数理解析研究所講究録
第 1842 巻 2013 年 22-39

22

VARIATIONAL CONVERGENCE OVER -UNIFORMLY
CONVEX SPACES$p$

KAZUHIRO KUWAE
ABSTRACT. We establish a variational convergence over -uniformly
convex spaces for $p\geq 2$ . Variational convergence for Cheeger type-maps into p–uniformly convex spaceenergy functionals overhaving NPC property of Busemann type and the existence of $r$
harmonic map for Cheeger type energy functionals with Dirichletboundary condition are also presented.
$p$$L^{p}$

1. INTRODUCTION AND MAIN RESULTThis article is a summary of a part of the paper [17] under preparation. We study a variational convergences over p–uniformly convex
spaces having NPC property in the sense of Busemann, where a puniformly convex space is a natural generalization of p–uniformly convex Banach space. Typical examples of -uniformly convex spaces are$U$ -spaces with $p\geq 2$ , CAT( )-spaces, more concretely, Hadamard
manifolds and trees, and so on. If the target space is a -uniformly
convex space having NPC property in the sense of Busemann, then the$U$ -mapping space is also a-uniformly convex geodesic spaces having NPC property in the sense of Busemann, and an energy functional
defined in a suitable way becomes convex and lower semi-continuous.
and $(H, d_{H})$ are all pThus, it is reasonable to consider thatuniformly convex geodesic spaces having the weak -convexity of Busemann type instead of such -mapping spaces (see Definition 3.1 below
and $E$ : $Harrow[O, \infty]$:
for the weak -convexity), and. For any
are convex lower semi-continuous functions withand $u\in H$ , there exists a unique minimizer, say$v\mapsto\lambda^{p-1}E(v)+d_{H}^{p}(u,v)$ . This defines a map: $Harrow H$ , calledofthe resolvent of $E$ (see Theorem 5.2 below and [9, 22, 20] for the case$:= \min_{v\in H}(\lambda^{p-1}E(v)+d_{H}^{p}(u, v))$ is called$p=2)$ . The minimum
the Moreau-Yosida approximation or the Hopf-Lax formula. Note that
if $X$ is a Hilbert space and if is a closed densely defined symmetric
$p$

$O$

$p$

$p$

$(H_{i}, d_{H_{i}})$$L$$I\mathscr{J}$$L$$E_{i}$$H_{i}arrow[0, \infty]$$E_{i},$$E\not\equiv+\infty$$J_{\lambda}^{E}(u)\in H,$$\lambda\geq 0$$J_{\lambda}^{E}$$E^{\lambda}(u)$$E$

2000 Mathematics Subject Classification. Primary $53C20$ ; Secondary $53C21,$$53C23.$

Key words and phmses. CAT(0)-space, CAT -space, -uniformly convex
space, -convex space of Busemann type, weak convergence, Moreau-Yosida approximation, Hopf-Lax formula, Mosco convergence, convergence of resolvents.
The authors are partially supported by a Grant-in-Aid for Scientific Research
No. 22340036 from the Japan Society for the Promotion of Science.
$(\kappa)$

$L$

$p$

23
quadratic form on $X$ , then we have
, where is theinfinitesimal generator associated with $E$ . The one-parameter family[
gives a deformation of a given map $u\in H$ to a$E$minimizer of (or a harmonic map),(if any). Jost [13]studied convergence of resolvents and Moreau-Yosida approximations.
Although his study is only on a fixed CAT(0)-space, we extend it fora sequence of -uniformly convex geodesic spaces having the weak Lconvexity of Busemann type with an asymptotic relation (Theorem 5.11
below). This is new even on a fixed p–uniformly convex geodesic spaces
having the weak -convexity of Busemann type.
We can apply our result in the following way. Let $(X_{i}, q_{i})arrow(X, q)$
and $(Y_{i}, 0_{i})arrow(Y, 0)(i=1,2,3, \ldots)$ be two pointed Gromov-Hausdorffconvergent sequences of proper metric spaces, where ‘ proper’ meansthat any bounded subset is relatively compact, and let us give a positive Radon measureonwith full support which converge to a$X$
positive Radon measure $m$ on(see the definition for the convergenceof measures in [20] . We are interested in the convergence and asymptotic behavior of maps :and also energy functionals
de$A$$J_{\lambda}^{E}=(I+\lambda A)^{-1}$$0,$ $+\infty[\ni\lambda\mapsto J_{\lambda}^{E}(u)$$\lim_{\lambdaarrow+\infty}J_{\lambda}^{E}(u)$

