Symmetry of minimizers with a level surface parallel to the boundary

Journal of the European Mathematical Society

Volumn 17, Issue 11, 2015, Pages 2789-2804

  Ciraolo, Giulio ^a   Magnanini, Rolando ^b   Sakaguchi, Shigeru ^c  

^a UNIVERSITY OF PALERMO   (Italy)

^b UNIVERSITY OF FLORENCE   (Italy)

^c TOHOKU UNIVERSITY   (Japan)

Author keywords

Minimizers of integral functionals; Overdetermined problems

Indexed keywords

EID: 84939296133     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/571     Document Type: Article

References (44)

1. Bardi, M., Mannucci, P.: On the Dirichlet problem for non-Totally degenerate fully nonlinear elliptic equations. Comm. Pure Appl. Anal. 5, 709-731 (2006) Zbl 1142.35041 MR 2246004
2. Barles, G., Busca, J.: Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Comm. Partial Differential Equations 26, 2323-2337 (2001) Zbl 0997.35023 MR 1876420
3. Brasco, L.: Global L1 gradient estimates for solutions to a certain degenerate elliptic equation. Nonlinear Anal. 74, 516-531 (2011) Zbl 1202.35103 MR 2733227
4. Brasco, L., Carlier, G., Santambrogio, F.: Congested traffic dynamics, weak flows and very degenerate elliptic equations. J. Math. Pures Appl. 93, 163-182 (2010) Zbl 1192.35079 MR 2584740
5. Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations. I. J. Fixed Point Theory Appl. 5, 353-395 (2009) Zbl 1215.35068 MR 2529505
6. Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations. II: symmetry and monotonicity via moving planes. In: Advances in Geometric Analysis, Int. Press, 97-105 (2012) Zbl 06467558 MR 2529505
7. Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations. III: viscosity solutions, including parabolic operators. Comm. Pure Appl. Math. 66, 109-143 (2013) Zbl 1279.35044 MR 2994551
8. Carstensen, C., Müller, S.: Local stress regularity in scalar nonconvex variational problems. SIAM J. Math. Anal. 34, 495-509 (2002) Zbl 1012.49027 MR 1951785
9. Cecchini, S., Magnanini, R.: Critical points of solutions of degenerate elliptic equations in the plane. Calc. Var. Partial Differential Equations 39, 121-138 (2010) Zbl 1197.35120 MR 2659682
10. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis: A Symposium in Honor of Salomon Bochner, Princeton Univ. Press, 195-199 (1970) Zbl 0212.44903 MR 0402831
11. Ciraolo, G.: A weak comparison principle for solutions of very degenerate elliptic equations. Ann. Mat. Pura Appl. 193, 1485-1490 (2014) Zbl 06363181 MR 3262643
12. Ciraolo, G., Magnanini, R., Sakaguchi, S.: Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration. J. Anal. Math., to appear; arXiv:1307.1257
13. Crandall, M. G., Ishii, H., Lions, P.-L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, 1-67 (1992) Zbl 0755.35015 MR 1118699
14. Crasta, G.: A symmetry problem in the calculus of variations. J. Eur. Math. Soc. 8, 139-154 (2006) Zbl 1114.49024 MR 2201279
15. Crasta, G.: Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems. Math. Z. 235, 569-589 (2000) Zbl 0965.49003 MR 1800213
16. Crasta, G., Gazzola, F.: Some estimates of the minimizing properties of web functions. Calc. Var. 15, 45-66 (2002) Zbl 1026.49010 MR 1920714
17. Da Lio, F., Sirakov, B.: Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations. J. Eur. Math. Soc. 9, 317-330 (2007) Zbl 1176.35068 MR 2293958
18. Demengel, F.: Some existence's results for noncoercive "1-Laplacian" operator. Asymptot. Anal. 43, 287-322 (2005) Zbl 1192.35036 MR 2160702
19. Fraenkel, L. E.: An Introduction to Maximum Principles and Symmetry in Elliptic Problems. Cambridge Univ. Press, Cambridge (2000) Zbl 0947.35002 MR 1751289
