A fully nonlinear Sobolev trace inequality

@article{Case2016AFN,
  title={A fully nonlinear Sobolev trace inequality},
  author={Jeffrey S. Case and Yi Wang},
  journal={arXiv: Analysis of PDEs},
  year={2016}
}
The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int u\sigma_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non… Expand
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References

SHOWING 1-10 OF 21 REFERENCES
On fractional GJMS operators
We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measureExpand
SOME ENERGY INEQUALITIES INVOLVING FRACTIONAL GJMS OPERATORS
Under a spectral assumption on the Laplacian of a Poincare--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2\gamma\in(0,2)$ orExpand
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫Expand
Fractional conformal Laplacians and fractional Yamabe problems
Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to- Neumann operators of uniformly degenerate elliptic boundary value problems observed by ChangExpand
On higher order extensions for the fractional Laplacian
The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with oneExpand
An Extension Problem Related to the Fractional Laplacian
The operator square root of the Laplacian (− ▵)1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to theExpand
Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary
One of the most celebrated theorems in mathematics is the Riemann mapping theorem. It says that an open, simply connected, proper subset of the plane is conformally diffeomorphic to the disk. InExpand
The Dirichlet Problem for Degenerate Hessian Equations
Abstract In this paper, we study the Dirichlet problem for a class of fully nonlinear degenerate elliptic equations which depend only on the eigenvalues of the Hessian matrix. We provide a new andExpand
The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
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
On a class of non-local operators in conformal geometry
In this expository article, the authors discuss the connection between the study of non-local operators on Euclidean space to the study of fractional GJMS operators in conformal geometry. TheExpand
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