Tangents to Subsolutions Existence and Uniqueness, II

Abstract

This second part of the paper (see Ann Math de Toulouse, arXiv:1408.5797 for part I) is concerned with questions of existence and uniqueness of tangents in the special case of $\varvec{\mathbb {G}}$-plurisubharmonic functions, where $\varvec{\mathbb {G}}\subset G(p,\mathbf{R}^n)$ is a compact subset of the Grassmannian of p-planes in $\mathbf{R}^n$. An u.s.c. function u on an open set $\Omega \subset \mathbf{R}^n$ is $\varvec{\mathbb {G}}$-plurisubharmonic if its restriction to $\Omega \cap W$ is subharmonic for every affine $\varvec{\mathbb {G}}$-plane W. Here $\varvec{\mathbb {G}}$ is assumed to be invariant under a subgroup $K\subset \mathrm{O}(n)$ which acts transitively on $S^{n-1}$. 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 $\varvec{\mathbb {G}}$-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 $\{\Theta (u,x)\ge c\}$ for $c>0$ are discrete. When the invariance group $K= \mathrm{O}(n), \mathrm{U}(n)$ or Sp(n) strong uniqueness holds for all but a small handful of cases. It also holds for essentially all interesting $\varvec{\mathbb {G}}$ 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.

Appendix: Further Discussion of Examples

In this appendix we examine specific subequations of Riesz characteristic p, in more detail. We consider two types: cone subequations and convex cone subequations, and in both cases the subequations will always be ST-invariant . It may be of some surprise that in each of these two categories there is a unique largest and smallest subequation.

1.1 The Largest/Smallest Characteristic p Subequation

We first consider the category of cone subequations. For $A\in \mathrm{Sym}^2(\mathbf{R}^n)$ let $\lambda _1(A) \le \cdots \le \lambda _n(A)$ denote the ordered eigenvalues of A, and set $\lambda _\mathrm{min}(A) \equiv \lambda _1(A)$ and $\lambda _\mathrm{max}(A)\equiv \lambda _n(A)$. We then define

$$\begin{aligned}&\mathcal{P}^\mathrm{min/max}_p \ \equiv \ \left\{ A : \lambda _\mathrm{min}(A) + (p-1) \lambda _\mathrm{max}(A) \ \ge \ 0 \right\} \end{aligned}$$

(6.1)

$$\begin{aligned}&\mathcal{P}^\mathrm{min/2}_p \ \equiv \ \left\{ A : \lambda _\mathrm{min}(A) + (p-1) \lambda _{2}(A) \ \ge \ 0 \right\} \end{aligned}$$

(6.2)

It is clear from Definition 3.2 in Part I that both of these subequations has Riesz characteristic p. These are the largest and smallest cone subequations with this property.

Lemma 6.1

Let F be an ST-invariant cone subequation of Riesz characteristic p. Then

$$\begin{aligned} \mathcal{P}^\mathrm{min/2}_p\ \subset \ F\ \subset \ \mathcal{P}^\mathrm{min/max}_p. \end{aligned}$$

Note One computes that the dual of this largest subequation $\mathcal{P}^\mathrm{min/max}_p$ is $\mathcal{P}^\mathrm{min/max}_q$ where $(p-1)(q-1)=1$. Compare this with (3.22) in Part I which says that $(p_F-1)(q_F-1)\ge 1$ for any subequation F. Also see Example 13.14 in Part I.

Proof of Lemma 6.1

Each $A\in \mathrm{Sym}^2(\mathbf{R}^n)$ can be written as a sum $A = \lambda _1 P_{e_1} + \cdots + \lambda _n P_{e_n}$ using the ordered eigenvalues of A. Set $B_0 \equiv \lambda _1 P_{e_1} + \lambda _2 P_{e_1^\perp }$, and $B_1 \equiv \lambda _1 P_{e_1} + \lambda _n P_{e_1^\perp }$, and note that $B_0\le A \le B_1$.

