The AE Theorem and addition theorems for quasi-convex functions

Theorem 6.4 and its generalization Theorem 6.8. A stronger form of the addition theorem is proved in §7. 2. Preliminaries. There are several (equivalent) ways of defining subsolutions. For the purposes of this paper the most convenient approach is as follows. Let Sym2(Rn) denote the set of symmetric n × n matrices. Definition 2.1. Given a real-valued function w defined on an open subset X ⊂Rn, a point x ∈X is an upper contact point for w if there exists (p, A) ∈Rn × Sym2(Rn) such that w(y) ≤w(x) +

Lemma 2.3. Suppose w is twice differentiable at x. If (p, A) is an upper contact jet for w at x, then p = Dxw is unique and A = D2 xw + P for some P ≥0. Conversely, for each P > 0 (Dxw, D2 xw + P) is an upper contact jet for w at x. Adding a smooth function ψ to w does not change the set of upper contact points but only the upper contact jets.

Lemma 2.4. Suppose ψ is smooth. Then x is an u. c. point for w ⇐⇒ x is an u. c. point for w + ψ (p, A) is an u. c. jet for w at x ⇐⇒ (p+Dxψ, A+D2 xψ) is an u. c. jet for w+ψ at x A third elementary fact needed here concerns convex functions.

Lemma 2.5. If w is convex and twice differentiable at x, then D2 xw ≥0. Example 2.6. Each function w ∈USC(X) determines a smallest primitive subequation, denoted J+(w), with the property that w is subharmonic. Namely, let J+(w) denote the set of tuples (x, w(x), p, A) such that (p, A) is an upper contact jet for w at x, and then take the closure J+(w) in X×R×Rn×Sym2(Rn). Obviously J+(w) satisfies condition (P), and hence the closure also satisfies (P). This example J+(w) is in some sense pathologic

Lemma 3.1. (D at UCP). Suppose w is quasi-convex. If x0 is an upper contact point for w, then w is differentiable at x0. Moreover, if (p, A) is any upper contact jet for w at x0, then p = Dx0w is unique. Another even more standard result is called partial continuity of the gradient, or first derivative.

Lemma 3.2. (PC of FD). Suppose w is quasi-convex and xj →x0. If w is differentiable at each xj and at x0, then Dxjw →Dx0w. Second Derivatives The results concerning the second-order part of upper contact jets of quasi-convex functions are of a deeper nature. The almost everywhere existence of the first derivative was omitted from the previous discussion because of the following stronger result.

THEOREM 3.3. (Alexandrov). A locally quasi-convex function is twice differentiable almost everywhere. For the next result we need two variations of the notion of an upper contact jet. First, we say that (p, A) is a strict upper contact jet for w ∈USC(X) at x0 ∈X if the upper contact inequality (2.1) is strict for y ̸= x0. An understanding of the strict upper contact jets will be adequate for our discussion since (p, A) is an upper contact jet if and only if (p, A + ǫI) is a strict upper contact j

THEOREM 3.6. (Jensen-Slodkowski). Suppose that w is a quasi-convex function possessing a strict upper contact jet (p0, A0) at x0. Let Bρ denote the ball of radius ρ about x0. Then there exists ¯ρ > 0 such that the measure |C(w, Bρ, A0)| > 0 ∀0 < ρ ≤¯ρ. (3.2) The four results above yield the following theorem which is easy to apply and adequate for many purposes. It is concerned with the upper contact jets of a quasi-convex function. The order two part of this theorem can be considered a “partial

THEOREM 3.7. (Upper Contact Jets). Suppose w is quasi-convex with an upper contact jet (p0, A0) at a point x0. Then (D at UCP) w is differentiable at x0 and Dx0w = p. Suppose E is a set of full measure in a neighborhood of x0. Then there exists a sequence {xj} ⊂E with xj →x0 such that w is twice differentiable at each xj and (PC of FD) Dxjw →Dx0w = p0, (PUSC of SD) D2 xjw →A ≤A0.

