 851-877. doi: 10.1016/j.jde.2012.04.015. Google Scholar [17] C. Castaing, P. R. de Fitte and M. Valadier, Young Measures on Topological spaces with Applications in Control Theory and Probability Theory, Mathematics and Its Applications (Kluwer Academic Publishers, Academic Press), Academic Press, 2004. doi: 10.1007/1-4020-1964-5. Google Scholar [18] M. G. Crandall, A visit with the 1-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, 75-122, Springer Lecture notes 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_3. Google Scholar [19] G. Croce, A differential inclusion: The case of an isotropic set, ESAIM Control Optim. Calc. Var., 11 (2005), 122-138. doi: 10.1051/cocv:2004035. Google Scholar [20] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd edition, Applied Mathematical Sciences, Springer, 2008. Google Scholar [21] B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems, Journal of Functional Analysis, 152 (1998), 404-446. doi: 10.1006/jfan.1997.3172. Google Scholar [22] B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 1999. doi: 10.1007/978-1-4612-1562-2. Google Scholar [23] B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case, Portugaliae Mathematica, 62 (2005), 421-436. Google Scholar [24] B. Dacorogna, G. Pisante and A. M. Ribeiro, On non quasiconvex problems of the calculus of variations, Discrete Contin. Dyn. Syst., 13 (2005), 961-983. doi: 10.3934/dcds.2005.13.961. Google Scholar [25] B. Dacorogna and A. M. Ribeiro, Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 907-921. doi: 10.1017/S0308210500003541. Google Scholar [26] B. Dacorogna and C. Tanteri, On the different convex hulls of sets involving singular values, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1261-1280. doi: 10.1017/S0308210500027311. Google Scholar [27] R. E. Edwards, Functional Analysis: Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995. Google Scholar [28] L. C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, 2nd edition, 2010. doi: 10.1090/gsm/019. Google Scholar [29] L. C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC press, 1992. Google Scholar [30] L. C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Metric Spaces, De Gruyter, 2012. doi: 10.1515/9783110280517. Google Scholar [31] I. Fonseca and G. Leoni, Modern methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, 2007. Google Scholar [32] R. A. Horn and Ch. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. Google Scholar [33] N. Katzourakis, $L^∞$-variational problems for maps and the Aronsson PDE system, J. Differential Equations, 253 (2012), 2123-2139. doi: 10.1016/j.jde.2012.05.012. Google Scholar [34] N. Katzourakis, $∞$-minimal submanifolds, Proceedings of the AMS, 142 (2014), 2797-2811. doi: 10.1090/S0002-9939-2014-12039-9. Google Scholar [35] N. Katzourakis, On the structure of $∞$-harmonic maps, Communications in PDE, 39 (2014), 2091-2124. doi: 10.1080/03605302.2014.920351. Google Scholar [36] N. Katzourakis, Explicit $2D$ $∞$-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. Google Scholar [37] N. Katzourakis, Optimal $∞$-quasiconformal immersions, ESAIM Control, Opt. and Calc. Var., 21 (2015), 561-582. doi: 10.1051/cocv/2014038. Google Scholar [38] N. Katzourakis, Nonuniqueness in vector-valued calculus of variations in $L^∞$ and some linear elliptic systems, Comm. on Pure and Appl. Anal., 14 (2015), 313-327. doi: 10.3934/cpaa.2015.14.313. Google Scholar [39] N. Katzourakis, An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in $L^∞$, Springer Briefs in Mathematics, 2015. doi: 10.1007/978-3-319-12829-0. Google Scholar [40] N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, Journal of Differential Equations, 263 (2017), 641-686. doi: 10.1016/j.jde.2017.02.048. Google Scholar [41] N. Katzourakis, Absolutely minimising generalised solutions to the equations of vectorial Calculus of Variations in $L^∞$, Calculus of Variations and PDE, 56 (2017), 1-25. doi: 10.1007/s00526-016-1099-z. Google Scholar [42] N. Katzourakis, A new characterisation of $∞$-Harmonic and $p$-Harmonic maps via affine variations in $L^∞$, Electronic Journal of Differential Equations, 2017 (2017), 1-19. Google Scholar [43] N. Katzourakis, Solutions of vectorial Hamilton-Jacobi equations are rank-one Absolute Minimisers in $L^∞$, Advances in Nonlinear Analysis, in press.Google Scholar [44] N. Katzourakis, Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients, preprint, arXiv: 1507.03042.Google Scholar [45] N. Katzourakis and J. Manfredi, Remarks on the validity of the maximum principle for the $∞$-Laplacian, Le Matematiche, 71 (2016), 63-74. Google Scholar [46] N. Katzourakis and T. 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