If $A\in \mathcal{P}^\mathrm{min/2}_p$, then $\lambda _1 + (p-1)\lambda _2\ge 0$. Thus, $B_0 \in \mathcal{P}^\mathrm{min/2}_p$. Since $\mathcal{P}^\mathrm{min/2}_p$ and F have the same increasing radial profile $E^\uparrow$ given by (3.1) in Part I (and $\lambda _2\ge 0$), we conclude that $B_0\in F$. However, $B_0\le A$ proving that $A\in F$.

For the other inclusion, pick $A \in F$. Since $F\subset \widetilde{\mathcal{P}}$, we have $\lambda _\mathrm{max} \ge 0$. Now $A\le B_1$ implies $B_1\in F$. Again F and $\mathcal{P}^\mathrm{min/max}_p$ have the same increasing radial profile $E^\uparrow$ given by (3.1) in Part I. Therefore, $B_1 \in \mathcal{P}^\mathrm{min/max}_p$. This implies by definition that $A \in \mathcal{P}^\mathrm{min/max}_p$. $\square$

The largest and smallest characteristic p subequations in the convex cone case are different in dimensions $\ge 3$ (see Sect. 4 in Part I for the definitions of $\mathcal{P}_p$ and $\mathcal{P}(\delta )$).

Lemma 6.2

Let F be an O(n)-invariant convex cone subequation of Riesz characteristic p. Then $1\le p\le n$ and

$$\begin{aligned} \qquad \qquad \qquad \mathcal{P}_p\ \subset \ F\ \subset \ \mathcal{P}(\delta ) \qquad \quad \mathrm{where \ \ \ } \delta _p\ =\ {(p-1)n \over n-p}. \end{aligned}$$

(One computes that the Riesz characteristic of $\mathcal{P}(\delta _p)$ is p). In fact, the first inclusion holds for any ST-invariant characteristic p convex cone subequation F.

Proof

The first inclusion follows from the fact that $-(p-1) P_e + P_{e^\perp }$ generate the extreme rays in $\mathcal{P}_p$. This is proved in [11, Theorem 5.1c]. The second inclusion is Proposition 13.9 in Part I. $\square$

1.2 O(n)-Invariant Subequations

Such a subequation F determines a subset $E\subset \mathbf{R}^n$ consisting of the n-tuples $(\lambda _1(A),\ldots ,\lambda _n(A))$ of eigenvalues of A. Consider $\lambda (A) = (\lambda _1(A),\ldots ,\lambda _n(A))$ as a multi-valued map $\lambda :\mathrm{Sym}^2(\mathbf{R}^n)\rightarrow \mathbf{R}^n$. Then we define $E\equiv \lambda (F)$. The set E is closed and symmetric (invariant under the permutation of coordinates in $\mathbf{R}^n$). In addition,

$$\begin{aligned} E \ \ \mathrm{is}\ \ \mathbf{R}^n_+\ \mathrm{(positive \ orphant) \ monotone, \ i.e., } \ \ E+ \mathbf{R}^n_+ = E, \end{aligned}$$

(6.3)

since the ordered eigenvalues are $\mathcal{P}$-monotone.

Definition 6.3

A closed symmetric subset $E\subset \mathbf{R}^n$ (with $\emptyset \ne E \ne \mathbf{R}^n$) will be called a universal eigenvalue subequation if E is $\mathbf{R}^n_+$-monotone.

Note that this is an abuse of language since E itself is not a subequation. The “universal” nature of E will be described later. However, such a set E determines the O(n)-invariant subequation

$$\begin{aligned} F\ =\ \lambda ^{-1}(E). \end{aligned}$$

(6.4)

Note that

$$\begin{aligned}&F\ \ \mathrm{is\ a\ cone\ }\quad \iff \quad E\ \ \mathrm{is\ a\ cone, \ and}\nonumber \\&F\ \ \mathrm{is\ convex\ }\quad \iff \quad E\ \ \mathrm{is\ convex.} \end{aligned}$$

(6.5)

Of course, $\mathcal{P}$ and $\mathbf{R}^n_+$ correspond, i.e., $\mathcal{P}= \lambda ^{-1}(\mathbf{R}^n_+)$. The Riesz characteristic of F is easily computed from its eigenvalue profile E.

