SOLUTIONS OF THE MONGE-AMP` ERE EQUATION
CORNELIU UDREA
We study compactness and uniform convergence in the class of solutions to the Dirichlet problem for the Monge-Amp`ere equation, using some equiconti- nuity tools.
AMS 2000 Subject Classification: 26B25, 46B50.
Key words: equicontinuity, relative compactness, Dirichlet problem, Monge- Amp`ere equation.
1. PRELIMINARIES
In this paper (E,k·k) is a real normed space. IfAis a nonempty subset of E we denote byC(A) (respectivelyU(A)) the set of all real valued continuous functions on A (respectively convex on the set A if A is convex). By D we denote an open nonempty subset of a given topological space.
We recall now the basic notions and results from [1], [2], [6], [11] (see also [13]) which will be used further. Some of these are presented with a sketch of the proof.
Lemma1.1. Suppose thatDis convex,H⊂ U(D)anda∈Dare such that H(a) :={h(a) :h∈H} is bounded. The following assertions are equivalent.
(i) H(x) is upper bounded for allx∈D.
(ii)H(x) is bounded for all x∈D.
Proof. (i) ⇒ (ii) Letm:= infH(a),r∈(1,∞) such thatz:=x+r(a− x)∈D, wherex∈D, andM an upper bound ofH(z). For allh∈Hwe have:
m≤h(a)≤ 1 rh(z) +
1−1
r
h(x)≤ 1 rM+
1− 1
r
h(x)⇒
h(x)≥ rm−M r−1 .
From now on, in this section,E =Rk, k∈N?,D is strictly convex and λis the Lebesgue measure on Rk. Moreover,M+(D) (respectively bM+(D))
REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 585–593
is the set of positive (respectively positive and bounded) Radon measures on D, and L∞(D) denotes the space of λ-essentially bounded functions on D.
Theorem1.2 ([6]). (i) For allu∈ U(D) there existsνu ∈ M+(D) such that if u is twice continuously differentiable and K ⊂D is compact, then
νu(K) = Z
K
det(D2u)dλ.
(ii)For all u, v∈ U(D) and α∈R?+ we have (a) νu+v ≥νu+νv, (b)να·u =αk·νu.
(iii) If (un)n∈N ⊂ U(D) and (un)n∈N? → u0 uniformly on the compact subsets of D, then (νun)n∈N?→(νu0) vaguely on U.
Remark 1.3. (i) For allu∈ U(D),νu is calledthe curvature measureofu.
(ii) If µ ∈ bM+(D), ϕ ∈ C(∂D) and u ∈ U(D) are defined as νu = µ and u|∂D = ϕ, then u is called the solution to the Dirichlet problem for the Monge-Amp`ere equation (the solution to the Dirichlet problem for short).
Proposition1.4 ([6]). For allu, v∈ U(D), we have the assertion below.
(i) (The minimum principle). Ifνu≤νvand for all a∈∂D,lim inf
D3x→au(x)
≥lim inf
D3x→av(x), then u≥v.
(ii) (The boundedness principle). Suppose m ∈ R is such that: for all a∈∂D, m≤lim inf
D3x→au(x). Then u≥m−(diamD)k
s νu(D)
ωk
, where ωk:=λ(B(0k,1).
Theorem1.5. Existence of the solution to the Dirichlet problem,[6]and [11]. For all µ∈bM+(D) and ϕ∈C(∂D)there is one and only one function M(µ;ϕ)∈ U(D)∩C(D) such that νM(µ;ϕ)=µ and (M(µ;ϕ))|∂D=ϕ.
Proposition 1.6 ([6]). Let µ1, µ2 ∈ bM+(D) and ϕ1, ϕ2 ∈ C(∂D).
Then
(i) M(µ1+µ2;ϕ1+ϕ2)≥M(µ1;ϕ1) +M(µ2;ϕ2);
(ii)if µ1 ≤µ2 and ϕ1 ≥ϕ2, it follows that M(µ1;ϕ1)≥M(µ2;ϕ2);
(iii) infϕ1−(diamD)k qµ1(D)
ωk ≤M(µ1, ϕ1)≤supϕ1.
