Singular solutions of Hessian fully nonlinear elliptic equations

Journal de Mathématiques Pures et Appliquées, 2013

Advances in Mathematics, 2012

Mathematische Zeitschrift, 2010

Journal of Algebra, 2004

Let k be a field of characteristic zero. For small n, we classify all f ∈ k [n] such that the Hessian of f is singular.

Journal of Functional Analysis, 2009

We solve the existence problem in the renormalized, or viscosity sense, and obtain global pointwise estimates of solutions for quasilinear and Hessian equations with measure coefficients and data, including the following model problems:

Advances in Nonlinear Analysis, 2012

We consider the homogeneous Dirichlet problem for a special k-Hessian equation of sub-linear type in a .k 1/-convex domain R n , 1 Ä k Ä n. We study the comparison between the solution of this problem and the (radial) solution of the corresponding problem in a ball having the same .k 1/-quermassintegral as . Next, we consider the eigenvalue problem for the k-Hessian equation and study a comparison between its principal eigenfunction and the principal eigenfunction of the corresponding problem in a ball having the same .k 1/-quermassintegral as . Symmetrization techniques and comparison principles are the main tools used to get these inequalities.

SIAM Journal on Optimization, 2018

We disclose an interesting connection between the gradient flow of a C 2smooth function ψ and strongly evanescent orbits of the second order gradient system defined by the square-norm of ∇ψ, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two C 2 , convex and bounded from below functions ψ 1 , ψ 2 : if ||∇ψ 1 || = ||∇ψ 2 ||, then ψ 1 = ψ 2 + k, for some k ∈ R.

Archive for Rational Mechanics and Analysis, 2003

In this note we construct new examples of quasiconvex functions defined on the set S n×n of symmetric matrices. They are built on the k-th elementary symmetric function of the eigenvalues, k = 1, 2, ..., n. The idea is motivated byŠverák's paper [S]. The proof of our result relies on the theory of the so-called k-Hessian equations, which have been intensively studied recently, see [CNS], [T], [TW1], [TW2].

Inventiones mathematicae, 2002

Siam Journal on Matrix Analysis and Applications, 2012

We define in the space of n × m matrices of rank n, n ≤ m, the condition Riemannian structure as follows: For a given matrix A the tangent space at A is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of A denoted σ n (A). When this smallest singular value has multiplicity 1, the function A → log(σ n (A) −2 ) is a convex function with respect to the condition Riemannian structure that is t → log(σ n (A(t)) −2 ) is convex, in the usual sense for any geodesic A(t). In a more abstract setting, a function α defined on a Riemannian manifold (M, , ) is said to be * Mathematics Subject Classification (MSC2000): 65F35 (Primary), 15A12 (Secondary).