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We introduce a fully nonlinear PDE with a differential form $\Lambda$, which unifies several important equations in K\"ahler geometry including Monge-Amp\`ere equations, J-equations, inverse $\sigma_{k}$ equations, and the deformed…
We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward…
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three…
In this paper, we show that the deformed Hermitian Yang-Mills (dHYM) equation on a rational homogeneous variety, equipped with any invariant K\"{a}hler metric, always admits a solution. In particular, we describe the Lagrangian phase, with…
We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional K\"ahler-Einstein equation with symmetry and the one-sided type-D…
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…
We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our…
This partially expository paper provides a view of Yang-Mills equations from the perspective of complex variables, operator theory, and $C^{*}$-algebras. Through operator-valued pluriharmonic and skew-Hermitian differential forms, it…
We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
In this paper, we present a generalization of a Hamilton--Jacobi theory to higher order implicit differential equations. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradsky approach…
Using the general notions of Batalin, Fradkin, Fradkina and Tyutin to convert second class systems into first class ones, we present a gauge invariant formulation of the massive Yang-Mills theory by embedding it in an extended phase space.…
A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…
In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the…
The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a…
The systematization of the purely Lagrangean approach to constrained systems in the form of an algorithm involves the iterative construction of a generalized Hessian matrix W taking a rectangular form. This Hessian will exhibit as many left…
We prove that if there exists a $C$-subsolution to a constant coefficients strictly $\Upsilon$-stable general inverse $\sigma_k$ equation, then there exists a unique solution. As a consequence, this result covers all the analytical results…
A globally converging numerical method to solve coupled sets of non-linear integral equations is presented. Such systems occur e.g. in the study of Dyson-Schwinger equations of Yang-Mills theory and QCD. The method is based on the knowledge…
The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in…