Corollary 1.7 ([13]). Let µ, µ1, µ2 ∈bM+(D) andϕ, ϕ1, ϕ2 ∈C(∂D).
Then
(i) |M(µ1;ϕ1)−M(µ2;ϕ2)| ≤ −M(|µ1−µ2|;− |ϕ1−ϕ2|);
(ii)|M(µ1;ϕ)−M(µ2;ϕ)| ≤ −M(|µ1−µ2|; 0);
(iii) |M(µ;ϕ1)−M(µ;ϕ2)| ≤ −M(0;− |ϕ1−ϕ2|).
Proof. (i) According to Proposition 1.6,
M(µ1;ϕ1) +M(|µ1−µ2|;− |ϕ1−ϕ2|)
≤M(µ1+|µ1−µ2|;ϕ1− |ϕ1−ϕ2|)≤M(µ2;ϕ2).
Hence
M(µ1;ϕ1)−M(µ2;ϕ2)≤ −M(|µ1−µ2|;− |ϕ1−ϕ2|).
The proof of (i) ends by noticing the symmetric role of the subscripts 1 and 2. Inequalities (ii) and (iii) are special cases of (i).
2. SOME REMARKS ON EQUICONTINUITY
In this section we shall present two criteria: the first one about uniform equicontinuity, and the second one about equicontinuity. Furthermore, we shall recall some results ([1], [2], [8]) concerning the equicontinuity of some families of convex function.
From now on, for all sets A, B and H ⊂BA, and for all a∈A we shall denote by H(a) the set{h(a) :h∈H}.
Proposition 2.1. Assume that X and L are Hausdorff uniform spaces, Y is a dense set in X, and H⊂C(X;L) is uniformly equicontinuous on Y.
Then
(i) H is uniformly equicontinuous on X;
(ii) if H(y) is a precompact (respectively a relatively compact subset) of L for all y ∈ Y, then H is a precompact (respectively a relatively compact) subset of C(X;L) with respect to the topology of the precompact convergence.
Proof. (i) Consider symmetric entourages V, V1 in L such that V1 ◦ V1 ◦V1 ⊂ V and symmetric entourages U, U1 in X such that U ◦U ◦U ⊂ U1 and (h(x), h(y))∈V1 for all (x, y)∈U1∩(Y ×Y) andh∈H.
If (a, b) ∈U ∩(X×X) and h ∈H, then by the continuity of h we can choose a symmetric entourage Uh in X such that
Uh⊂U, (x, a)∈Uh, (y, b)∈Uh for all (x, y)∈Y ×Y ⇒ (h(x), h(a))∈V1, (h(y), h(b))∈V1.
Take ya, yb ∈ Y, (ya, a) ∈ Uh,(yb, b) ∈ Uh. It follows that (ya, yb)∈ U ◦U◦U ⊂U1, and
(h(ya), h(a))∈V1, (h(yb), h(b))∈V1, (h(ya), h(yb))∈V1 ⇒ (h(a), h(b))∈V1◦V1◦V1⊂V.
HenceH is uniformly equicontinuous onX.
(ii) On account of (i), the following topologies coincide onH: the point- wise convergence on Y, the pointwise convergence on X, and the precompact
convergence onX. By the hypothesis,His precompact (respectively relatively compact) in C(X;L) with respect to the pointwise convergence, hence it is precompact (respectively relatively compact) with respect to the precompact convergence on X.
Remark 2.2. (i) In the framework of Proposition 2.1, if we suppose thatX is precompact, thenHis a precompact (respectively relatively compact) subset of C(X;L) with respect to the topology of the uniform convergence onX.
(ii) We shall use Proposition 2.1 in the case of on open subset D of X and H ⊂C(D;L) uniformly equicontinuous onD.
