# On Axially Symmetric Solutions of Fully Nonlinear Elliptic Equations

@inproceedings{Nadirashvili2010OnAS, title={On Axially Symmetric Solutions of Fully Nonlinear Elliptic Equations}, author={Nikolai S. Nadirashvili and Serge Vladuts}, year={2010} }

Here, uij denotes the partial derivative ∂ u/∂xi∂xj . A function u is called a classical solution of (1) if u ∈ C(Ω) and u satisfies (1). Actually, any classical solution of (1) is a smooth (C) solution, provided that F is a smooth (C) function of its arguments. For a matrix S ∈ S(R) we denote by λ(S) = {λi : λ1 ≤ ... ≤ λn} ∈ R the (ordered) set of eigenvalues of the matrix S. Equation (1) is called a Hessian equation ([T1],[T2] cf. [CNS]) if the function F (S) depends only on the eigenvalues…

## 5 Citations

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

- Mathematics
- 2015

We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if $u$ is a viscosity solution of a rotationally invariant equation of the form…

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

- Mathematics
- 2015

This is a continuation of Ikoma and Ishii (Ann Inst H Poincaré Anal Non Linéaire 29:783–812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous…

Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations

- Mathematics
- 2012

We prove here the existence of boundary blow up solutions for fully nonlinear equations in general domains, for a nonlinearity satisfying Keller-Osserman type condition. If moreover the nonlinearity…

H\"older regularity of the gradient for solutions of fully nonlinear equations with sublinear first order terms

- Mathematics
- 2013

In this paper we shall establish some regularity results of solutions of a class of fully nonlinear equations, with a first order term which is sub-linear. We prove local H\"older regularity of the…

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls

- Mathematics
- 2012

Abstract We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for…

## References

SHOWING 1-9 OF 9 REFERENCES

On the Dirichlet problem for Hessian equations

- Mathematics
- 1995

in domains f~ in Euclidean n-space, R n, where f is a given symmetric function on R n, A denotes the eigenvalues A1, ..., An of the Hessian matrix of second derivatives D2u and r is a given function…

Elliptic Partial Differential Equations of Second Order

- Physics
- 1997

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations…

The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

- Mathematics
- 1985

On etudie le probleme de Dirichlet dans un domaine borne Ω de R n a frontiere lisse ∂Ω:F(D 2 u)=ψ dans Ω, u=φ sur ∂Ω

Weak solutions of hessian equations

- Mathematics
- 1997

We consider the existence, uniqueness and Holder regularity of weak solutions of Hessian equations, determined by the elementary symmetric functions, with Lp inhomogeneous terms. The notion of weak…

Fully Nonlinear Elliptic Equations

- Mathematics
- 1995

Introduction Preliminaries Viscosity solutions of elliptic equations Alexandroff estimate and maximum principle Harnack inequality Uniqueness of solutions Concave equations $W^{2,p}$ regularity…

Nonlinear Elliptic and Parabolic Equations of the Second Order Equations

- Mathematics, Physics
- 1987

An electron spectrometer having a wide bandwidth and a high signal-to-noise ratio. Each electron channel sensor is the termination of a 50-ohm transmission line thereby eliminating series inductance…

User ’ s guide to viscosity solutions of second order partial differential equations

- Partial Differential Equations
- 1995

Lions, User’s guide to viscosity solutions of second order partial differential equations

- Bull. Amer. Math. Soc. (N.S.),
- 1992

Interscience Publisher

- New York-london-Sydney,
- 1964