arXiv research

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

Study solves equations on tori for Calabi-Yau problems.

Solves geometric problems using fully nonlinear equations and Morse theory.

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

Study improves understanding of solutions to complex equations in geometry.

Solves open problems for fully nonlinear elliptic equations on manifolds.

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…

This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations.

Study fully nonlinear elliptic equations on complex manifolds.

Solves nonlinear problems on metric structures through eigenvalue counting.

We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

Estimates for complex equations on manifolds derived from a conjecture.

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

Study on maximum principles for nonlinear equations on Riemannian manifolds.

Uniform estimates for complex equations on compact manifolds found.

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

Paper establishes estimates for nonlinear equations on compact manifolds.

We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.

Introduces a new PDE involving differential forms for Kähler geometry.

We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.

Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.

Paper develops techniques to solve complex PDEs involving higher cohomology forms.

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $σ_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for solutions to these equations and establish an existence result associated to them.

Study proves radial symmetry of solutions to certain nonlinear equations in space forms.

Solves a specific Dirichlet problem on Riemannian manifolds.

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and opt…

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

Uniform bounds for complex equations using Monge-Ampère method.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number …

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models …

Note on advancements in nonlinear elliptic equations' regularity theory.

In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solution…

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.

We present some new ideas to derive {\em a priori} second order estiamtes for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in $\bfR^n$, are powerful enough to work in general Riemannian manifolds.

This paper proves Liouville theorems for conformally invariant fully nonlinear equations.