Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries

Mathematische Nachrichten

Volumn 284, Issue 14-15, 2011, Pages 1872-1879

  Enache, Cristian ^a,b   Sakaguchi, Shigeru ^a  

^a HIROSHIMA UNIVERSITY   (Japan)

^b OVIDIUS UNIVERSITY   (Romania)

Author keywords

35B50; 35R35; Ellipsoids; Free boundary; Fully nonlinear elliptic equations; MSC (2010) 35J65; Overdetermined problems

Indexed keywords

EID: 80052950125     PISSN: 0025584X     EISSN: 15222616     Source Type: Journal    
DOI: 10.1002/mana.200810170     Document Type: Article

References (22)

1. C. Anneda and G. Porru, Problems on Monge-Ampère Equation in the plane, in: Recent Progress on some Problems in Several Complex Variables and Partial Differential Equations Vol. 400 (Contemp. Math., AMS, 2006), pp. 1-9.
2. B. Brandolini, C. Nitsch, P. Salani, and C. Trombetti, Serrin-type overdetermined problems: an alternative proof, Arch. Ration. Mech. Anal. 190, 267-280 (2008).
3. L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations Vol. 43 (Colloquium Publications, AMS, 1995).
4. L. A. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations III. Functions of the eigenvalues of the Hessian, Acta Math. 155(3-4), 261-301 (1985).
5. L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math. 35, 333-363 (1982).
6. A. Farina and B. Kawohl, Remarks on an overdetermined boundary value problem, Calc. Var. 31 351-357 (2008).
7. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1988).
8. A. Henrot and G. Philippin, Some overdetermined boundary value problems with elliptical free boundaries, SIAM J. Math. Anal. 29(2), 309-320 (1998).
9. L. G. Makar-Limanov, Solutions of Dirichlet's problem for the equation Δu = -1 in a convex region, Math. Notes Acad. Sci. URSS 9, 52-53 (1971).
10. L. E. Payne, Some remarks on overdetermined systems in linear elasticity, J. Elasticity 18, 181-189 (1987).
11. L. E. Payne and G. A. Philippin, On some maximum principles involving harmonic functions and their derivatives, SIAM J. Math. Anal. 10, 96-104 (1979).
12. L. E. Payne and G. A. Philippin, On the conformal capacity problem in "Geometry of Solutions to PDE" (Symposia Math, Academic Press, 1989) pp. 119-136.
13. L. E. Payne and G. A. Philippin, Some isoperimetric inequalities for capacity, polarization and virtual mass, Appl. Anal. 23, 43-61 (1986).
14. L. E. Payne and G. A. Philippin, On two free boundary problems in potential theory, J. Math. Anal. Appl. 161, 332-342 (1991).
15. L. E. Payne and G. A. Philippin, On gradient maximum principles for quasilinear elliptic equations, Nonlinear Anal. 23, 387-398 (1994).
16. L. E. Payne and P. W. Schaefer, Some nonstandard problems for the Poisson equation, Q. Appl. Math. 51, 81-90 (1993).
17. G. A. Philippin, On a free boundary value problem in electrostatics, Math. Methods Appl. Sci. 12, 387-392 (1990).
18. G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's, Z. Angew. Math. Phys. 54, 739-755 (2003).
19. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer Verlag, 1975).
20. P. W. Schaefer, On nonstandard overdetermined boundary value problems, Nonlinear Anal. 47, 2203-2212 (2001).
21. J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43, 304-318 (1971).
22. H. F. Weinberger, Remarks on the preceding paper of Serrin, Arch. Ration. Mech. Anal. 43, 319-320 (1971).