Proposition 2.3. Suppose that X is a Hausdorff topological space, Y a nonempty subset ofX,La topological vector lattice andH⊂C(X;L)such that (2.1) ∃u, v∈C(X;L), u|X\Y = 0L with|h−v| ≤ |u| ∀h∈H.
Then
(i) H is equicontinuous onX\Y;
(ii) if H is equicontinuous on Y and H(y) is a precompact (respectively a relatively compact) subset of L for all y ∈Y, then H is a precompact (res- pectively a relatively compact) subset of C(X;L) with respect to the topology of the compact convergence on X.
Proof. (i) For all a∈X\Y,x∈X andh∈H we have
|h(x)−h(a)| ≤ |h(x)−v(x)|+|v(x)−v(a)|+|v(a)−h(a)| ≤ |u(x)|+|v(x)−v(a)|.
By the continuity of u, vand of the absolute value function onL, the equicon- tinuity of the familyH at the point afollows.
(iii) We use Ascoli’s theorem and the fact that H(a) ={ν(a)} for all a∈X\Y.
Remark 2.4. (i) Assume that under the assumptions of Proposition 2.1 the family H has the property that
(2.2) ∃u∈C(X;L), u|X\Y = 0L such that|h1−h2| ≤ |u| ∀h1, h2 ∈H.
It follows that H also has the properties stated in the conclusion of Proposition 2.1.
(ii) The hypothesis of Proposition 2.3 (ii) is satisfied when H is a pre- compact (respectively a relatively compact) subset of C(Y, L) with respect to the topology of the compact convergence onY.
(iii) We shall apply the result of Proposition 2.3 in the following case:
D⊂X is a nonempty open subset,H ⊂C(D;L) and either:
(2.3) ∃u, v∈C(D, L), u|∂D= 0Lsuch that |h−v| ≤ |u| ∀h∈H,
or
(2.4) ∃u∈(D;L), u|∂D = 0L such that|h1−h2| ≤ |u| ∀h1, h2∈H.
Corollary2.5. Given a compact spaceX andu, v, H as in Proposition 2.3 (respectivelyu, H as in Remark2.4 (i)), then
(i) H is equicontinuous onX\Y;
(ii)ifH is equicontinuous onY andL=R, thenH is relatively compact with respect to the topology of the uniform convergence on X.
Theorem 2.6 ([1] and [2]). Given an open convex subset D of E and H ⊂ U(D) such thatH(x) is bounded for all x∈D, then for everya∈D the following assertions are equivalent:
(i) H is equicontinuous ata;
(ii)H is upper bounded on a neighbourhood of a.
Proof (sketch). (ii) ⇒ (i) Let r, M ∈ (0,∞) such that B(a, r) ⊂D and h(x) ≤ M, |h(a)| ≤ M for all x ∈ B(a, r), h ∈ H. If ε ∈ (0,∞) choose α ∈(1,∞) such that 2M/α≤ε.
For all y∈B(0E, r/α), since (t7→ (u(a+th)−u(a−th))/t) : [−1,0)∪ (0,1]→R is increasing ∀a, e∈E,∀u∈ U([a−e, a+e]), we have
−2M ≤ h(a−αy)−h(a)
−1 ≤ h(a+y)−h(a)
1 α
≤ h(a+αy)−h(a)
1 ≤2M
⇒ |h(a+y)−h(a)| ≤2M α≤ε.
Corollary 2.7. For any family H as in Proposition 2.3 the following assertions are equivalent:
(i) H is equicontinuous onD;
(ii)H is upper bounded on an open nonempty subset of D.
Proof (sketch). (ii) ⇒ (i) Let V ⊂ D be an open nonempty set, and M ∈ (0,∞) such that h(x) ≤ M for all x ∈ V, h ∈ H. For all a ∈ D\V, chooseb∈V,r∈(1,∞) such thatz:=b+r(a−b)∈D, andM0∈(0,∞) such that H(z)⊂(−∞, M0]. Since (1−1r)V +1rz ∈ V(a) and y:= (1−1r)x+1rz, for allx∈V, for allh∈H we have h(y)≤(1−1r)h(x) +1rh(z)≤(1−1r)M+
1
rM0, i.e., H is upper bounded on a neighbourhood of a and we can apply Theorem 2.6.
