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Second-order L variational problems and the ∞-polylaplacian

  • Nikos Katzourakis ORCID logo EMAIL logo and Tristan Pryer

Abstract

In this paper we initiate the study of second-order variational problems in L, seeking to minimise the L norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given HC1(sn×n), for the functional

E(u,𝒪)=H(D2u)L(𝒪),uW2,(Ω),𝒪Ω,

the associated equation is the fully nonlinear third-order PDE

A2u:=(HX(D2u))3:(D3u)2=0.

Special cases arise when H is the Euclidean length of either the full hessian or of the Laplacian, leading to the -polylaplacian and the -bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.


Communicated by Juan Manfredi


Acknowledgements

Nikos Katzourakis would like to thank Craig Evans, Robert Jensen, Roger Moser, Juan Manfredi and Jan Kristensen for their inspiring mathematical discussions and especially their illuminating remarks on 𝒟-solutions and on second-order L variational problems.

References

[1] H. Abugirda, B. Ayanbayev and N. Katzourakis, Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian, preprint (2017), https://arxiv.org/pdf/1704.04492. 10.1216/rmj.2020.50.383Search in Google Scholar

[2] H. Abugirda and N. Katzourakis, Existence of 1D vectorial absolute minimisers in L under minimal assumptions, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2567–2575. 10.1090/proc/13421Search in Google Scholar

[3] L. Ambrosio and J. Malý, Very weak notions of differentiability, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 3, 447–455. 10.1017/S0308210505001344Search in Google Scholar

[4] G. Aronsson, Minimization problems for the functional supxF(x,f(x),f(x)), Ark. Mat. 6 (1965), 33–53. 10.1007/BF02591326Search in Google Scholar

[5] G. Aronsson, Minimization problems for the functional supxF(x,f(x),f(x)). II, Ark. Mat. 6 (1966), 409–431. 10.1007/BF02590964Search in Google Scholar

[6] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561. 10.1007/BF02591928Search in Google Scholar

[7] G. Aronsson, On the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=0, Ark. Mat. 7 (1968), 395–425. 10.1007/BF02590989Search in Google Scholar

[8] G. Aronsson, Minimization problems for the functional supxF(x,f(x),f(x)). III, Ark. Mat. 7 (1969), 509–512. 10.1007/BF02590888Search in Google Scholar

[9] G. Aronsson, On certain singular solutions of the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=0, Manuscripta Math. 47 (1984), no. 1–3, 133–151. 10.1007/BF01174590Search in Google Scholar

[10] G. Aronsson, Construction of singular solutions to the p-harmonic equation and its limit equation for p=, Manuscripta Math. 56 (1986), no. 2, 135–158. 10.1007/BF01172152Search in Google Scholar

[11] G. Aronsson and E. N. Barron, L variational problems with running costs and constraints, Appl. Math. Optim. 65 (2012), no. 1, 53–90. 10.1007/s00245-011-9151-zSearch in Google Scholar

[12] B. Ayanbayev and N. Katzourakis, A pointwise characterisation of the PDE system of vectorial calculus of variations in L, Proc. Roy. Soc. Edinburgh Sect. A, to appear. Search in Google Scholar

[13] E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of L functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 4, 495–517. 10.1016/s0294-1449(01)00070-1Search in Google Scholar

[14] E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of L functionals, Arch. Ration. Mech. Anal. 157 (2001), no. 4, 255–283. 10.1007/PL00004239Search in Google Scholar

[15] C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory, Math. Appl. 571, Kluwer Academic Publishers, Dordrecht, 2004. 10.1007/1-4020-1964-5Search in Google Scholar

[16] M. G. Crandall, A visit with the -Laplace equation, Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Math. 1927, Springer, Berlin (2008), 75–122. 10.1007/978-3-540-75914-0_3Search in Google Scholar

[17] G. Croce, N. Katzourakis and G. Pisante, 𝒟-solutions to the system of vectorial calculus of variations in L via the singular value problem, Discrete Contin. Dyn. Syst. 37 (2017), no. 12, 6165–6181. 10.3934/dcds.2017266Search in Google Scholar

[18] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Appl. Math. Sci. 78, Springer, New York, 2008. Search in Google Scholar

[19] B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1999. 10.1007/978-1-4612-1562-2Search in Google Scholar

[20] B. Dacorogna and G. Pisante, A general existence theorem for differential inclusions in the vector valued case, Port. Math. (N.S.) 62 (2005), no. 4, 421–436. Search in Google Scholar

[21] J. M. Danskin, The theory of max–min, with applications, SIAM J. Appl. Math. 14 (1966), 641–664. 10.1137/0114053Search in Google Scholar

[22] C. De Lellis and L. Székelyhidi, Jr., The Euler equations as a differential inclusion, Ann. of Math. (2) 170 (2009), no. 3, 1417–1436. 10.4007/annals.2009.170.1417Search in Google Scholar

[23] R. E. Edwards, Functional Analysis. Theory and Applications, Dover Publications, New York, 1995. Search in Google Scholar

[24] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Search in Google Scholar

[25] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar

[26] C. L. Fefferman, A sharp form of Whitney’s extension theorem, Ann. of Math. (2) 161 (2005), no. 1, 509–577. 10.4007/annals.2005.161.509Search in Google Scholar

[27] L. C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Measure Spaces, De Gruyter, Berlin, 2012. 10.1515/9783110280517Search in Google Scholar

[28] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monogr. Math., Springer, New York, 2007. Search in Google Scholar

[29] A. Gastel and C. Scheven, Regularity of polyharmonic maps in the critical dimension, Comm. Anal. Geom. 17 (2009), no. 2, 185–226. 10.4310/CAG.2009.v17.n2.a2Search in Google Scholar

