# Seminars & Events for Joint Princeton Rutgers Geometric PDEs Seminar

##### Blow up analysis of solutions of conformally invariant fully nonlinear elliptic equations

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points

is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are

comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem. This is a joint work with Luc Nguyen.

##### A fully nonlinear Sobolev trace inequality

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 $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition

$u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $\int -u \sigma_k(D^2 u)$. 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-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$. This is joint work with Jeffrey Case.

##### A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time.

##### Some existence and non-existence results for Poincare-Einstein metrics

I will begin with a brief overview of the existence question for conformally compact Einstein manifolds with prescribed conformal infinity. After stating the seminal result of Graham-Lee, I will discuss a non-existence result (joint with Qing Han) for certain conformal classes on the 7-dimensional sphere. I will also mention some ongoing work (with Gabor Szekelyhidi) on a version of "local existence" of Poincare-Einstein metrics.