• The increasing Riesz characteristic of F equals $\sup \{p : (-(p-1), 1,\ldots , 1) \in E\}$.

• The decreasing Riesz characteristic of F equals $\sup \{q : (-1,\ldots , -1, (q-1)) \in E\}$.

It is also worth noting that if E and F correspond, the $\widetilde{E}$ and $\widetilde{F}$ correspond.

1.3 The Complex and Quaternionic Analogues of an O(n)-Invariant Subequation

As described in Example 4.7 in Part I, each O(n)-invariant subequation F on $\mathbf{R}^n$ canonically determines a U(n)-invariant subequation $F^\mathbf{C}$ on $\mathbf{C}^n$ and an Sp(n)-invariant subequation $F^\mathbf{H}$ on $\mathbf{H}^n$. In both cases the subequation is given by requiring that the n eigenvalues of the hermitian symmetric part of the matrix lie in $E_F$. That is, if E is defined by $A\in F \iff \lambda (A) \in E$, then

$$\begin{aligned} A\in F^\mathbf{C}\ \ \iff \ \ \lambda ^\mathbf{C}_k(A) \in E \qquad \mathrm{and} \qquad A\in F^\mathbf{H}\ \ \iff \ \ \lambda ^\mathbf{H}_k(A) \in E, \end{aligned}$$

(6.6)

where $\lambda ^\mathbf{C}_k(A) = \lambda _k(A_\mathbf{C})$ and $\lambda ^\mathbf{H}_k(A) = \lambda _k(A_\mathbf{H})$

The associated Riesz characteristics are doubled for the complex analogues (see Lemma 4.8 in Part I).

This classifies all the U(n)-invariant subequations F with the property that

$$\begin{aligned} F \ =\ \pi _\mathbf{C}^{-1}\left( F^\mathbf{C}\right) \qquad \mathrm{where} \ \ \pi _\mathbf{C}(A) \ =\ A_\mathbf{C}\end{aligned}$$

(6.7)

and $F^\mathbf{C}$ is a subset of the hermitian symmetric matrices. Similarly, it defines all the Sp(n)-invariant subequations F with the property that

$$\begin{aligned} F \ =\ \pi _\mathbf{H}^{-1}\left( F^\mathbf{H}\right) \qquad \mathrm{where} \ \ \pi _\mathbf{H}(A) \ =\ A_\mathbf{H}\end{aligned}$$

(6.8)

and $F^\mathbf{H}$ is a subset of the quaternionic hermitian symmetric matrices (here the Riesz characteristic is quadrupled).

The largest/smallest results (Lemmas 6.1 and 6.2 in the $\mathbf{R}$ case) have counterparts in the $\mathbf{C}$ and $\mathbf{H}$ cases. The precise statements and their proofs are left to the reader.

Example 6.4 (Lagrangian Pluripotential Theory) A notable new example of a U(n)-invariant subequation not satisfying (6.8) comes from Lagrangian geometry, namely the geometrically defined subequation $F(\mathrm{LAG})$ where $\mathrm{LAG}\subset G^\mathbf{R}(n, \mathbf{C}^n)$ is the set of Lagrangian n-planes in $\mathbf{C}^n$. The $F(\mathrm{LAG})$-subharmonics can be characterized as the upper semi-continuous functions whose restrictions to Lagrangian planes are subharmonic (parallel to classical pluripotential theory). The eigenvalues of the skew-hermitian part $A_\mathbf{C}^\mathrm{skew}$ come in pairs $\lambda _1, - \lambda _1, \lambda _2, -\lambda _2,\ldots , \lambda _n, -\lambda _n$. The subequation $F(\mathrm{LAG})$ is determined by a constraint on these eigenvalues together with the real trace

$$\begin{aligned} \mu \ \equiv \ {1\over 2} \mathrm{tr}(A), \end{aligned}$$

namely

$$\begin{aligned} \mu \ \pm \ \lambda _1 \ \pm \ \lambda _2\ \pm \ \cdots \ \pm \ \lambda _n\ \ge \ 0 \end{aligned}$$

