Convex integration and the Lp theory of elliptic equations

Annali della Scuola Normale - Classe di Scienze

Volumn 7, Issue 1, 2008, Pages 1-50

  Astala, Kari ^a   Faraco, Daniel ^b   Székelyhidi Jr , László ^c  

^a UNIVERSITY OF HELSINKI   (Finland)

^b UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

^c ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 52249122621     PISSN: 0391173X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (43)

1. K. ASTALA, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37-60.
2. K. ASTALA, T. IWANIEC and G. MARTIN, Pucci's conjecture and the Alexandrov inequality for elliptic PDEs in the plane, J. Reine Angew. Math. 591 (2006), 49-74.
3. K. ASTALA, T. IWANIEC and G. MARTIN, "Elliptic Partial Diffential Equations and Quasiconformal Mappings in the Plane", to appear.
4. K. ASTALA, T. IWANIEC and E. SAKSMAN, Beltrami operators in the plane, Duke Math. J. 107 (2001), 27-56.
5. A. BAERNSTEIN II and L. V. KOVALEV On Hölder regularity for elliptic equations of non-divergence type in the plane, Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) (2005), 295-317.
6. B. BOJARSKI, Homeomorphic solutions of Beltrami systems, Dold. Akad. Nauk SSSR (N.S.) 102 (1955), 661-664.
7. B. BOJARSKI, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43 (1957), 451-503.
8. 1 estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal. 175 (2005), 287-300.
9. S. CONTI, D. FARACO, F. MAGGI and S. MÜLLER, Rank-one convex functions on 2 × 2 symmetric matrices and laminates on rank-three lines, Calc. Var. Partial Differential Equations 24 (2005), 479-493.
10. B. DACOROGNA, "Direct Methods in the Calculus of Variations", Applied Mathematical Sciences, Vol. 78, Springer-Verlag, Berlin, 1989.
11. B. DACOROGNA and P. MARCELLINI, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997), 1-37.
12. B. DACOROGNA and P. MARCELLINI, "Implicit partial differential equations", Progress in Nonlinear Differential Equations and their Applications, Vol. 37, Birkhäuser Boston Inc., Boston, MA, 1999.
13. O. DRAGIČEVIĆ and A. VOLBERG, Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2003), 415-435.
14. D. FARACO, "Quasiconvex Hulls and the Homogenization of Differential Operators", Licenciate thesis, University of Jyväskylä, 2000.
15. D. FARACO, Milton's conjecture on the regularity of solutions to isotropic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 889-909.
16. D. FARACO, Tartar conjecture and Beltrami operators, Michigan Math. J. 52 (2004), 83-104.
17. D. FARACO and L. SZÉKELYHIDI JR., in preparation.
18. M. GROMOV, "Partial differential relations", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 9, Springer-Verlag, Berlin, 1986.
19. T. IWANIEC and G. MARTIN, "Geometric Function Theory and Non-Linear Analysis", Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2001.
20. T. IWANIEC and C. SBORDONE, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143-161.
21. B. KIRCHHEIM, "Rigidity and Geometry of Microstructures", Habilitation thesis, University of Leipzig, 2003.
22. B. KIRCHHEIM, S. MÜLLER and V. ŠVERÁK, Studying nonlinear PDE by geometry in matrix space, In: "Geometric analysis and Nonlinear partial differential equations", S. Hildebrandt and H. Karcher (eds.), Springer-Verlag, 2003, 347-395.
23. O. LEHTO and K. I. VIRTANEN, "Quasiconformal Mappings in the Plane", second ed. Springer-Verlag, New York, 1973. Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126.
24. 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), 109-124.
25. J. MALÝ, Examples of weak minimizers with continuous singularities, Expo. Math. 13 (1995), 446-454.
26. p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa C1. Sci. (3) 17 (1963), 189-206.
27. G. W. MILTON, Modelling the properties of composites by laminates, In: "Homogenization and Effective Moduli of Materials and Media" (Minneapolis, Minn., 1984/1985), IMA Vol. Math. Appl., Vol. 1, Springer, New York, 1986, 150-174.
28. S. MÜLLER, Variational models for microstructure and phase transitions, In: "Calculus of Variations anal Geometric Evolution Problems", (Cetraro, 1996), Lecture Notes in Math., Vol. 1713, Springer, Berlin, 1999, 85-210.
29. S. MÜLLER and V. ŠVERÁK, Attainment results for the two-well problem by convex integration, In: "Geometric Analysis and the Calculus of Variations", Internat. Press, Cambridge, MA, 1996, 239-251.
30. S. MÜLLER and V. ŠVERÁK, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2) 157 (2003), 715-742.
31. J. C. OXTOBY, "Measure and Category", second ed., Graduate Texts in Mathematics, Vol. 2, Springer-Verlag, New York, 1980.
32. P. PEDREGAL, Laminates and microstructure, European J. Appl. Math. 4 (1993), 121-149.
33. P. PEDREGAL, Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 72-78.
34. S. PETERMICHL and A. VOLBERG, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), 281-305.
35. L. C. PICCININI and S. SPAGNOLO, On the Hölder continuity of solutions of second order elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa C1. Sci. (3) 26 (1972), 391-402.
36. C. PUCCI, Operatori ellittici estremanti, Ann. Mat. Pura Appl. (4) 72 (1966), 141-170.
37. J. SERRIN, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 385-387.
38. M. A. SYCHEV, Comparing two methods of resolving homogeneous differential inclusions, Calc. Var. Partial Differential Equations 13 (2001), 213-229.
39. M. A. SYCHEV, Few remarks on differential inclusions, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 649-668.
40. L. SZÉKELYHIDI JR., The regularity of critical points of polyconvex functionals, Arch. Ration. Mech. Anal. 172 (2004), 133-152.
41. L. TARTAR, L. Compensated compactness and applications to partial differential equations, In: "Nonlinear Analysis and Mechanics: Heriot-Watt Symposium", Vol. IV, Res. Notes in Math., Vol. 39, Pitman, Boston, Mass., 1979, 136-212.
42. B. YAN, A linear boundary value problem for weakly quasiregular mappings in space, Calc. Var. Paxtial Differential Equations 13 (2001), 295-310.
43. B. YAN, A Baire's category method for the Dirichlet problem of quasiregular mappings, Trans. Amer. Math. Soc. 355 (2003), 4755-4765.