arXiv research

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…

Richberg technique adapted for nonlinear subequations.

We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E in X have the property that every F-subharmonic function (subsolution) on…

Introduces generalized equations based on subequations and Dirichlet duality.

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in $\mathbb{C}^n.$ Existence of geodesics in the space of positive Lagrangians is an important step in…

In this paper, we study the singular sets of $F$-subharmonic functions $u: B_{2}(0^{n})\rightarrow\mathbf{R}$, where $F$ is a subequation. The singular set $\mathcal{S}(u)\subset B_{2}(0^{n})$ has a stratification $\mathcal{S}^{0}(u)\subset\mathcal{S}^{1}(u)\subset\cdots\subset\mathcal{S}^{k}(u)\subset\cdots\subset\mat…

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting…

Estimates for polynomial operators using determinant majorization and subharmonics.

New findings on convexity of special Lagrangian geodesics.

We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this cas…

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact …

This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in ${\mathbb R}^n$. An upper semi-continuous function u on an open set $Ω$ in ${\mathbb R}^n$ is G-plurisubharmon…

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. I…