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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.

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., Comment: Further applications to diagonal operators and to operators depending on the ordered eigenvalues have been given. Examples showing the preciseness of the results are presented. There are two new sections and two new appendices

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 pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between nonlinear potential theory; that is, the study of subharmonics with respect to a general constraint set in the 2-jet bundle and the study of subsolutions and supersolutions of a nonlinear (degenerate) elliptic PDE. The main results include (but are not limited to) the validity of the comparison principle and the existence and uniqueness to solutions to the relevant Dirichlet problems on domains which are suitably "pseudoconvex". The methods employed are geometric and flexible as well as being very general on the potential theory side, which is interesting in its own right. Moreover, in many important geometric contexts no natutral operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structual conditions on a given differential operator., Comment: 42 pages

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 comparison principles for nonlinear differential operators, a program seemingly different from the first. However, we shall marry these two points of view, for a wide variety of equations, under something called the correspondence principle. In potential theory one is given a constraint set F on the 2-jets of a function, and the boundary of F gives a differential equation. There are many differential operators, suitably organized around F, which give the same equation. So potential theory gives a great strengthening and simplification to the operator theory. Conversely, the set of operators associated to F can have much to say about the potential theory. An object of central interest here is that of monotonicity, which explains and unifies much of the theory. We shall always assume that the maximal monotonicity cone for a potential theory has interior. This is automatic for gradient-free equations where monotonicity is simply the standard degenerate ellipticity and properness assumptions. We show that for each such potential theory F there is an associated canonical operator, defined on the entire 2-jet space and having all the desired properties. Furthermore, comparison holds for this operator on any domain which admits a regular strictly M-subharmonic function, where M is a monotonicity subequation for F. On the operator side there is an important dichotomy into the unconstrained cases and constrained cases, where the operator must be restricted to a proper subset of 2-jet space. These two cases are best illustrated by the canonical operators and Dirichlet-Garding operators, respectively. The article gives many, many examples from pure and applied mathematics, and also from theoretical physics., Comment: 170 pages

This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the $F$-potential theory associated to a general nonlinear convex subequation $F \subset J^2(X)$ on a manifold $X$. The main theorem is the following "local to global" result. Suppose $u$ is a continuous strictly $F$-subharmonic function such that each point $x\in X$ has a fundamental neighborhood system consisting of domains for which a "quasi" form of $C^\infty$ approximation holds. Then for any positive $h\in C(X)$ there exists a strictly $F$-subharmonic function $w\in C^\infty(X)$ with $u< w< u+h$. Applications include all convex constant coefficient subequations on ${\bf R}^n$, various nonlinear subequations on complex and almost complex manifolds, and many more.

The Special Lagrangian Potential Equation for a function $u$ on a domain $\Omega\subset {\bf R}^n$ is given by ${\rm tr}\{\arctan(D^2 \,u) \} = \theta$ for a contant $\theta \in (-n {\pi\over 2}, n {\pi\over 2})$. For $C^2$ solutions the graph of $Du$ in $\Omega\times {\bf R}^n$ 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 $\sum_k \arctan\, \lambda_k^{{\mathfrak g}} = \theta$ 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 $\sum_j \arctan \kappa_j = \theta$ where $\kappa_1, ... , \kappa_n$ are the principal curvatures of the graph of $u$ in $\Omega\times {\bf R}$. We also discuss the inhomogeneous Dirichlet Problem ${\rm tr}\{\arctan(D^2_x \,u)\} = \psi(x)$ where $\psi : \overline{\Omega}\to (-n {\pi\over 2}, n {\pi\over 2})$. This equation has the feature that the pull-back of $\psi$ to the Lagrangian submanifold $L\equiv {\rm graph}(Du)$ is the phase function $\theta$ of the tangent spaces of $L$. On $L$ it satisfies the equation $\nabla \psi = -JH$ where $H$ is the mean curvature vector field of $L$., Comment: Several things were added to the paper. More examples were given, some new results were proved, and the historical discussion was expanded

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 $\Omega\subset {\mathbb R}^n$ for solutions to ${{\mathbb H}}$ with prescribed boundary function $\varphi \in C(\partial \Omega)$. 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 $\partial \Omega$ must be assumed. Many examples of generalized equations are discussed, including the constrained Laplacian, the twisted Monge-Amp\`ere 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., Comment: Many new examples have been added as well as some new material

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = \psi(x)$ 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-Amp\`ere operator ${\rm det}({\rm Hess}\,u) = \psi(x)$ on a riemannian manifold $X$, where ${\rm Hess}$ is the riemannian Hessian, the restricted domain is $F = \{{\rm Hess} \geq 0\}$, and $\psi$ is continuous with $\psi\geq0$. 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-Amp\`ere 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 $\psi$ is a delta function., Comment: Some minor additions

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., Comment: Change of title

The general purpose of this paper is to investigate the notion of "pluriharmonics" for the general potential theory associated to a convex cone $F\subset {\rm Sym}^2({\bf R}^n)$. For such $F$ there exists a maximal linear subspace $E\subset F$, called the edge, and $F$ decomposes as $F=E \oplus F_0$. The pluriharmonics or edge functions are $u$'s with $D^2u \in E$. Many subequations $F$ have the same edge $E$, but there is a unique smallest such subequation. These are the focus of this investigation. Structural results are given. Many examples are described, and a classification of highly symmetric cases is given. Finally, the relevance of edge functions to the solutions of the Dirichlet problem is established., Comment: We expanded some introductory material

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: $\Theta_j K_p(x - x_j)$, 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 $\Theta_1,..., \Theta_k$. 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 $\Theta_j |x - x_j|^{2-p}$ 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)., Comment: Material has been added concerning the Dirichlet Problem with prescribed values at marked interior points when the Riesz characteristic is strictly between 1 and 2

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.

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 $\Omega$ in ${\mathbb R}^n$ is G-plurisubharmonic if its restriction to $\Omega\cap W$ 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 $\Theta K_p$ where $K_p$ is the Riesz kernel and $\Theta$ is the density of u at the point. Strong uniqueness is a form of regularity which implies that the sets $\{\Theta(u,x)\geq c\}$ 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-Amp\`ere equations (convex functions, and complex and quaternionic plurisubharmonic functions) tangents, which are far from unique, are then systematically studied and classified.