Wild solutions to scalar Euler-Lagrange equations

Johansson, Carl

doi:10.1090/tran/9090

Wild solutions to scalar Euler-Lagrange equations
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by Carl Johan Peter Johansson;

Trans. Amer. Math. Soc. 377 (2024), 4931-4960

DOI: https://doi.org/10.1090/tran/9090

Published electronically: May 15, 2024

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Abstract:

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether $W^{1,1}$ solutions are necessarily $W^{1,2}_{\operatorname {loc}}$, which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl’s classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist $W^{1,1}$ solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a $W^{1,1}$ solution to be improved to $W^{1,2}_{\operatorname {loc}}$ under suitable assumptions on the functional and solution.

References

Kari Astala, Daniel Faraco, and László Székelyhidi Jr., Convex integration and the $L^p$ theory of elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 1, 1–50. MR 2413671
Kari Astala, Tadeusz Iwaniec, and Eero Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27–56. MR 1815249, DOI 10.1215/S0012-7094-01-10713-8
Alano Ancona, Elliptic operators, conormal derivatives and positive parts of functions, J. Funct. Anal. 257 (2009), no. 7, 2124–2158. With an appendix by Haïm Brezis. MR 2548032, DOI 10.1016/j.jfa.2008.12.019
Kari Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), no. 1, 37–60. MR 1294669, DOI 10.1007/BF02392568
Haïm Brezis, On a conjecture of J. Serrin, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19 (2008), no. 4, 335–338. MR 2465684, DOI 10.4171/RLM/529
Diego Cordoba, Daniel Faraco, and Francisco Gancedo, Lack of uniqueness for weak solutions of the incompressible porous media equation, Arch. Ration. Mech. Anal. 200 (2011), no. 3, 725–746. MR 2796131, DOI 10.1007/s00205-010-0365-z
Sergio Conti, Daniel Faraco, and Francesco Maggi, A new approach to counterexamples to $L^1$ estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 287–300. MR 2118479, DOI 10.1007/s00205-004-0350-5
S. Conti, D. Faraco, F. Maggi, and S. Müller, Rank-one convex functions on $2\times 2$ symmetric matrices and laminates on rank-three lines, Calc. Var. Partial Differential Equations 24 (2005), no. 4, 479–493. MR 2180863, DOI 10.1007/s00526-005-0343-8
Maria Colombo and Riccardo Tione, Non-classical solutions of the $p$-Laplace equation, J. Eur. Math. Soc. (JEMS) (2022), to appear.
Camillo De Lellis, Guido De Philippis, Bernd Kirchheim, and Riccardo Tione, Geometric measure theory and differential inclusions, Ann. Fac. Sci. Toulouse Math. (6) 30 (2021), no. 4, 899–960 (English, with English and French summaries). MR 4350101, DOI 10.5802/afst.1691
Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 93649
Oliver Dragičević and Alexander Volberg, Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2003), no. 2, 415–435. MR 1992955, DOI 10.1307/mmj/1060013205
Lawrence C. Evans, A new proof of local $C^{1,\alpha }$ regularity for solutions of certain degenerate elliptic p.d.e, J. Differential Equations 45 (1982), no. 3, 356–373. MR 672713, DOI 10.1016/0022-0396(82)90033-X
Daniel Faraco, Milton’s conjecture on the regularity of solutions to isotropic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 20 (2003), no. 5, 889–909 (English, with English and French summaries). MR 1995506, DOI 10.1016/S0294-1449(03)00014-3
Xavier Fernández-Real and Xavier Ros-Oton, Regularity theory for elliptic PDE, Zurich Lectures in Advanced Mathematics, vol. 28, EMS Press, Berlin, [2022] ©2022. MR 4560756, DOI 10.4171/zlam/28
Xavier Fernández-Real and Riccardo Tione, Improved regularity of second derivatives for subharmonic functions, Proc. Amer. Math. Soc. 151 (2023), no. 12, 5283–5297. MR 4648925, DOI 10.1090/proc/16490
David Gilbarg and Neil S Trudinger, Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften, vol. 224, Springer Berlin / Heidelberg, Berlin, Heidelberg, 2015.
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
R. A. Hager and J. Ross, A regularity theorem for linear second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 283–290. MR 374643
Jonas Hirsch and Riccardo Tione, On the constancy theorem for anisotropic energies through differential inclusions, Calc. Var. Partial Differential Equations 60 (2021), no. 3, Paper No. 86, 52. MR 4248559, DOI 10.1007/s00526-021-01981-z
T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143–161. MR 1288682, DOI 10.1515/crll.1994.454.143
Tianling Jin, Vladimir Maz’ya, and Jean Van Schaftingen, Pathological solutions to elliptic problems in divergence form with continuous coefficients, C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 773–778 (English, with English and French summaries). MR 2543981, DOI 10.1016/j.crma.2009.05.008
Carl Johan Peter Johansson, Nonsmooth and nonenergetic solutions of PDE’s through convex integration (2022), available at https://infoscience.epfl.ch/record/300369?&ln=en.
Carl Johan Peter Johansson and Riccardo Tione, ${T}_5$ configurations and hyperbolic systems, Comm. Cont. Math. , posted on (2023)., DOI 10.1142/S021919972250081X
John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
John L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), no. 6, 849–858. MR 721568, DOI 10.1512/iumj.1983.32.32058
F. Leonetti and V. Nesi, Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton, J. Math. Pures Appl. (9) 76 (1997), no. 2, 109–124. MR 1432370, DOI 10.1016/S0021-7824(97)89947-3
S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2) 157 (2003), no. 3, 715–742. MR 1983780, DOI 10.4007/annals.2003.157.715
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841
Stefanie Petermichl and Alexander Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. MR 1894362, DOI 10.1215/S0012-9074-02-11223-X
J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), no. 1, 257–282 (German). MR 1545448, DOI 10.1007/BF01170635
J. Schauder, Numerische abschätzungen in elliptischen linearen differentialgleichungen., Stud. Math. (in German), Lwów, Poland: Polska Akademia Nauk. Instytut Matematyczny 5 (1937), 34–42.
James Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 385–387. MR 170094
Massimo Sorella and Riccardo Tione, The four-state problem and convex integration for linear differential operators, J. Funct. Anal. 284 (2023), no. 4, Paper No. 109785, 44. MR 4515812, DOI 10.1016/j.jfa.2022.109785
Vladimír Sverák and Xiaodong Yan, Non-Lipschitz minimizers of smooth uniformly convex functionals, Proc. Natl. Acad. Sci. USA 99 (2002), no. 24, 15269–15276. MR 1946762, DOI 10.1073/pnas.222494699
László Székelyhidi Jr., The regularity of critical points of polyconvex functionals, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 133–152. MR 2048569, DOI 10.1007/s00205-003-0300-7
Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219–240. MR 474389, DOI 10.1007/BF02392316
N. N. Ural′ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222 (Russian). MR 244628
Alexis F. Vasseur, The De Giorgi method for elliptic and parabolic equations and some applications, Lectures on the analysis of nonlinear partial differential equations. Part 4, Morningside Lect. Math., vol. 4, Int. Press, Somerville, MA, 2016, pp. 195–222. MR 3525875