$p$

$L$

$X_{i}$$m_{i}$$)$$X_{i}arrow Y_{i}$$u_{i}$

fined on a family of maps from$:=L_{o}^{p}(X, Y, m)$ .
andFor$X_{i}arrow Y_{i}$$L^{p}$$d_{L_{i}^{p}}(u_{i}, v_{i})$$:=\Vert d_{Y_{i}}(u_{i}, v_{i})\Vert_{L_{i}^{p}}$(resp.
) is the(resp. $m$ ). Consider$\Vert$$\Vert$$\Vert_{L_{i}^{p}}$$\Vert_{L^{p}}$$E_{i}$$u_{i},$. We set$v_{i}\in L_{i}^{p}$$:=L_{o_{i}}^{p}(X_{i}, Y_{i}, m_{i})$we set$(resp. u, v\in L^{p})$ ,$(resp. d_{L^{p}}(u, v)$$L^{p}$$L_{i}^{p}$$:=\Vert d_{Y}(u, v)\Vert_{L^{p}})$, where-norm with respect to the measure$m_{i}$$\mathcal{L}^{p}:=u_{i}L_{i}^{p}\sqcup L^{p}$and endowed the -topology defined in [20] with . The -topologyonhas some nice properties involving the -metric structure of, such as, if
and$v_{i}arrow u,$ $v\in L^{p}$ respectively in, then. By their properties we present a set of axioms
for a topology on
forand. We call such a topology
satisfying the axioms the asymptotic relation between(seeandDefinition 4.3). Since and are typically improper, the asymptotic
relation can be thought as a non-uniform variant of Gromov-Hausdorffconvergence.
We now assume thatandare -uniformly convex spaces with$k\in$$0,2]$
common parameter]
having NPC in the sense of Busemannand satisfying (B) and (C). Then
(resp. $E$ )andare so. Let
be Cheeger type -energy functional on(resp.
$H^{1,p}(X, Y;m)(\subset U))$ . Here$(resp. H^{1,p}(X, Y;m))$ is
the Cheeger-type -Sobolev space for -maps with respect to(resp. $m$ )from(resp. $X$ to ) (see Section 6 below). Then
to(resp. $E$ )is a lower semi-continuous convex functional on(resp.
). As acorollary of Theorem 5.11 below, we have the following:$L^{p}$$L^{p}$$\mathcal{L}^{p}$$L^{p}$$\mathcal{L}^{p}$$L^{p}$$L_{i}^{p}$$L^{p}$$L_{i}^{p}\ni u_{i},$$d_{L_{i}^{p}}(u_{i}, v_{i})arrow d_{Lp}(u, v)$$\mathcal{L}^{p}$$(L_{i}^{p}, d_{L_{i}^{p}})$$(L^{p}, d_{L^{p}})$$L^{p}$$\{L_{i}^{p}\}$$L^{p}$$L_{i}^{p}$$Y_{i}$$Y$$p$$L_{i}^{p}$$L^{p}$$E_{i}$$H^{1,p}(X_{i}, Y_{i};m_{i})(\subset L_{i}^{p})$$p$$H^{1,p}(X_{i}, Y_{i};m_{i})$$L^{p}$$p$$X_{i}$$Y_{i}$$m_{i}$$Y$$E_{i}$$L_{i}^{p}$$L^{p}$

24
converges to $E$ in the Mosco sense, then for any
Theorem 1.1. If
we have the following (1) and (2).
(1)
strongly converges to(2)strongly converges to$E_{i}$$\lambda>0$$E^{\lambda}$$E_{i}^{\lambda}$$J_{\lambda}^{E}.$$J_{\lambda}^{E_{i}}$Under a suitable condition like uniform Ricci lower bound condition$X$ , we can expect that the Mosco convergence ofto $E$
forholds. At present, we are still in progress to deduce it.
As an addendum, we also show the existence of -harmonic map for
Cheeger type energy functionals over -maps into -uniformly convex
space having NPC in the sense of Busemann with Dirichlet boundary
condition (see Theorem 6.20 below).
$\{E_{i}\}$$X_{i},$$p$$L^{p}$

2.

