Related papers: Hessian estimates for the sigma-2 equation in dime…

In space dimension $n\geq3$, we consider the electromagnetic Schr\"odinger Hamiltonian $H=(\nabla-iA(x))^2-V$ and the corresponding Helmholtz equation $(\nabla-iA(x))^2u+u-V(x)u=f \in \mathbb{R}^n$. We extend the well known $L^p$-$L^q$…

This paper establishes the global curvature estimate for the $n-2$ curvature equation with the general right hand side which partially solves this longstanding problem.

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension $n\geq3$. We prove, by multiplier techniques in the sense of Morawetz, a family of a priori estimates from which the limiting…

We survey quadratic Hessian equations: definition, background, rigidity of entire solutions, regularity of viscosity solutions, a priori Hessian estimates, and open problems.

We derive an a priori real Hessian estimate for solutions of a large family of geometric fully non-linear elliptic equations on compact Hermitian manifolds, which is independent of a lower bound for the right-hand side function. This…

We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases…

We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with…

We are concerned with the Dirichlet problem for a class of Hessian type equations. Applying some new methods we are able to establish the $C^2$ estimates for an approximating problem under essentially optimal structure conditions. Based on…

After explicitly constructing the symmetric space sigma model lagrangian in terms of the coset scalars of the solvable Lie algebra gauge in the current formalism we derive the field equations of the theory.

We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…

We give a geometric formulation of 3D incompressible Euler that contains the Eulerian and Lagrangian gauges as special cases. In the Lagrangian gauge, incompressible Euler is a real analytic ODE in Banach space; a short proof of this known…

Let $(X,\alpha)$ be a K\"ahler manifold of dimension n, and let $[\omega] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic…

After a Hessian computation, we quickly prove the 3D simplex mean width conjecture using classical methods. Then, we generalize some components to $d$ dimensions.

We establish interior $W^{2,\delta}$ type estimates for a class of degenerate fully nonlinear elliptic equations with $L^n$ data. The main idea of our approach is to slide $C^{1,\alpha}$ cones, instead of paraboloids, vertically to touch…

The Zakharov system in dimension $d=2,3$ is shown to have a local unique solution for any initial values in the energy space $H^{s} \times H^{l} \times H^{l-1}$, where the range of regularity $(s, l)$ is extended, especially at $s=l-1$. The…

We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on $\mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any…

We construct the Lagrangian formulation of massive higher spin on-shell (1,0) supermultiplets in three dimensional anti-de Sitter space. The construction is based on description of the massive three dimensional fields in terms of frame-like…

We formulate closed-form Hessian distances of information entropies in one-dimensional probability density space embedded with the L2-Wasserstein metric.

In this note, we prove interior a priori first- and second-order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that $u$ is…

In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean…