Abstract
In this note we present a new proof for the following Phragmen–Lindelöf type result: If \({{0 \leq u \in S_{\lambda,\Lambda}}}\) in the half space \({H^{+}_{n}}\) and u vanishes on the boundary \({\partial H^{+}_{n}}\), then necessarily \(u(x) = u(e_{n}) \cdot x_{n}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong S., Sirakov B., Smart C.: Singular solutions of fully nonlinear elliptic equations and applications. Arch. Ration. Mech. Anal. 205(2), 345–394 (2012)
S. Axler, P. Bourdon, W. Ramey. Harmonic function theory. Second edition. Graduate Texts in Mathematics 137. Springer-Verlag, New York, 2001.
J.E.M. Braga, D. Moreira. Inhomogeneous Hopf–Oleĭnik lemma and applications. Part I: Regularity of the normal mapping. Preprint.
Caffarelli L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. of Math. (2) 130(1), 189–213 (1989)
L. Caffarelli, X. Cabré. Fully Nonlinear Elliptic Equations. Amer. Math. Soc. Coll. Publ. 43. Amer. Math. Soc., Providence (RI), 1995.
Caffarelli L., Fabes E., Mortola S., Salsa S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. Journal 30((4), 621–640 (1981)
L. Caffarelli, S. Salsa. A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics 68. Amer. Math. Soc., Providence (RI), 2005.
Gilbarg D.: The Pragmén–Lindelöf theorem for elliptic partial differential equations. J. Rational Mech. Anal. 1, 411–417 (1952)
D.Gilbarg, N. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1977 (reprint 1998).
Hopf E.: Remarks on the proceding paper by D. Gilbarg. J. Rational Mech. Anal. 1, 418–424 (1952)
Huber A.: A theorem of Phragmén–Lindelöf type. Proc. Amer. Math. Soc. 4, 852–857 (1953)
Kilpeläinen T., Shahgholian H., Zhong X.: Growth estimates through scaling for quasilinear partial differential equations. Ann. Acad. Sci. Fenn. Math. 32, 595–599 (2007)
Serrin J.: On Pragmém–Lindelöf Principle for elliptic partial differential equations. J. Rational Mech. Anal. 3, 395–413 (1954)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Braga, J.E.M. A New Proof of the Phragmén–Lindelöf Theorem for Fully Nonlinear Equations. Milan J. Math. 85, 247–256 (2017). https://doi.org/10.1007/s00032-017-0272-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00032-017-0272-y