(6.9)

for all $2^n$ choices of ±. There is a polynomial operator M on $\mathrm{Sym}^2_\mathbf{R}(\mathbf{C}^n)$ analogous to the determinants $\mathrm{det}_\mathbf{R}A$, $\mathrm{det}_\mathbf{C}A$, and $\mathrm{det}_\mathbf{H}A$, namely

$$\begin{aligned} M_{\mathrm{LAG}}(A) \ =\ \prod _{2^n\ { \mathrm times}} \left( \mu \ \pm \ \lambda _1 \ \pm \ \lambda _2\ \pm \ \cdots \ \pm \ \lambda _n\right) \end{aligned}$$

(6.10)

(see [8, page 433]) and [15, Def. 5.1]. Of course since $\mathrm{LAG}\subset G(n,\mathbf{R}^{2n})$ and F is geometrically defined by $\mathrm{LAG}$, F has Riesz characteristic n. The set $\mathrm{LAG}$ (for $n>1$) has the transitivity property, so that Theorem 3.2 applies to yield strong uniqueness of tangents to Lagrangian plurisubharmonic functions.

Example 6.4 $_p$. (Isotropic Subharmonic) The previous example can be generalized as follows. For each integer p, $1\le p\le n$ we consider the set

$$\begin{aligned} \mathrm{ISO}_p \ =\ \{W\in G^\mathbf{R}(p, \mathbf{C}^n) : W \ \mathrm{is\ an\ isotropic\ } p\ \mathrm{plane}\}. \end{aligned}$$

Recall that a real p-plane W in $\mathbf{C}^n=\mathbf{R}^{2n}$ is isotropic if

$$\begin{aligned} v\perp Jw\qquad \forall \, v,w\in W, \end{aligned}$$

i.e., the Kähler form $\omega$ satisfies $\omega \bigr |_W=0$. Note that $\mathrm{ISO}_n = \mathrm{LAG}$ and $\mathrm{ISO}_1=\mathcal{P}$. Except for $\mathrm{ISO}_1 = \mathcal{P}$ the set $\mathrm{ISO}_p$ has the transitivity property, and so Theorem 3.2 applies to yield strong uniqueness of tangents.

1.4 Subequations Arising from Gårding Operators

Gårding’s beautiful theory of hyperbolic polynomials provides a surprisingly rich collection of non-linear operators. (This connection is mentioned in Krylov [17]). Moreover, associated with each such “Gårding operator” there are many actual subequations. Here we provide a brief overview. We first start with an operator and discuss how the many associated subequations are universally constructed. We then describe three of the basic ways of constructing new Gårding operators from a given one. These repeatable processes lead to a vast array of Gårding operators, starting with just one.

See for example [9] for a self-contained development of Gårding’s theory. His two fundamental results can be summarized by saying that:

$$\begin{aligned}&\mathrm{The \ Garding\ cone\ } \Gamma \ \mathrm{is\ convex,\ and} \end{aligned}$$

(6.11)

$$\begin{aligned}&\mathrm{The \ Garding\ eigenvalues\ are\ } \Gamma -\mathrm{monotone.} \end{aligned}$$

(6.12)

Definition 6.5

A homogeneous real polynomial M of degree m on the space $\mathrm{Sym}^2(\mathbf{R}^n)$ of second derivatives, with $M(I)>0$, is a Gårding operator if:

• (1) For each $A\in \mathrm{Sym}^2(\mathbf{R}^n)$ the polynomial $M(sI+A)$ has m real roots (M is I-hyperbolic), and

• (2) The Gårding cone $\Gamma$, defined as the connected component of I in $\{M(A)>0\}$, satisfies positivity $\Gamma +\mathcal{P}\subset \Gamma$.