Proposition 2.8. Let H⊂ U(D), whereD, is an open convex subset of E,V an open nonempty subset ofDandr∈(0,∞)such thatV+B(0E, r)⊂D and H is bounded on V +B(0E, r). Then H is an equi-Lipschitz family onV (In particular H is uniformly equicontinuous on V).
Proof. LetM ∈(0,∞) such that{|h(x)|:h∈Handx∈V+B(0E, r)} ⊂ [0, M]. For all x, y ∈ V, x 6=y, put z := y+ ky−xkr (y−x) and remark that
z∈V +B(0E, r) and y= (1−t)x+tz, wheret:= r+ky−xkky−xk . For everyh∈H we then have
h(y)≤(1−t)h(x) +th(z)⇔h(y)−h(x)≤t(h(z)−h(x)).
Therefore,
h(y)−h(x)≤ ky−xk
r+ky−xk2M ≤ 2M
r ky−xk.
Corollary 2.9. Given a nonempty open convex subset D of Rk and H ⊂ U(D) such that H(x) is bounded for all x∈D,
(i) H is equicontinuous onD;
(ii) for any compact subset K of D: (a) H is bounded on K; (b) H is an equi-Lipschitz family on K (in particular H is uniformly equicontinuous on K).
Proposition 2.10. If H ⊂ U(D)∩C(D) (D an open convex subset of E) satisfies condition (2.1), then
(i) H is equicontinuous onD.
(ii) H is relatively compact in C(D) with respect to the topology of the compact convergence.
(iii) Suppose D is relatively compact. Then H is relatively compact in C(D) with respect to the topology of the uniform convergence.
Proof. Apply Theorem 2.6 and Proposition 2.3.
3. FAMILIES OF SOLUTIONS TO THE DIRICHLET PROBLEM
In this sectionDis a strictly convex and bounded open subset ofRk. We shall consider the solutions to the Dirichlet problem for the Monge-Amp`ere equation.
Theorem 3.1. Let M ⊂ bM+(D) and µ0 ∈bM+(D) be such that for every µ ∈ M, µ≤ µ0. For any ϕ ∈C(∂D), the set {M(µ :ϕ) :µ ∈ M} is relatively compact in the topology of the uniform convergence on the set D.
Proof. By Corollary 1.8, for allµ1, µ2 ∈ M we have
|M(µ1;ϕ)−M(µ2;ϕ)| ≤ −M(|µ1−µ2|; 0)≤ −M(2µ0; 0).
Hence, by Corollary 2.5, the set {M(µ;ϕ) : µ∈ M} is relatively compact in the topology of the uniform convergence on the set D.
Theorem 3.2. If F ⊂ C(∂D) is a bounded set in (C(∂D),k · k∞) and µ ∈ bM+(D), then {M(µ;ϕ) : µ ∈ F } is relatively compact in C(D) with respect to the topology of compact convergence.
Proof. Letc∈R+ such that|ϕ| ≤cfor all ϕ∈ F. We have M(µ;−c)≤ M(µ;ϕ)≤M(µ;c) and by Corollary 2.9 and Ascoli’s theorem the set{M(µ: ϕ) : µ ∈ F } is relatively compact in C(D) with respect to the topology of compact convergence.
Proposition 3.3. Let F ⊂C(∂D) be such that |ϕ| ≤ϕ0 for all ϕ∈ F, where ϕ0 ∈C(∂D), and let M ⊂bM+(D) be such thatµ≤µ0 for all µ∈M, where µ0 ∈ bM+(D). The set {M(µ;ϕ) : µ ∈ M and ϕ ∈ F } is relatively compact in C(D) with respect to the topology of the compact convergence.
Proof. By Proposition 1.4 (ii), for all µ∈ M and ϕ∈ F we have
−supϕ0−(diamD)k s
µ0(D) ωk
≤M(µ0;−ϕ0)≤M(µ;ϕ)≤M(µ;ϕ0)≤supϕ0.
