Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 111–115

A NOTE ON THE CONVEXITY THEOREM FOR MEAN VALUES OF SUBTEMPERATURES

Miroslav Brzezina

Technical University of Liberec, Department of Mathematics Hálkova 6, 461 17 Liberec 1, Czech Republic; miroslav.brzezina@vslib.cz

Abstract. In this note we prove the finiteness of mean values of subtemperatures over level surfaces of the Green function and we give a capacitary interpretation of the corresponding “mean value measure”.

1. Introduction

In a series of papers, N.A. Watson has extended the classical result of F. Riesz that the integral mean of subharmonic functions in R^{d}, d ≥ 3, over concentric spheres of radius r is a convex function of r^{2−d}, for subsolutions of both second order, linear elliptic partial differential equations with variable coefficients ([8], [10]) and the heat equation ([6], [7], [9], [10]).

In the case of potential theory for the heat operator, we show that the function φ in Theorem 4 of N.A. Watson’s paper [10, p. 253], is actually finite. To prove this, we use some ideas of [6, p. 249], and the fact that the corresponding mean value measure μ_{p_{0},c}^{D} on the heat sphere ∂Ω_{D}(p_{0}, c) is the balayage of the Dirac measure ε_{p_{0}} concentrated at p_{0} onto the complement of the heat ball Ω_{D}(p_{0}, c) as well as the capacitary measure (for the adjoint operator) for \overline{Ω_{D}(p_{0}, c)}, see Theorem 2 below.

Following N.A. Watson, [10, p. 252], we give some definitions and notations. Let Θ denote the heat operator ∑_{i=1}^{d} ∂^{2}/∂x_{i}^{2} −∂/∂t in R^{d+1} , d ∈ N, and let Θ^{∗} denote its adjoint ∑_{i=1}^{d} ∂^{2}/∂x_{i}^{2} + ∂/∂t. In what follows, all notions concerning the adjoint heat operator will be denoted by ^{∗}, e.g., ^{∗} regular, ^{∗} heat potential etc. We put

∇_{x}u := (\frac{∂}{∂x_{1}} u, …, \frac{∂}{∂x_{d}} u), ∇u := (∇_{x}u, \frac{∂}{∂t}u),

and use ∥·∥ to denote the Euclidean norm in both R^{d} and R^{d+1} . A temperature is a solution of the heat equation, and subtemperatures and supertemperatures are corresponding subsolutions and supersolution, cf. [4].

1991 Mathematics Subject Classification: Primary 31B35, 31C15, 35K05.

Financial support by the Konferenz der deutschen Akademien der Wissenschaften is gratefully acknowledged.

Let D be an open subset of R^{d+1} which is ^{∗} Dirichlet regular, i.e., if ∂D ≠ ∅ then for every continuous function f: ∂D → R there exists exactly one function ∗H_{f}^{D}: \overline{D} → R such that

Θ^{∗}(^{∗}H_{f}^{D}) = 0 on D, ^{∗}H_{f}^{D} |_{∂D}= f

and ^{∗}H_{f}^{D} ≥ 0 whenever f ≥ 0.

Further, let G_{D} denote the Green function for D in the sense of [4], cf. [3]. If p_{0} ∈ D, then G_{D}(p_{0}, ·) is a non-negative ^{∗} supertemperature on D, and a ∗ temperature on D \ {p_{0}}. It follows from Sard’s theorem that the set

(1)  {p ∈ D; G_{D}(p_{0}, p) = (4πc)^{−d/2}}

is a smooth regular d-dimensional manifold for almost every c > 0. We call such a value of c a regular value. For an arbitrary positive value of c, we put

Ω_{D}(p_{0}, c) := {p ∈ D; G_{D}(p_{0}, p) > (4πc)^{−d/2}}.