[30] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd ed., Appunti. Sc. Norm. Super. Pisa (N. S.) 11, Edizioni della Normale, Pisa, 2012. 10.1007/978-88-7642-443-4Search in Google Scholar

[31] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar

[32] P. Hornung and R. Moser, Intrinsically p-biharmonic maps, Calc. Var. Partial Differential Equations 51 (2014), no. 3–4, 597–620. 10.1007/s00526-013-0688-3Search in Google Scholar

[33] N. Katzourakis, L variational problems for maps and the Aronsson PDE system, J. Differential Equations 253 (2012), no. 7, 2123–2139. 10.1016/j.jde.2012.05.012Search in Google Scholar

[34] N. Katzourakis, Explicit 2D -harmonic maps whose interfaces have junctions and corners, C. R. Math. Acad. Sci. Paris 351 (2013), no. 17–18, 677–680. 10.1016/j.crma.2013.07.028Search in Google Scholar

[35] N. Katzourakis, -minimal submanifolds, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2797–2811. 10.1090/S0002-9939-2014-12039-9Search in Google Scholar

[36] N. Katzourakis, On the structure of -harmonic maps, Comm. Partial Differential Equations 39 (2014), no. 11, 2091–2124. 10.1080/03605302.2014.920351Search in Google Scholar

[37] N. Katzourakis, An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L, Springer Briefs Math., Springer, Cham, 2015. 10.1007/978-3-319-12829-0Search in Google Scholar

[38] N. Katzourakis, Nonuniqueness in vector-valued calculus of variations in L and some linear elliptic systems, Commun. Pure Appl. Anal. 14 (2015), no. 1, 313–327. 10.3934/cpaa.2015.14.313Search in Google Scholar

[39] N. Katzourakis, Optimal -quasiconformal immersions, ESAIM Control Optim. Calc. Var. 21 (2015), no. 2, 561–582. 10.1051/cocv/2014038Search in Google Scholar

[40] N. Katzourakis, A characterisation of -harmonic and p-harmonic maps via affine variations in L, Electron. J. Differential Equations 2017 (2017), Paper No. 29. Search in Google Scholar

[41] N. Katzourakis, Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in L, Calc. Var. Partial Differential Equations 56 (2017), no. 1, Article ID 15. 10.1007/s00526-016-1099-zSearch in Google Scholar

[42] N. Katzourakis, Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, J. Differential Equations 263 (2017), no. 1, 641–686. 10.1016/j.jde.2017.02.048Search in Google Scholar

[43] N. Katzourakis, Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L, Adv. Nonlinear Anal. (2017), 10.1515/anona-2016-0164. 10.1515/anona-2016-0164Search in Google Scholar

[44] N. Katzourakis and T. Pryer, On the numerical approximation of -harmonic mappings, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 6, Article ID 51. 10.1007/s00030-016-0415-9Search in Google Scholar

[45] N. Katzourakis and T. Pryer, On the numerical approximation of p-biharmonic and -biharmonic functions, preprint (2017), https://arxiv.org/abs/1701.07415. Search in Google Scholar

[46] J. Kristensen and F. Rindler, Characterization of generalized gradient Young measures generated by sequences in W1,1 and BV, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 593–598; erratum, Arch. Ration. Mech. Anal. 203 (2012), 693–700. 10.1007/s00205-009-0287-9Search in Google Scholar

[47] O. Lakkis and T. Pryer, A finite element method for nonlinear elliptic problems, SIAM J. Sci. Comput. 35 (2013), no. 4, A2025–A2045. 10.1137/120887655Search in Google Scholar

[48] O. Lakkis and T. Pryer, An adaptive finite element method for the infinity Laplacian, Numerical Mathematics and Advanced Applications – ENUMATH 2013, Lect. Notes Comput. Sci. Eng. 103, Springer, Cham (2015), 283–291. 10.1007/978-3-319-10705-9_28Search in Google Scholar

[49] B. Malgrange, Ideals of Differentiable Functions, Tata Inst. Fund. Res. Stud. Math. 3, Tata Institute of Fundamental Research, Bombay, 1967. Search in Google Scholar

[50] R. Moser, Regularity of minimizing extrinsic polyharmonic maps in the critical dimension, Manuscripta Math. 131 (2010), no. 3–7, 475–485. 10.1007/s00229-010-0331-ySearch in Google Scholar

[51] R. Moser and H. Schwetlick, Minimizers of a weighted maximum of the Gauss curvature, Ann. Global Anal. Geom. 41 (2012), no. 2, 199–207. 10.1007/s10455-011-9278-9Search in Google Scholar

[52] P. Pedregal, Parametrized Measures and Variational Principles, Progr. Nonlinear Differential Equations Appl. 30, Birkhäuser, Basel, 1997. 10.1007/978-3-0348-8886-8Search in Google Scholar

[53] T. Pryer, On the finite element approximation of infinity-harmonic functions, Proc. Roy. Soc. Edinburgh Sect. A, to appear. Search in Google Scholar

[54] Z. N. Sakellaris, Minimization of scalar curvature in conformal geometry, Ann. Global Anal. Geom. 51 (2017), no. 1, 73–89. 10.1007/s10455-016-9524-2Search in Google Scholar

[55] M. Valadier, Young measures, Methods of Nonconvex Analysis (Varenna 1989), Lecture Notes in Math. 1446, Springer, Berlin (1990), 152–188. 10.1007/BFb0084935Search in Google Scholar

[56] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. 10.1007/978-1-4612-2972-8_14Search in Google Scholar

Received: 2016-11-04
Revised: 2018-01-03
Accepted: 2018-01-05
Published Online: 2018-01-27
Published in Print: 2020-04-01

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