Critical points of solutions of degenerate elliptic equations in the plane

Calculus of Variations and Partial Differential Equations

Volumn 39, Issue 1, 2010, Pages 121-138

  Cecchini, Simone ^a,b   Magnanini, Rolando ^a  

^a UNIVERSITY OF FLORENCE   (Italy)

^b UNIVERSITY OF MODENA AND REGGIO EMILIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77954424135     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-009-0304-8     Document Type: Article

References (29)

1. Alessandrini G.: Critical points of solutions to the p-Laplace equation in dimension two. Boll. Univ. Math. Ital. A (7) 1(2), 239-246 (1987).
2. Alessandrini G., Magnanini R.: The index of isolated critical points and solutions of elliptic equations in the plane. Ann. Scuola Norm. Super. Pisa Cl. Sci. (4) 19(4), 567-589 (1992).
3. Alessandrini G., Magnanini R.: Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal. 25(5), 1259-1268 (1994).
4. Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (2000).
5. yy = 0. Ark. Mat. 7, 395-425 (1968).
6. Aronsson G.: On certain p-harmonic functions in the plane. Manuscr. Math. 61(1), 79-101 (1988).
7. Aronsson G., Lindqvist P.: On p-harmonic functions in the plane and their stream functions. J. Differ. Equ. 74(1), 157-178 (1988).
8. De Silva, D., Savin, O.: Minimizers of convex functionals arising in random surfaces. Preprint. http://arxiv. org/abs/0809. 3816 (2008).
9. Evans, L. C.: Weak convergence methods for nonlinear partial differential equations, vol 74 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1990).
10. Evans L. C., Gariepy R. F.: Some remarks concerning quasiconvexity and strong convergence. Proc. R. Soc. Edinb. Sect. A 106(1-2), 53-61 (1987).
11. Evans L. C., Gariepy R. F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
12. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I, volume 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin, Cartesian Currents (1998).
13. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin, Reprint of the 1998 edition (2001).
14. Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co. Inc., River Edge, NJ (2003).
15. Hardt R., Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N.: Critical sets of solutions to elliptic equations. J. Differ. Geom. 51(2), 359-373 (1999).
16. p norms. SIAM J. Math. Anal. 33(3), 545-569 (electronic) (2001).
17. Ishii H., Crandall M. G., Lions P. L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N. S.) 27(1), 1-67 (1992).
18. Kawohl B.: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1-22 (1990).
19. Koike S.: A Beginner's Guide to the Theory of Viscosity Solutions, volume 13 of MSJ Memoirs. Mathematical Society of Japan, Tokyo (2004).
20. 1-smooth functions of two variables. Sibirsk. Mat. Zh. 47(5), 1083-1091 (2006).
21. Langenbach, A.: Verallgemeinerte und exakte Lösungen des Problems der elastisch-plastischen Torsion von Stäben. Math. Nachr. 28, 219-234 (1964/1965).
22. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2-D eikonal equation. I. Qualitative properties. In Nonlinear partial differential equations (Evanston, IL, 1998), volume 238 of Contemporary Mathematics, pp. 203-229. American Mathematical Society, Providence, RI (1999).
23. Magnanini, R., Talenti, G.: On complex-valued solutions to a two-dimensional eikonal equation. II. Existence theorems. SIAM J. Math. Anal. 34(4), 805-835 (electronic) (2003).
24. Magnanini R., Talenti G.: On complex-valued solutions to a 2D eikonal equation. III. Analysis of a Bäcklund transformation. Appl. Anal. 85(1-3), 249-276 (2006).
25. Payne L. E., Philippin G. A.: Some applications of the maximum principle in the problem of torsional creep. SIAM J. Appl. Math. 33(3), 446-455 (1977).
26. Philippin G. A.: A minimum principle for the problem of torsional creep. J. Math. Anal. Appl. 68(2), 526-535 (1979).
27. Schwarz G.: Hodge Decomposition-A Method for Solving Boundary Value Problems, Springer, Berlin (1995).
28. Simon L.: Lectures on Geometric Measure Theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University Centre for Mathematical Analysis, Canberra (1983).
29. Talenti G.: Equazioni lineari ellittiche in due variabili. Matematiche (Catania) 21, 339-376 (1966).