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H. Blaine Lawson Jr

Publications and source records attributed to H. Blaine Lawson Jr.

At least 19 recordsLinked to original sources
arXiv2026

Duality for Garding operators and a new inequality of GM/AM type

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 space S(n) of symmetric $n \times n$ polynomials, which are hyperbolic with respect to any positive definite matrix. (In particular, any polynomial on ${\mathbb R}^n$, which is hyperbolic with respect to elements in $({\mathbb R_+})^n$, gives rise to a Garding operator on S(n), in many ways.) Here a new duality theorem is developed with many special cases and examples. The dual $g^*$ of a Garding operator $g$ is used to prove a new basic inequality for $g$. An important Lemma of Guo-Phong-Tong is improved upon as a final application.

math.AP↗
arXiv2024

A definitive majorization result for nonlinear operators

Let ${\mathfrak g}$ be a Garding-Dirichlet operator on the set S(n) of symmetric $n\times n$ matrices. We assume that ${\mathfrak g}$ is $I$-central, that is, $D_I {\mathfrak g} = k I$ for some $k>0$. Then $$ {\mathfrak g}(A)^{1\over N} \ \geq\ {\mathfrak g}(I)^{1\over N} (\det\, A)^{1\over n} \qquad \forall\, A>0. $$ From work of Guo, Phong, Tong, Abja, Dinew, Olive and many others, this inequality has important applications.

math.AP↗
arXiv2022

Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive

The objective of this note is to establish the Determinant Majorization Formula $F(A)^{1\over N} \geq \det(A)^{1\over n}$ for all operators $F$ determined by an invariant Garding-Dirichlet polynomial of degree $N$ on symmetric $n \times n$ matrices. Here "invariant" means under the group O$(n)$, U$(n)$ or Sp$(n)$ when the matrices are real symmetric, Hermitian symmetric, or quaternionic symmetric respectively. This greatly expands the applicability of the recent work of Guo-Phong-Tong and Guo-Phong for differential equations on complex manifolds. It also relates to the work of Abja-Olive on interior regularity. Further applications to diagonal operators and to operators depending on the ordered eigenvalues are given. Examples showing the preciseness of the results are presented. For the application to Abja-Olive's work, and other comments in the paper, we establish some results for Garding-Dirichlet operators in appendices. One is an exhaustion lemma for the Garding cone. Another gives bounds for higher order derivatives, which result from their elegant expressions as functions of the Garding eigenvalues. There is also a discussion of the crucial assumption of the Central Ray Hypothesis.

math.AP↗
arXiv2020

Pseudoconvexity for the Special Lagrangian Potential Equation

The Special Lagrangian Potential Equation for a function $u$ on a domain $Ω\subset {\bf R}^n$ is given by ${\rm tr}\{\arctan(D^2 \,u) \} = θ$ for a contant $θ\in (-n {π\over 2}, n {π\over 2})$. For $C^2$ solutions the graph of $Du$ in $Ω\times {\bf R}^n$ is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem for this equation, but the existence result relies on explicitly computing the associated boundary conditions (or, otherwise said, computing the pseudo-convexity for the associated potential theory). This is done in this paper, and the answer is interesting. The result carries over to many related equations -- for example, those obtained by taking $\sum_k \arctan\, λ_k^{\mathfrak g} = θ$ where ${\mathfrak g} : {\rm Sym}^2({\bf R}^n)\to {\bf R}$ is a Garding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformed Hermitian-Yang-Mills equation which appears in mirror symmetry. Another example is $\sum_j \arctan κ_j = θ$ where $κ_1, ... , κ_n$ are the principal curvatures of the graph of $u$ in $Ω\times {\bf R}$. We also discuss the inhomogeneous Dirichlet Problem ${\rm tr}\{\arctan(D^2_x \,u)\} = ψ(x)$ where $ψ: \overlineΩ\to (-n {π\over 2}, n {π\over 2})$. This equation has the feature that the pull-back of $ψ$ to the Lagrangian submanifold $L\equiv {\rm graph}(Du)$ is the phase function $θ$ of the tangent spaces of $L$. On $L$ it satisfies the equation $\nabla ψ= -JH$ where $H$ is the mean curvature vector field of $L$.

math.AP↗
arXiv2020

A generalization of pde's from a Krylov point of view

We introduce and investigate the notion of a `generalized equation' of the form $f(D^2 u)=0$, based on the notions of subequations and Dirichlet duality. Precisely, a subset ${\mathbb H}\subset {\rm Sym}^2({\mathbb R}^n)$ is a generalized equation if it is an intersection ${\mathbb H} = {\mathbb E}\cap (-\widetilde{\mathbb G})$ where ${\mathbb E}$ and ${\mathbb G}$ are subequations and $\widetilde{\mathbb G}$ is the subequation dual to ${\mathbb G}$. We utilize a viscosity definition of `solution' to ${\mathbb H}$. The mirror of ${\mathbb H}$ is defined by ${\mathbb H}^* \equiv {\mathbb G}\cap (-\widetilde {\mathbb E})$. One of the main results here concerns the Dirichlet problem on arbitrary bounded domains $Ω\subset {\mathbb R}^n$ for solutions to ${\mathbb H}$ with prescribed boundary function $φ\in C(\partial Ω)$. We prove that: (A) Uniqueness holds $\iff$ ${\mathbb H}$ has no interior, and (B) Existence holds $\iff$ ${\mathbb H}^*$ has no interior. For (B) the appropriate boundary convexity of $\partial Ω$ must be assumed. Many examples of generalized equations are discussed, including the constrained Laplacian, the twisted Monge-Ampère equation, and the $C^{1,1}$-equation. The closed sets ${\mathbb H}$ which can be written as generalized equations are intrinsically characterized. For such an ${\mathbb H}$ the set of subequation pairs with ${\mathbb H} = {\mathbb E}\cap (-\widetilde{\mathbb G})$ is partially ordered, and there is a canonical least element, contained in all others. Harmonics for the canonical equation are harmonic for all others giving ${\mathbb H}$. A general form of the main theorem, which holds on any manifold, is also established.

math.AP↗
arXiv2019

The inhomogeneous Dirichlet Problem for natural operators on manifolds

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = ψ(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically from a given geometry on $X$. An important point is that the equation need not be convex and can be highly degenerate. Furthermore, the inhomogeneous term can take values at the boundary of the restricted domain $F$ of the operator $f$. A simple example is the real Monge-Ampère operator ${\rm det}({\rm Hess}\,u) = ψ(x)$ on a riemannian manifold $X$, where ${\rm Hess}$ is the riemannian Hessian, the restricted domain is $F = \{{\rm Hess} \geq 0\}$, and $ψ$ is continuous with $ψ\geq0$. A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption. The main theorem applies to pairs $(F,f)$, which are locally jet-equivalent to a given constant coefficient pair $({\bf F}, {\bf f})$. This covers a large family of geometric equations on manifolds: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex manifolds, and operators, such as the Lagrangian Monge-Ampère operator, on symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established with existence requiring the same boundary assumptions as in the homogeneous case [10]. We also have results where the inhomogeneous term $ψ$ is a delta function.

math.AP↗
arXiv2018

Lagrangian potential theory and a Lagrangian equation of Monge-Ampère type

The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in ${\bf C}^n$. However, it applies quite generally -- perhaps most importantly to symplectic manifolds equipped with a Gromov metric. The Lagrange Monge-Ampere operator is an explicit polynomial on ${\rm Sym}^2(TX)$ whose principle branch defines the space of Lag-harmonics. Interestingly the operator depends only on the Laplacian and the SKEW-Hermitian part of the Hessian. The Dirichlet problem for this operator is solved in both the homogeneous and inhomogeneous cases. The homogeneous case is also solved for each of the other branches. This paper also introduces and systematically studies the notions of Lagrangian plurisubharmonic and harmonic functions, and Lagrangian convexity. An analogue of the Levi Problem is proved. In ${\bf C}^n$ there is another concept, Lag-plurihamonics, which relate in several ways to the harmonics on any domain. Parallels of this Lagrangian potential theory with standard (complex) pluripotential theory are constantly emphasized.

math.DG↗
arXiv2016

The AE Theorem and addition theorems for quasi-convex functions

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 F, then u is F-subharmonic, i.e., u is a viscosity F-subsolution. This AE Theorem makes otherwise difficult results transparent. Some instances of this are presented, including two versions of addition, and a comparison theorem.

math.AP↗
arXiv2016

Notes on the differentiation of quasi-convex functions

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 apply and adequate in many cases. Along the way we establish the equivalence of the basic lemmas of Jensen and Slodkowski.

math.AP↗
arXiv2016

The Dirichlet Problem with Prescribed Asymptotic Singularities

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 case it is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel: $Θ_j K_p(x - x_j)$, where $K_p(x) = - {1\over|x|^{p-2}}$ for $p>2$ and $K_2(x) = \log |x|$, at any prescribed finite set of points $x_1,...,x_k$ in the domain and any finite set of positive real numbers $Θ_1,..., Θ_k$. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions. Uniqueness and existence results are also established for finite-type singularities such as $Θ_j |x - x_j|^{2-p}$ for $1\leq p<2$. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).

math.AP↗
arXiv2016

Tangents to subsolutions -- existence and uniqueness, Part I

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 in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the "Riesz characteristic", which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Ampère equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $\geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Hölder continuity of subsolutions when the Riesz characteristic p satisfies $1 \leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the "geometric cases". A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Ampère case are completely classified.

math.AP↗
arXiv2016

Characterizing the Strong Maximum Principle

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 easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral \int dy / f(y) near 0 is infinite or finite. This complements our previous work characterizing when the (ordinary) maximum principle holds. Along the way we characterize radial subsolutions.

math.AP↗
arXiv2014

Tangents to subsolutions -- existence and uniqueness, Part II

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-plurisubharmonic if its restriction to $Ω\cap W$ is subharmonic for every affine G-plane $W$. Here G is assumed to be invariant under a subgroup K of O(n) which acts transitively on the sphere $S^{n-1}$. Tangents to u at a point x are the cluster points of u under a natural flow (or blow-up) at x. They always exist and are G-harmonic at all points of continuity. A homogeneity property is established for all tangents in these geometric cases. This leads to principal results concerning the Strong Uniqueness of Tangents, which means that all tangents are unique and of the form $ΘK_p$ where $K_p$ is the Riesz kernel and $Θ$ is the density of u at the point. Strong uniqueness is a form of regularity which implies that the sets $\{Θ(u,x)\geq c\}$ for $c>0$ are discrete. When the invariance group K= O(n), U(n) or Sp(n), strong uniqueness holds for all but a small handful of cases. It also holds for essentially all interesting G which arise in calibrated geometry. When strong uniqueness fails, homogeneity implies that tangents are characterized by a subequation on the sphere, which is worked out in detail. In the cases corresponding to the real, complex and quaternionic Monge-Ampère equations (convex functions, and complex and quaternionic plurisubharmonic functions) tangents, which are far from unique, are then systematically studied and classified.

math.AP↗
arXiv2014

Removable singularities 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 X-E, which is locally bounded across E, extends to an F-subharmonic function on X. We also study the related question for F-harmonic functions (solutions) which are continuous across E. Main results assert that if there exists a convex cone subequation M such that F+M is contained in F, then any closed set E which is M-polar has these properties. To be M-polar means that E = {f = -\infty} where f is M-subharmonic on X and smooth outside of E. Many examples and generalizations are given. These include removable singularity results for all branches of the complex and quaternionic Monge-Ampere equations, and a general removable singularity result for the harmonics of geometrically defined subequations. For pure second-order subequations in R^n with monotonicity cone M, the Riesz characteristic p = p(M) is introduced, and extension theorems are proved for any closed singular set E of locally finite Hausdorff (p-2)-measure (or, more generally, of (p-2)-capacity zero). This applies for example to branches of the equation s_k(D^2 u) = 0 (kth elementary function) where p(M) = n/k. For convex cone subequations themselves, several removable singularity theorems are proved, independent of the results above.

math.AP↗
arXiv2013

p-convexity, p-plurisubharmonicity and the Levi problem

Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables, namely: local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the p-hull of the boundary union with the "core" of the space. Lastly, the exteme rays in the convex cone of p-positive matrices are characterized. This is a basic result with many applications.

math.DG↗
arXiv2013

Potential Theory on Almost Complex Manifolds

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. In the second theorem, using our "viscosity" definitions, the Dirichlet problem is solved for the complex Monge-Ampère equation in both the homogeneous and inhomogeneous forms. These two results are based on theorems found in two recent papers of the authors. Finally, it is shown that the plurisubharmonic functions considered here agree with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.

math.CV↗
arXiv2013

The Restriction Theorem for Fully Nonlinear Subequations

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 the same constraints. We first prove an elementary result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any "geometrically defined" subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.

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