Theorem 3.7 can be stated succinctly in terms of the subset J+(w) (see Example 2.6) representing the upper contact jets of w, and another subset depending on E Define the subset J(w, E) of the 2-jet bundle J2(X) to be the set of tuples (x, w(x), Dxw, D2 xw + P) such that x ∈E, w is twice differentiable at x, and P ≥0. Then Theorem 3.7 condenses to: If w is quasi−convex and E has full measure, then J+(w) ⊂J(w, E). (3.4) 5

THEOREM 4.1. Suppose that F is a primitive subequation on a manifold X and u : X →R is a locally quasi-convex function. Then J2 xu ∈Fx for almost all x ∈X ⇒ u is F subharmonic on X. (4.1)

THEOREM 5.1. Suppose that F and G are primitive subequations on a manifold X. Then H ≡F + G is also a primitive subequation. Moreover, if u and v are quasi-convex functions on X, then u ∈F(X) and v ∈G(X) ⇒ u + v ∈H(X). Moreover, If F and G are constant coefficient (translation invariant) on an open subset X ⊂Rn, then the assumption that u and v are quasi-convex can be dropped.

Lemma 6.1. ([HL1]). w ∈eP(X) ⇐⇒ w is a subaffine function on X. Now putting these results together we have two theorems.

THEOREM 6.2. (The Subaffine Theorem). If u ∈F(X) and v ∈eF(X), then the sum w = u + v is a subaffine function on X.

Lemma 6.3. Let Ω⊂Rn be any domain, and set ∂Ω≡Ω−Ω. Then w ≤a on ∂Ω ⇒ w ≤a on Ω (6.4) for all functions w ∈USC(Ω) which are subaffine on Ω.

THEOREM 6.4. (Comparison). Suppose Ω⊂Rn is a domain, and u, v ∈USC(Ω). If u ∈F(Ω) and v ∈eF(Ω), then u + v ≤0 on ∂Ω ⇒ u + v ≤0 on Ω.

Lemma 6.6. One has w ∈( g R−× P)(X) ⇐⇒ w ∈USC(X) and w ≤a+ on ∂K ⇒ w ≤a+ on K ∀Kcpt ⊂X and ∀a+ ∈Aff+ Such functions will be referred to as subaffine plus functions on X. Here as before, if Ω⊂Rn is any domain and w is any upper semi-continuous function on Ωwhich is subaffine plus on Ω, then w ≤a+

THEOREM 6.7 (The Subaffine Plus Theorem). If u ∈F(X) and v ∈eF(X), then the sum w ≡u + v is subaffine plus on X.

THEOREM 6.8. (Gradient-Free Comparison). Suppose Ω⊂Rn is a domain, and u, v ∈USC(Ω). If u ∈F(Ω) and v ∈eF(Ω), then u + v ≤0 on ∂Ω ⇒ u + v ≤0 on Ω.

THEOREM 7.1. Suppose u and v are quasi-convex and that the sum w ≡u + v has an upper contact jet (p0, A0) at x0 ∈Rn. Then x0 is an upper contact point for both u and v. Furthermore: First Derivatives (D at UCP). Both u and v are differentiable at x0, and the upper contact jets are all of the form (Dx0u, −) and (Dx0w, −) respectively. (DPC at FD). For any sequence xj →x0 with both u and v differentiable at each xj, Dxju →Dx0u and Dxjv →Dx0v Second Derivatives (PUSC of SD). For each set E of full me

THEOREM 8.1. (Strict Comparison). Suppose G and F are subequations on a manifold X with G ⊂IntF. Then for any domain Ω⊂⊂X, if u, v ∈USC(Ω) are locally quasi-convex on Ωwith u ∈G(Ω) and v ∈eF(Ω), comparison holds, i.e., u + v ≤0 on ∂Ω ⇒ u + v ≤0 on Ω (ZMP) Moreover, if F and G are constant coefficient on Rn, then the assumption that u and v are quasi-convex can be dropped.