The primary subequation associated with the Gårding operator M is the closure of the Gårding cone $\Gamma$, which is a convex cone subequation. However, there are many others.

Definition 6.6

The negatives of the roots of $M(sI+A)=0$ are called the M-eigenvalues of A, and are denoted by $\Lambda (A) \equiv (\Lambda _1(A),\ldots \Lambda _m(A))$. Thus, $M(sI+A) = M(I) \prod _j \left( s+ \Lambda _j(A) \right)$. These M-eigenvalues of A are well defined up to permutations. Note that $M(A) = M(I) \Lambda _1(A) \cdots \Lambda _n(A)$.

Definition 6.7

The kth branch of the equation $M(A)=0$ is defined to be the set

$$\begin{aligned} \{\Lambda _k(A)\ \ge \ 0\}, \end{aligned}$$

where $\Lambda _1(A) \le \cdots \le \Lambda _m(A)$ are the ordered M-eigenvalues of A.

An important part of the theory shows that the ordered M-eigenvalues are strictly $\Gamma$-monotone. Since $\mathcal{P}\subset \overline{\Gamma }$ by (6.13), we have that each of the m branches of $\{M(A)=0\}$ is a subequation.

Note that $\overline{\Gamma }= \{\Lambda _\mathrm{min}(A)\ge 0\}$ is the first and smallest branch, and that its dual subequation $\{\Lambda _\mathrm{max}(A)\ge 0\}$ is the largest branch.

The branches $\{\Lambda _k(A)\ge 0\}$, $k=1\ldots ,m$, are the subequations most intimately associated with the Gårding operator M in that if a $C^2$-function u is harmonic for one of these subequations, then

$$\begin{aligned} M\left( D_x^2 u\right) \ =\ 0. \end{aligned}$$

(6.13)

However, there are many others, all constructed exactly as in the O(n)-invariant case.

Now we make full use of the concept (Definition 6.3) of a universal eigenvalue subequation.

Proposition 6.8

Given a universal eigenvalue subequation $E\subset \mathbf{R}^m$, each Gårding operator of degree m on $\mathrm{Sym}^2(\mathbf{R}^n)$ determines a subequation on $\mathbf{R}^n$, namely

$$\begin{aligned} F_E \ \equiv \ \{ A : \Lambda (A) \in E\}. \end{aligned}$$

This subequation is $\overline{\Gamma }$-monotone (not just $\mathcal{P}$-monotone). Moreover, $F_E$ is a cone if and only if E is a cone., and $F_E$ is convex if and only if E is convex.

Proof

This is straightforward except for the last assertion which is due to [4].

For example, $E = \mathbf{R}^m_+$ is the universal “Monge–Ampère subequation” inducing the subequation $\overline{\Gamma }$ for each degree m Gårding operator $M(A)= M(I) \Lambda _1(A) \cdots \Lambda _m(A)$.

We complete this discussion of Gårding operators by describing three of the basic methods of constructing new Gårding operators from a given Gårding operator M of degree m. To be specific the reader may want to start with one of the basic operators $\mathrm{det}(A_K)$ for $K=\mathbf{R},\mathbf{C}$ or $\mathbf{H}$.

I. The Derived or Elementary Symmetric Operator With $k=1,\ldots ,m$ fixed and $\ell \equiv m-k$ we define

$$\begin{aligned} \sigma _k(A) \ \equiv \ {1\over (\ell )!} {d^{\ell }\over dt^{\ell }} M(A+tI)\bigr |_{t=0} \qquad \mathrm{and} \qquad \sigma _k(A) \ \equiv \ \ =\ \sum _{i_1<\cdots i_k} \Lambda _{i_1}(A) \ldots \Lambda _{i_k}(A) \end{aligned}$$

and note that they are equal.