20. Fragalà, I., Gazzola, F., Kawohl, B.: Overdetermined problems with possibly degenerate ellipticity, a geometric approach. Math. Z. 254, 117-132 (2006) Zbl 1220.35077 MR 2232009
21. Gazzola, F.: Existence of minima for nonconvex functionals in spaces of functions depending on the distance from the boundary. Arch. Ration. Mech. Anal. 150, 57-76 (1999) Zbl 0949.49010 MR 1738167
22. Gidas, B., Ni, W. M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979) Zbl 0425.35020 MR 0544879
23. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977). Zbl 0361.35003 MR 0473443
24. Jensen, R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Ration. Mech. Anal. 101, 1-27 (1988) Zbl 0708.35019 MR 0920674
25. Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin. 44, 659-667 (2003) Zbl 1105.35029 MR 2062882
26. Kawohl, B., Kutev, N.: Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations. Funkcial. Ekvac. 43, 241-253 (2000) Zbl 1142.35315 MR 1795972
27. Kawohl, B., Kutev, N.: Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Comm. Partial Differential Equations 32, 1209-1224 (2007) Zbl 1185.35084 MR 2354491
28. Magnanini, R., Sakaguchi, S.: Nonlinear diffusion with a bounded stationary level surface. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 937-952 (2010) Zbl 1194.35209 MR 2629887
29. Magnanini, R., Sakaguchi, S.: Matzoh ball soup revisited: the boundary regularity issue. Math. Methods Appl. Sci. 36, 2023-2032 (2013) Zbl 1279.35062 MR 3108824
30. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties. In: Nonlinear Partial Differential Equations, Contemp. Math. 238, Amer. Math. Soc., 203-229 (1999) Zbl 0940.35100 MR 1724665
31. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part two: existence theorems. SIAM J. Math. Anal. 34, 805-835 (2003) Zbl 1126.35367 MR 1969603
32. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part three: analysis of a Backlund transformation. Appl. Anal. 85, 249-276 (2006) Zbl 1131.35033 MR 2198842
33. Magnanini, R., Talenti, G.: On complex-valued 2D eikonals. Part four: continuation past a caustic. Milan J. Math. 77, 1-66 (2009) Zbl 1205.35301 MR 2578872
34. Protter, M. H.,Weinberger, H. F.: Maximum Principles in Differential Equations. Springer, New York (1984) Zbl 0549.35002 MR 0762825
35. Pucci, P., Serrin, J.: The Maximum Principle, Birkhauser, Basel (2007). Zbl 1134.35001 MR 2356201
36. Santambrogio, F., Vespri, V.: Continuity in two dimensions for a very degenerate elliptic equation. Nonlinear Anal. 73, 3832-3841 (2010) Zbl 1202.35107 MR 2728558
37. Serrin, J.: On the strong maximum principle for quasilinear second order differential inequalities. J. Funct. Anal. 5, 184-193 (1970) Zbl 0188.41701 MR 0259328
38. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304-318 (1971) Zbl 0222.31007 MR 0333220
39. Shahgholian, H.: Diversifications of Serrin's and related symmetry problems. Complex Var. Elliptic Equations 57, 653-665 (2012) Zbl 1269.35021 MR 2916825
40. Silvestre, L., Sirakov, B.: Overdetermined problems for fully nonlinear elliptic equations. Calc. Var. Partial Differential Equations 54, 989-1007 (2015) Zbl 06481196 MR 3385189
41. Sirakov, B.: Symmetry for exterior elliptic problems and two conjectures in potential theory. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 135-156 (2001) Zbl 0997.35014 MR 1808026
42. Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51, 126-150 (1984) Zbl 0488.35017 MR 0727034
43. Tolksdorf, P.: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Comm. Partial Differential Equations 8, 773-817 (1983) Zbl 0515.35024 MR 0700735
44. Trudinger, N.: Comparison principles and pointwise estimates for viscosity solutions. Rev. Mat. Iberoamer. 4, 453-468 (1988) Zbl 0695.35007 MR 1048584