64 citations · 212 across the 30 of their papers we have counts for
22 papers · 1 filter
Duality for Garding operators and a new inequality of GM/AM type
F. Reese Harvey, H. Blaine Lawson
This article concerns Garding Dirichlet operators, and also the theory of Garding polynomials. Garding Dirichlet operators are those which arise from homogeneous polynomials on the…
A definitive majorization result for nonlinear operators
F. Reese Harvey, H. Blaine Lawson
Let be a Garding-Dirichlet operator on the set S(n) of symmetric matrices. We assume that is -central, that is, $D_I {\mathfrak g} =…
Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive
F. Reese Harvey, H. Blaine Lawson
The objective of this note is to establish the Determinant Majorization Formula for all operators determined by an invariant Garding-D…
Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs
F. Reese Harvey, Kevin R. Payne
We discuss one of the many topics that illustrate the interaction of Blaine Lawson's deep geometric and analytic insights. The first author is extremely grateful to have had the pl…
Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and PDEs
Marco Cirant, F. Reese Harvey, H. Blaine Lawson +1
We prove comparison principles for nonlinear potential theories in euclidian spaces in a very straightforward manner from duality and monotonicity. We shall also show how to deduce…
The Richberg technique for subsolutions
F. Reese Harvey, H. Blaine Lawson, Szymon Pliś
This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the -potential theory associated to a general nonlinear convex subequation $F…
Pseudoconvexity for the Special Lagrangian Potential Equation
F. Reese Harvey, H. Blaine Lawson
The Special Lagrangian Potential Equation for a function on a domain is given by for a contant $θ\in (-n {π\over 2}, n…
A generalization of pde's from a Krylov point of view
F. Reese Harvey, H. Blaine Lawson
We introduce and investigate the notion of a `generalized equation' of the form , based on the notions of subequations and Dirichlet duality. Precisely, a subset ${\mat…
Pluriharmonics in general potential theories
F. Reese Harvey, H. Blaine Lawson
The general purpose of this paper is to investigate the notion of "pluriharmonics" for the general potential theory associated to a convex cone . F…
The Dirichlet Problem with Prescribed Asymptotic Singularities
F. Reese Harvey, H. Blaine Lawson
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…
Tangents to subsolutions -- existence and uniqueness, Part II
F. Reese Harvey, H. Blaine Lawson
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 th…
Tangents to subsolutions -- existence and uniqueness, Part I
F. Reese Harvey, H. Blaine Lawson
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation . These include all the potential theories attached to calibr…
Notes on the differentiation of quasi-convex functions
F. Reese Harvey, H. Blaine Lawson
This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to a…
The AE Theorem and addition theorems for quasi-convex functions
F. Reese Harvey, H. Blaine Lawson
The main point of this paper is to prove the following useful result: If the almost everywhere 2-jet of a locally quasi-convex function u satisfies a degenerate elliptic constraint…
Characterizing the Strong Maximum Principle
F. Reese Harvey, H. Blaine Lawson
In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easi…
Existence, Uniqueness and Removable Singularities for Nonlinear Partial Differential Equations in Geometry
F. Reese Harvey, H. Blaine Lawson
This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats rest…
Removable singularities for nonlinear subequations
F. Reese Harvey, H. Blaine Lawson
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 t…
The equivalence of viscosity and distributional subsolutions for convex subequations - a strong Bellman principle
F. Reese Harvey, H. Blaine Lawson
There are two useful ways to extend nonlinear partial differential inequalities of second order: one uses viscosity theory and the other uses the theory of distributions. This pape…
The Restriction Theorem for Fully Nonlinear Subequations
F. Reese Harvey, H. Blaine Lawson
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…
Hyperbolic polynomials and the Dirichlet problem
F. Reese Harvey, H. Blaine Lawson
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 inequa…
Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds
F. Reese Harvey, H. Blaine Lawson
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 t…
Dirichlet Duality and the Nonlinear Dirichlet Problem
F. Reese Harvey, H. Blaine Lawson
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is re…