II. The p-Convexity Operator For each real number p with $1\le p\le m$ set

$$\begin{aligned} \Sigma _p(A) \ \equiv \ \prod \left\{ \Lambda _{i_1}(A) + \cdots + \Lambda _{i_{[p]}} (A)+ (p-[p]) \Lambda _j(A)\right\} , \end{aligned}$$

where the product is taken over all increasing multi-indices $I=(i_1,\ldots , i_{[p]})$ and all $j\notin I$.

III. The $\delta$-Uniformly Elliptic Regularization Operator With $0\le \delta \le \infty$ (and renormalizing at $\delta =\infty$) set

$$\begin{aligned} M^\delta (A)\ \equiv \ M\left( A+{\delta \over n} (\mathrm{tr}A) I\right) \ =\ M(I) \prod _{j=1}^m \left\{ \Lambda _j(A) + {\delta \over n}(\mathrm{tr}A)\right\} . \end{aligned}$$

$\square$

Remark 6.9

(Iteration) The first process lowers the degree of the operator. The second process raises the degree of the operator, and the degree remains the same in the third process. One can apply any sequence of the three operations, thereby producing a huge collection of Gårding operators all dependent on the primary operator.

1.5 Riesz Characteristics of Branches

Suppose that M is an ST-invariant Gårding operator (Definition 6.5) with ordered eigenvalues $\Lambda _1(A)\le \cdots \le \Lambda _m(A)$. The $k\mathrm{th}$ branch, defined by $\Lambda _k(A)\ge 0$, and the $(m-k+1)^\mathrm{st}$ branch, defined by $\Lambda _{m-k+1}(A)\ge 0$, are dual subequations, since $\Lambda _{m-k+1}(A) = -\Lambda _k(-A)$ is the operator dual to $\Lambda _k(A)$. Note that $0\le \Lambda _k(P_e) \le 1$, $k=1,\ldots ,m$ since $0<P_e<I$.

Proposition 6.10

The Reisz characteristic of the $k\mathrm{th}$ branch equals $\Lambda _{m-k+1}(P_e)^{-1}$. That is, with $\Lambda _1(P_e)\le \cdots \Lambda _m(P_e)$, the quantities $\Lambda _m(P_e)^{-1} \le \cdots \Lambda _1(P_e)^{-1}$ are the Riesz characteristics of the ordered branches starting with $\Lambda _m(P_e)^{-1} =$ the Riesz characteristic of the smallest branch.

Proof

If p is the Riesz characteristic of the $k\mathrm{th}$ branch, then $0= \Lambda _k(I-pP_e) = 1 + p\Lambda _k(-P_e) = 1- p\Lambda _{m-k+1}(P_e)$. $\square$

Here are some examples. Let $\lambda _1(A) \le \cdots \le \lambda _n(A)$ denote the (standard) ordered eigenvalues of A.

I. The Basic Elementary Symmetric Function Take $M =\sigma _k$, the $k\mathrm{th}$ elementary symmetric function of the $\lambda _j(A)$. In general it is hard to compute the M-eigenvalues $\Lambda _1(A)\le \cdots \Lambda _k(A)$ of A. However, when $A=P_e$ it is a straightforward computation that

$$\begin{aligned} \sigma _k(P_e+t I)\ =\ \sigma _k(I) t^{k-1}\left( t+{k\over n} \right) , \end{aligned}$$

so the $\sigma _k$-eigenvalues of $P_e$ are $0\ldots ,0,{k\over n}$, and hence the Riesz characteristic of the branches are $p_1 = {n\over k}$ and $p_2=\cdots = p_k=\infty$.

