Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates

Advances in Differential Equations

Volumn 4, Issue 2, 1999, Pages 137-161

  Garofalo, Nicola ^a   Sartori, Elena ^b  

^a PURDUE UNIVERSITY   (United States)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0013038125     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. A. Aftalion and J. Busca, Radial symmetry of overdetermined boundary value problems in exterior domains, (April 1998)., preprint.
2. G. Alessandrini, A symmetry problem for condensers, Math. Meth. Appl. Sci., 15 (1992), 315-320.
3. G. Alessandrini and N. Garofalo, Symmetry for degenerate parabolic equations, Arch. Rat. Mech. Anal., 108 (1989), 161-174.
4. A.D. Alexandroff, A characteristic property of the spheres, Ann. Mat. Pura Appl., 58 (1962), 303-354.
5. H.W. Alt and L A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.
6. H.W. Alt, L.A. Caffarelli, and A. Friedman, A free boundary problem for quasilinear elliptic equations, Ann. Sc. Norm. Sup. Pisa., 11 (1984), 1-44.
7. L. Caffarelli, N. Garofalo, and F. Segala, A gradient bound for entire solutions of quasilinear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473.
8. J.I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl., 177 (1993), 263-286.
9. 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.
10. N. Garofalo and J.L. Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math., 111 (1989), 9-33.
11. N. Garofalo and E. Sartori, Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains., (October 1997), preprint.
12. F. Gehring, Inequalities for condensers, hyperbolic capacity, and extremal length, Michigan Math. J., 18 (1971), 1-20.
13. J.J. Gergen, Note on the Green function of a star-shaped three dimensional region, Amer. J. Math., 42 (1931), 746-752.
14. B. Gidas, W.M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
15. B. Gidas, W M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations, Math. Anal. Appl., Part A, Adv. in Math. Suppl. Studies., 7A (1981), 369-402.
16. D. Gilbarg and N.S. Trudinger, "Elliptic PartialDifferential Equations of Second Order", 2nd edition, Springer Verlag, Berlin, Heidelberg ,1983.
17. J. Heinonen, T. Kilpeläinen, and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations", Oxford University Press, Oxford, New York, Tokyo, 1993.
18. S. Kichenassamy and L. Veron, Singular solutions of the p-Laplace equation, Math. Ann., 275 (1986), 559-615.
19. O. Ladyzhenskaya and N. Ural'tseva, "Linear and Quasilinear Elliptic Equations", Academic Press, New York, London, 1968.
20. J. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858.
21. G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
22. G.A. Philippin, On a free boundary problem in electrostatic Math. Methods Appl. Sci., 12 (1990), 387-392.
23. G.A. Philippin and L.E. Payne, On some maximum principles involving harmonic functions and their derivatives, SIAM J Math. Anal., 10 (1979), 96-104.
24. G.A. Philippin and L.E. Payne, On the conformal capacity problem in "Geometry of Solutions to PDE", Symposia Math., Academic Press, New York, London, 1989, 119-136.
25. G.A. Philippin and L.E. Payne, Some isoperimetric inequalities for capacity, polarization and virtual mass, Appl. Anal., 23 (1986), 43-61.
26. G.A. Philippin and L.E. Payne, On two free boundary problems in potential theory, J. Math. Anal. Appl., 161 (1991), 332-342.
27. G.A. Philippin and L.E.Payne, On gradient maximum principles for quasilinear elliptic equations, Nonlinear Anal., 23 (1994), 387-398.
28. M.H. Protter and H.F. Weinberger, "Maximum Principles inDifferential Equations", Springer Verlag, Berlin, Heidelberg, 1984.
29. (Lord) J W.S. Rayleigh, "The Theory of Sound," Macmillan, New York, 1877 (reprinted: Dover, New York, 1945).
30. W. Reichel, Radial symmetry for elliptic boundary value problems on exterior domains, Arch. Rat. Mech. Anal., to appear.
31. W. Reichel, Radial symmetry for an elettrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains, Zeitschrift Analysis Anwerdungen., 15 (1996), 3, 619-635.
32. E. Sartori, On a free boundary problem for the p-Laplacian, J. Math. Anal. Appl., 218 (1998), 117-126.
33. J. Serrin, Local behavior of solutions of quasi-linear elliptic equations, Acta Math., 111 (1964), 247-302.
34. J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240.
35. J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal., 43 (1971), 304-318.
36. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations J.Diff. Eq., 51 (1984), 126-150.
37. P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points Comm. Par.Diff. Eqns., 8 (7) (1983), 773-817.
38. A. Vogel, Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems, Atti Sem. Mat. Fis. Univ. Modena., 40 (1992), 443-484.
39. S.E. Warschawski, On the Green function of a star-shaped three dimensional region, Amer. Math. Monthly., 57 (1950), 471-473.
40. H.F. Weinberger, Remark on a preceding paper by Serrin, Arch. Rat. Mech. Anal., 43 (1971), 319-320.