For any regular value of c, the union of {p_{0}} and the set in (1) is equal to ∂Ω_{D}(p_{0}, c). The set \overline{Ω}_{D}(p_{0}, c) is a compact subset of D; see [10, p. 252]. The set Ω_{D}(p_{0}, c) is called the heat ball and ∂Ω_{D}(p_{0}, c) the heat sphere (with respect to D). If c is a regular value, we put for p ∈ ∂Ω_{D}(p_{0}, c) \ {p_{0}}

K_{D}(p_{0}, p) := ∥∇_{x}G_{D}(p_{0}, p)∥^{2}∥∇G_{D}(p_{0}, p)∥^{−1}

and

ℳ_{D}(u, p_{0}, c) := ∫_{∂Ω_{D}(p_{0},c)} uK_{D}(p_{0}, ·) dσ

whenever the integral exists. Here σ denotes the surface area measure on ∂Ω_{D}(p_{0}, c). The measure μ_{p_{0},c}^{D} := K_{D}(p_{0}, ·)σ will be called the mean value measure. If D = R^{d+1} , the measure μ_{p_{0},c}^{D} is called the Fulks–Pini measure; see, e.g.,

[1] for details.

If c_{1} , c_{2} are regular values, c_{1} < c_{2} , we put

A_{D}(p_{0}, c_{1}, c_{2}) := Ω_{D}(p_{0}, c_{2}) \ \overline{Ω_{D}(p_{0}, c_{1})}.

The following result is due to N.A. Watson, [9, p. 176]; for a simpler proof, see [10, p. 253].

Theorem 1. Let u be a subtemperature on an open superset E of \overline{A_{D}(p_{0}, c_{1}, c_{2})}. Then there is a function φ, either finite and convex or identically −∞, such that

ℳ_{D}(u, p_{0}, c) = φ(c^{−d/2})

for all regular values c in [c_{1}, c_{2}].

A note on the convexity theorem for mean values of subtemperatures 113

In the case D = R^{d+1} , the finiteness of ℳ_{D}(u, p_{0}, c) was proved in [6, p. 249]. We will show that this also holds when D ≠ R^{d+1} .

First we give some preliminary results. The ^{∗} heat potential on D (adjoint heat potential) of a positive Radon measure μ on D is defined by

∗G_{D}^{μ}(p) := ∫_{D} G_{D}(q, p) μ(dq).

From [3, p. 348], it follows that for every compact set K ⊂ D there exists a uniquely determined Radon measure ^{∗}μ_{D,K} (called the ^{∗} equilibrium measure for K with respect to D) such that

∗G_{D}^{^{∗}μ_{D,K}} = ^{∗}R̂_{1}^{K};

here, of course, ^{∗}R̂_{1}^{K} denotes the balayage of 1 on K with respect to D and the adjoint heat theory. The number ^{∗}μ_{D,K}(K) is called the ^{∗} capacity of K (with respect to D).

If U ⊂ R^{d+1} is a set and p_{0} ∈ R^{d+1} , then ε_{p_{0}}^{U} stands for the balayage of the Dirac measure ε_{p_{0}} concentrated at p_{0} on U.

For c > 0, we consider the function

w(p) := min{(4πc)^{−d/2} ; G_{D}(p_{0}, p)}, p ∈ D.

Lemma 1. Let p_{0} ∈ D and c be a regular value. Put Ω := Ω_{D}(p_{0}, c) and ν := ε_{p_{0}}^{∁Ω} . Then

∗G_{D}^{ν} = w.

Lemma 2. Let p_{0} ∈ D and c be a regular value. Put Ω := Ω_{D}(p_{0}, c). Then

∗G_{D}^{^{∗}μ_{D,\overline{Ω}}} = (4πc)^{−d/2}w.

We omit the proofs of Lemma 1 and 2, since they differ from those of [2, pp. 472, 473] only in minor details.

Theorem 2. Let p_{0} ∈ D and c be a regular value. Put Ω := Ω_{D}(p_{0}, c). Then

μ_{p_{0},c}^{D} = ε_{p_{0}}^{∁Ω} = (4πc)^{−d/2} ^{∗}μ_{D,\overline{Ω}}.

Proof. The first equality was proved in [9, p. 181]. The second one follows from the uniqueness of the representation of ^{∗} heat potentials, see [3, p. 305], and from Lemma 1 and Lemma 2.