II. The Basic p-Convexity Operator Take $M=\Sigma _p$ and set $\alpha =p-[p]$. The cases $\alpha =0$ and $\alpha >0$ are different. If $\alpha =0$, then $m={n\atopwithdelims ()p}$ and $\Lambda _I(A) = {1\over p}\{\lambda _{i_1} + \cdots + \lambda _{i_p}\}$. Therefore, $A\equiv P_e$ has its smallest ${n-1\atopwithdelims ()p}$ eigenvalues $\Lambda _I(P_e)=0$, and the remaining ${n-1\atopwithdelims ()p-1}$ eigenvalues $\Lambda _I(P_e)={1\over p}$. Hence the Riesz characteristic of each of the first ${n-1\atopwithdelims ()p-1}$ branches is p, and for the remaining larger branches it is $\infty$.

When $\alpha >0$, the degree is $m= {n\atopwithdelims ()[p]} (n-[p])$. In the case $n=4$, $[p]=2$ and $0<\alpha <1$, there are 12 branches. The smallest 3 have characteristic $2+\alpha$, the next 3 have characteristic $1+{2\over \alpha }$, and the rest have characteristic $\infty$.

III. The Basic $\delta$-Uniformly Elliptic Operator Take $M= \mathrm{det}^\delta$. Thus, $M(A) = \mathrm{det}\left( A+{\delta \over n}(\mathrm{tr}A)I \right)$ so that $M(I) = (1+\delta )^n$ and $\Lambda _j(A) = {1\over 1+\delta }\{ \lambda _j(A) + {\delta \over n} \mathrm{tr}A\}$. Thus the first $n-1$ eigenvalues of $P_e$ are ${\delta \over n(1+\delta )}$, and the largest is $\Lambda _n(P_e) = {n+\delta \over n(1+\delta )}$. Therefore, ${n(1+\delta )\over n+\delta }$ is the Riesz characteristic of the smallest branch $\mathcal{P}(\delta )$, and all the remaining branches have Riesz characteristic $n(1+{1\over \delta })$.

IV. The Lagrangian Operator Take $M\equiv M_\mathrm{LAG}$ as defined by (6.11). Since (Definition 6.6) $M(sI+A) = M(I) \prod (s+\Lambda (A))$ define the $2^n$ Gårding eigenvalues, they can be calculated to be

$$\begin{aligned} \Lambda (A)\ =\ {1\over n} \left( \mu \pm \lambda _1 \pm \cdots \pm \lambda _n\right) , \end{aligned}$$

where $\pm \lambda _1,\ldots , \pm \lambda _n$ are the eigenvalues of the skew-hermitian part $A^\mathrm{skew}_\mathbf{C}$ of A, and $\mu \equiv \frac{1}{2}\mathrm{tr}A$. If $A=P_e$, then $\mu = \frac{1}{2}$, and $A^\mathrm{skew}_\mathbf{C}= \frac{1}{2}(P_e -P_{Je})$ has eigenvalues $\lambda _1 =\pm \frac{1}{2}$ and $\lambda _j=0$ for $j>1$. Hence the M-eigenvalues of $P_e$ are zero ($2^{n-1}$ times) and ${1\over n}$ ($2^{n-1}$ times). Therefore, the Riesz characteristic of the smallest $2^{n-1}$ branches is n, and of the largest $2^{n-1}$ branches, it is $\infty$.

1.6 Elliptic Regularization: Subequation Expansion/Contraction

For each $r>0$ consider the linear map

$$\begin{aligned} \Phi _r(A)\ \equiv \ rA + (1-r)(\mathrm{tr}\,A)\frac{1}{n} I \ =\ r\left( A - (\mathrm{tr}\,A)\frac{1}{n} I\right) + (\mathrm{tr}\,A)\frac{1}{n} I. \end{aligned}$$

(6.14)

The restriction of $\Phi _r$ to each affine hyperplane $\{\mathrm{tr}\,A=\lambda \}$ is the r-homothety (multiplication by r) about the center ${\lambda \over n}I$. This follows from the second equality. The inverse is

$$\begin{aligned} \Phi _{1\over r} \ =\ {1\over r} \left( A + (r-1) (\mathrm{tr}\,A)\frac{1}{n} I \right) . \end{aligned}$$

(6.15)

Definition 6.11

Suppose $\delta \equiv r-1\ge 0$ and F is a cone subequation. Then

$$\begin{aligned} F(\delta ) \ =\ \Phi _r(F) \ =\ \{A : r\Phi _{1\over r} (A) = A+ \delta (\mathrm{tr}\,A)\frac{1}{n} I\in F\} \end{aligned}$$

is called the rth expansion of F.