We use again Corollary 2.9 and Ascoli’s theorem to complete the proof.
Corollary3.4. Suppose that(µn)n∈N?,µ, γare positive Radon bounded measures on D such that (µn)n → µ in the vague topology, and µn ≤ γ for all n ∈ N?. Then for all ϕ ∈ C(∂D) we have (M(µn;ϕ))n → M(µ;ϕ) uniformly on D.
Proof. Since {M(µn;ϕ) : n∈N?} is relatively compact in the topology of the uniform convergence onD(Theorem 3.1), the sequence (M(µn;ϕ))n∈N?
has a limit point with respect to this topology. Letu∈ U(D)∩C(D) be such a limit point and suppose that (M(µn;ϕ))n is uniformly convergent to the map u on the set D. By Theorem 1.2 (iii),
(µn)n= (νM(µn;ϕ))n→νu and u|∂D =ϕ,
and, by Theorem 1.5, u=M(µ;ϕ). Hence {M(µn;ϕ) :n∈N?}has a unique limit point with respect to the topology of the uniform convergence on D, which is M(µ;ϕ), i.e., (M(µn;ϕ))n→M(µ;ϕ) uniformly onD.
Corollary 3.5. Let (fn)n ⊂ L∞+(D) be bounded with respect to k · k∞ and f ∈ L∞+(D) be such that (fn)n→f λ a.e. on D. Then
(M(fn·λ;ϕ))n→M(f ·λ;ϕ) uniformly on D, for all ϕ∈C(∂D).
Proof. Since (fn·λ)n→f ·λvaguely, we have fn·λ≤(supfn)λfor all n∈N?, and we apply Corollary 3.4.
Proposition 3.6. Let (µn)n, µ be positive bounded Radon measures such that (µn)n → µ strongly on the space Cc(D) and let (ϕn)n ⊂C(∂D) be
such that (ϕn)n → ϕ uniformly on ∂D. Then (M(µn;ϕn))n → M(µ;ϕ) uni- formly on D.
Proof. If ε ∈ (0,∞) and nε ∈ N are such that for all n ≥ nε we have kϕn−ϕk∞≤ε, then
|M(µn;ϕn)−M(µ;ϕ)| ≤ −M(|µn−µ|;−|ϕn−ϕ|)≤ −M(|µn−µ|;−ε), and, by Proposition 1.6 (iii),
−M(|µn−µ|;−ε)≤ε+ (diamD)k s
|µn−µ|(D) ωk .
Since the convergence of (µn)n is strong, and the measures are bounded, we have lim
n→∞|µn−µ|(D) = 0. Moreover, for alln∈N?,
kM(µn;ϕn)−M(µ;ϕ)k∞≤ kM(|µn−µ|;−ε)k∞⇒ lim sup
n→∞
kM(µn;ϕn)−M(µ;ϕ)k∞≤lim sup
n→∞
kM(|µn−µ|;−εk∞≤
≤ε+ (diamD) lim sup
n→∞
k
s
|µn−µ|(D) ωk =ε, hence lim
n→∞kM(µn;ϕn))n−M(µ;ϕ)k∞= 0.
Corollary 3.7. Suppose that (fn)n ⊂ L∞+(D) is bounded and λ a.e.
convergent on D to the map f. If (ϕn)n ⊂C(∂D) is uniformly convergent to the functionϕon∂D, thenM(fn·λ;ϕn))n is uniformly convergent to the map M(f ·λ;ϕ) onD.
Proof. Since (fn·λ)n ⊂bM+(D) is strongly convergent to the measure f ·λ, the assertion is a consequence of Proposition 3.6.
Acknowledgements.Research supported by National Authority for Scientific Re- search (NASR) CEX-D11-23, 05.10.2005.
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Received 9 January 2009 University of Pite¸sti
Faculty of Mathematics and Computer Sciences Str. Tˆargu din Vale nr. 1
110040 Pite¸sti, Romania corneliu udrea@yahoo.com