In the case D = R^{d+1} , the first equality was established by H. Bauer in [1], the second one in [2].

Corollary. Let p_{0}, p_{1} ∈ D and c be a regular value. Then

ℳ_{D}(G_{D}(·, p_{1}), p_{0}, c) = min{(4πc)^{−d/2}, G_{D}(p_{0}, p_{1})}.

Proof. By definitions we have

ℳ_{D}(G_{D}(·, p_{1}), p_{0}, c) = ∫_{D} G_{D}(·, p_{1}) dμ_{p_{0},c}^{D} = ^{∗}G_{D}^{μ_{p_{0},c}^{D}}(p_{1}).

According to Lemma 1 and Theorem 2

ℳ_{D}(G_{D}(·, p_{1}), p_{0}, c) = w(p_{1}) = min{(4πc)^{−d/2}, G_{D}(p_{0}, p_{1})}.

In the case D = R^{d+1} , the assertion of the Corollary was proved by N.A. Watson in [6, p. 248].

Now, we are in position to prove the finiteness of the mean values ℳ_{D}(u, p_{0}, c) in Theorem 1.

Theorem 1′. Let u be a subtemperature on an open superset E of \overline{A_{D}(p_{0}, c_{1}, c_{2})}. Then there is a real-valued convex function φ such that

ℳ_{D}(u, p_{0}, c) = φ(c^{−d/2})

for all regular values c in [c_{1}, c_{2}].

Proof. Let c ∈ [c_{1}, c_{2}] be a regular value. By Theorem 1, it is sufficient to prove that ℳ_{D}(u, p_{0}, c) is finite. Let V be an open superset of \overline{A_{D}(p_{0}, c_{1}, c_{2})}, \overline{V} ⊂ E ∩ D, \overline{V} compact. By the Riesz decomposition theorem, see [4, p. 279], there exists a finite measure μ and a temperature h on V such that u = h − G_{V}^{μ} on V . According to [9, p. 168], ℳ_{D}(h, p_{0}, c) = h(p_{0}); obviously, h(p_{0}) is finite. Since, for all p, q ∈ V ,

0 ≤ G_{V} (p, q) ≤ G_{D}(p, q)

we have

∫_{V} G_{V} (p, q) μ(dq) ≤ ∫_{V} G_{D}(p, q) μ(dq).

By Fubini’s theorem and Corollary, we have

ℳ_{D}(G_{V}^{μ} , p_{0}, c) ≤ ∫_{V} ℳ_{D}(G_{D}(·, q), p_{0}, c) μ(dq) ≤ (4πc)^{−d/2} · μ(V) < ∞.

A note on the convexity theorem for mean values of subtemperatures 115

References

[1] Bauer, H.: Heat balls and Fulks measures. - Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 1985, 62–82.

[2] Brzezina, M.: Capacitary interpretation of the Fulks measure. - Exposition. Math. 11, 1993, 469–474.

[3] Doob, J.L.: Classical potential theory and its probabilistic counterpart. - Springer-Verlag, Berlin, 1984.

[4] Watson, N.A.: Green functions, potentials, and the Dirichlet problem for the heat equation. - Proc. London Math. Soc. (3) 33, 1976, 251–298.

[5] Watson, N.A.: Thermal capacity. - Proc. London Math. Soc. (3) 37, 1978, 342–362.

[6] Watson, N.A.: A convexity theorem for local mean value of subtemperatures. - Bull. London Math. Soc. 22, 1990, 245–252.

[7] Watson, N.A.: Mean values and thermic majorization of subtemperatures. - Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 1991, 113–124.

[8] Watson, N.A.: Mean values of subharmonic functions over Green spheres. - Math. Scand. 69, 1991, 307–319.

[9] Watson, N.A.: Mean values of subtemperatures over level surfaces of Green functions. - Ark. Mat. 30, 1992, 165–185.

[10] Watson, N.A.: Generalization of the spherical mean convexity theorem on subharmonic functions. - Ann. Acad. Sci. Fenn. Ser. A I Math. 17, 1992, 241–255.

Received 4 August 1994