Note that $F(\delta )$ is also a subequation for all $\delta >0$ since the homothety factor $r\ge 1$. Note also that if F is a convex cone contained in $\mathrm{Int}\Delta = \{\mathrm{tr}\,A> 0\}$, then $F(\delta )$ ranges from F to $\Delta$ as $\delta$ ranges from 0 to $\infty$. Finally, note that $\partial F(\delta ) = \Phi _r(\partial F)$.

Proposition 6.12

Suppose F is a cone subequation with (Riesz) characteristic $p=p_F$ and dual (Riesz) characteristic $q=q_F$. Then the $\delta$-uniformly elliptic cone subequation $F(\delta ) = \Phi _r(F)$ ($\delta \equiv r-1\ge 0$) has its two characteristics given by the same function

$$\begin{aligned} p_{F(\delta )}\&= \ {n(1+\delta )p \over n+\delta p} \ =\ p + {\delta p (n-p) \over n + \delta p} \nonumber \\ q_{F(\delta )}\&= \ {n(1+\delta )q \over n+\delta q} \ =\ q + {\delta q (n-q) \over n + \delta q}. \end{aligned}$$

(6.16)

These formulas hold when $p_F=\infty$ or when $q_F=\infty$, that is

$$\begin{aligned} p_{F}\&=\ \infty \quad \Rightarrow \quad p_{F(\delta )} = {n(1+\delta ) \over \delta } \nonumber \\ q_{F}\&=\ \infty \quad \Rightarrow \quad q_{F(\delta )} = {n(1+\delta ) \over \delta }. \end{aligned}$$

(6.17)

Proof

Note that $A\equiv P_{e^\perp } -(p-1) P_e \in \partial F \iff \Phi _r(A) \in \partial \Phi _r(F) = \partial F(\delta )$ and

$$\begin{aligned} \Phi _r(A)= & {} (1+\delta ) P_{e^\perp } - (1+\delta ) (p-1) P_e - {\delta (n-p)\over n} I \\= & {} {n+\delta p \over n} \left[ P_{e^\perp } - \left( {n(1+\delta )p \over n+\delta p} -1 \right) P_e \right] . \end{aligned}$$

Finally, since $-A\in \partial F \iff -\Phi _r(A) \in \partial \Phi _r(F)= \partial F(\delta )$, the formula for $q_{F(\delta )}$ as a function of $q_F$ is the same as the formula for $p_{F(\delta )}$ as a function of $p_F$. $\square$

Proposition 6.13

If F is M-monotone, then $F(\delta )$ is $M(\delta )$-monotone.

Proof

Straightforward.

Example 6.14

As $\delta$ ranges from 0 to $\infty$, $\mathcal{P}(\delta )$ increases from $\mathcal{P}$ to $\Delta$. Each $\mathcal{P}(\delta )$ is a convex cone, and with $\delta >0$ small, these subequations form a “fundamental system” of conical neighborhoods of $\mathcal{P}$. Consequently, they provide one of the nicer definitions of uniform ellipticity. Namely, a subequation F is $\delta$-uniformly elliptic if

$$\begin{aligned} F+\mathcal{P}(\delta ) \ \subset \ F. \end{aligned}$$

(6.18)

Since $F(\delta ) + \mathcal{P}(\delta ) \subset F(\delta )$, each $F(\delta )$ is automatically $\delta$-uniformly elliptic. For this reason, $F(\delta )$ is also called the $\delta$-elliptic regularization of F (cf. [17]).