Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems

Sakaguchi, Shigeru

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 14 (1987) no. 3, pp. 403-421

[1] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, New York Heidelberg Berlin, 1982. | Zbl | MR

[2] M.S. Berger, Nonlinearity And Functional Analysis Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, New York San Francisco London, 1977. | Zbl | MR

[3] H.J. Brascamp and E.H. Lieb, On extension of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366-389. | Zbl | MR

[4] L.A. Caffarelli and A. Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), 431-456. | Zbl | MR

[5] L.A Caffarelli and J. Spruck, Convexity properties of solutions to some classical varational problems, Comm. P. D. E. 7 (1982), 1337-1379. | Zbl | MR

[6] J.I. Díaz, Nonlinear partial differential equations and free boundaries volume I Elliptic equations, Research Notes in Mathematics 106, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1985. | Zbl | MR

[7] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies Number 105, Princeton University Press, Princeton, New Jersey, 1983. | Zbl | MR

[8] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46. | Zbl | MR

[9] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg -New York, 1977. | Zbl | MR

[10] B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. P. D. E. 10 (1985), 1213-1225. | Zbl | MR

[11] B. Kawohl, A remark on N. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem, Math. Methods Appl. Sci. 8 (1986), 93-101. | Zbl | MR

[12] B. Kawohl, When are superharmonic functions concave? Applications to the St. Venant torsion problem and to the fundamental mode of the clamped membrane, Z. Angew. Math. Mech. 64 (1984), 364-366. | Zbl | MR

[13] A.U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), 687-704. | Zbl | MR

[14] N. Korevaar, Capillary surface convexity above convex domains, Indiana Univ. Math. J. 32 (1983), 73-81. | Zbl | MR

[15] N. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), 603-614. | Zbl | MR

[16] N. Korevaar, and J.L. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Rational Mech. Anal. 97 (1987), 19-32. | Zbl | MR

[17] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York. and London, 1968. | Zbl | MR

[18] J.L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201- 224. | Zbl | MR

[19] F. Riesz and B. Sz-Nagy, Functional Analysis, Translated from the 2nd French edition by Leo F. Boron, Frederick Ungar Publishing Co., New York 1955. | MR

[20] F. De Thelin, Sur l'espace propre associé à la première valeur propre du pseudo-Laplacien, C.R. Acad. Sc. Paris, Sér. A 303 (1986), 355-358. | Zbl | MR

[21] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Equations 51 (1984), 126-150. | Zbl | MR

[22] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. P. D. E. 8 (1983), 773-817. | Zbl | MR

[23] N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. | Zbl | MR