Second-order L^∞ variational problems and the ∞-polylaplacian

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.2020.50.383Suche 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/13421Suche 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/S0308210505001344Suche in Google Scholar

[4] G. Aronsson, Minimization problems for the functional supx⁢F⁢(x,f⁢(x),f′⁢(x)), Ark. Mat. 6 (1965), 33–53. 10.1007/BF02591326Suche in Google Scholar

[5] G. Aronsson, Minimization problems for the functional supx⁢F⁢(x,f⁢(x),f′⁢(x)). II, Ark. Mat. 6 (1966), 409–431. 10.1007/BF02590964Suche in Google Scholar

[6] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561. 10.1007/BF02591928Suche in Google Scholar

[7] G. Aronsson, On the partial differential equation ux2⁢ux⁢x+2⁢ux⁢uy⁢ux⁢y+uy2⁢uy⁢y=0, Ark. Mat. 7 (1968), 395–425. 10.1007/BF02590989Suche in Google Scholar

[8] G. Aronsson, Minimization problems for the functional supx⁢F⁢(x,f⁢(x),f′⁢(x)). III, Ark. Mat. 7 (1969), 509–512. 10.1007/BF02590888Suche in Google Scholar

[9] G. Aronsson, On certain singular solutions of the partial differential equation ux2⁢ux⁢x+2⁢ux⁢uy⁢ux⁢y+uy2⁢uy⁢y=0, Manuscripta Math. 47 (1984), no. 1–3, 133–151. 10.1007/BF01174590Suche 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/BF01172152Suche 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-zSuche 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. Suche 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)00070-1Suche 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/PL00004239Suche 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. 10.1007/1-4020-1964-5Suche 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_3Suche 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.2017266Suche in Google Scholar

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

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

[21] J. M. Danskin, The theory of max–min, with applications, SIAM J. Appl. Math. 14 (1966), 641–664. 10.1137/0114053Suche 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.2009.170.1417Suche in Google Scholar

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

[24] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Suche 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. Suche 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.2005.161.509Suche in Google Scholar

[27] L. C. Florescu and C. Godet-Thobie, Young Measures and Compactness in Measure Spaces, De Gruyter, Berlin, 2012. 10.1515/9783110280517Suche 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. Suche 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.a2Suche 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. 10.1007/978-88-7642-443-4Suche in Google Scholar

[31] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Suche 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/s00526-013-0688-3Suche 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.2012.05.012Suche 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.2013.07.028Suche in Google Scholar

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

[36] N. Katzourakis, On the structure of ∞-harmonic maps, Comm. Partial Differential Equations 39 (2014), no. 11, 2091–2124. 10.1080/03605302.2014.920351Suche 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. 10.1007/978-3-319-12829-0Suche 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.2015.14.313Suche in Google Scholar

[39] N. Katzourakis, Optimal ∞-quasiconformal immersions, ESAIM Control Optim. Calc. Var. 21 (2015), no. 2, 561–582. 10.1051/cocv/2014038Suche 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. Suche 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-zSuche 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.2017.02.048Suche 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-2016-0164Suche 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/s00030-016-0415-9Suche 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. Suche 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/s00205-009-0287-9Suche 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. 10.1137/120887655Suche 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_28Suche in Google Scholar

[49] B. Malgrange, Ideals of Differentiable Functions, Tata Inst. Fund. Res. Stud. Math. 3, Tata Institute of Fundamental Research, Bombay, 1967. Suche 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-ySuche 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/s10455-011-9278-9Suche in Google Scholar

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

[53] T. Pryer, On the finite element approximation of infinity-harmonic functions, Proc. Roy. Soc. Edinburgh Sect. A, to appear. Suche 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/s10455-016-9524-2Suche 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/BFb0084935Suche 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_14Suche in Google Scholar