$p$

$p$

-UNIFORMLY CONVEX SPACES

map
Definition 2.1 (Geodesics). Let $(Y, d)$ be a metric space.
$IarrowY$:is said to be a curve if it is continuous, whereof a curve : $Iarrow Y$ is defined to
is a closed interval. The lengthbe$A$$I=[a, b]\subset \mathbb{R}$$\gamma$$L(\gamma)$$L(\gamma)$$\gamma$$:= \sup\{\sum_{i=1}^{n}d(\gamma(t_{i-1}), \gamma(t_{i}))|a=t_{0}<t_{1}<\cdots<t_{n-1}<t_{n}=b\}.$: $Iarrow Y$ is said to be a minimal geodesic if $L(\gamma|_{[s,t]})=$
holds for any $t\in I,$ $s<t$ , equivalently
for any $r<s<t.$ curve : $Iarrow Y$ is said to be a geodesic if
for any $t\in I,$ $s<t$ with sufficiently small $|t-s|,$ifholds. metric space $(Y, d)$ is called a -geodesic space for $R\in$ ]any two points in whose distance is strictly less than can bejoinedby a minimal geodesic. We simply say that $(Y, d)$ is a geodesic space if
it is an -geodesic space. Throughout this paper, for given $y\in Y,$
denote by: $[0,1]arrow Y$ a minimal geodesics from $x=:\gamma_{xy}(0)$ to$y=:\gamma_{xy}(1)$ provided $(Y, d)$ is an-geodesic space and $d(x, y)<R.$

A curve$\gamma$$d(\gamma_{s}, \gamma_{t})$$d(\gamma_{r}, \gamma_{t})=d(\gamma_{r}, \gamma_{s})+$

$s,$

$A$

$d(\gamma_{s}, \gamma_{t})$$\gamma$$L(\gamma|_{[s,t]})=d(\gamma_{s}, \gamma_{t})$

$s,$

$R$

$A$

$0,$

$\infty]$

$R$

$Y$

$x,$

$\infty$$\gamma_{xy}$$R$, we denote by. Let
constant curvature$:=\infty$ if
and

For

$n\in \mathbb{N}$$R_{\kappa}$$\kappa\leq 0$

the -dimensional space form of, that is,
be the diameter of$\kappa>0.$if$\mathbb{M}^{n}(\kappa)$$\kappa\in \mathbb{R}$$R_{\kappa}$$R_{\kappa}$$:=\pi/\sqrt{\kappa}$$n$$\mathbb{M}^{n}(\kappa)$

Definition 2.2 (CAT -Inequality, see [2]). Let $(Y, d)$ be a metricspace anda geodesic triangle in with perimeter strictly less than. Let
in. We say thatbe a comparison triangle of$CAT(\kappa)$
pointscorresponding-inequality
if
all
and
its
satisfies
satisfy$(\kappa)$$Y$$\triangle$$2R_{\kappa}$$1S$$\triangle$$\mathbb{M}^{2}(\kappa)$$\triangle$$p,$ $q\in\triangle$$\tilde{p},\tilde{q}\in\triangle\sim$$d(p, q)\leq d(\tilde{p},\tilde{q})$.

Definition 2.3 (CAT -Space, see [2]). metric space $(Y, d)$ is said
-geodesic space and all geodesic
to be a $CAT(\kappa)$ -space if $(Y, d)$ is asatisfy CAT
with perimeter strictly less than 2
triangles in$A$$(\kappa)$$R_{\kappa}$$Y$inequality.

$R_{\kappa}$$(\kappa)-$

25
Definition 2.4 -Uniformly Convex Geodesic Space; cf. Naor-Silberman [25] . metric space $(Y, d)$ is called -uniformly convex with pammeter $k>0$ if $(Y, d)$ is a geodesic space and for any three points$z\in Y$ , any minimal geodesicinwith, and all $t\in[0,1],$
$(p$

$A$

$)$

$p$

$x,$ $y,$$Y$$:=(\gamma_{t})_{t\in[0,1]}$$\gamma$$\gamma_{0}=x,$$\gamma_{1}=y$(2.1)$d^{p}(z, \gamma_{t})\leq(1-t)d^{p}(z, x)+td^{p}(z, y)-\frac{k}{2}t(1-t)d^{p}(x, y)$.

By definition, putting, we see $k\in$ ] $0,2]$ and $p\in[2,$. Theinequality (2.1) yields the (strict) convexity of $Y\ni x\mapsto d^{p}(z, x)$ for a
fixed $z\in Y$ . Any closed convex subset of a -uniformly convex space
is again a -uniformly convex space with the same parameter. Any
space over a measurable space is -uniformly convex with parameter
provided $p\geq 2$ . Every CAT(0)-space is a -uniformly
convex space with parameter
for $p>2$ (we can take
$k=2$ if $p=2)$ , because
is isometrically embedded into $L^{p}([0,1])$
for $p>1$ (see [5],[25]) and any -space is -uniformly convex for$p\geq 2$ .
Ohta [28] proved that for $\kappa>0$ any CAT -spacewithdiam $(Y)<R_{\kappa}/2$ is a 2-uniformly convex space with parameter $\{(\pi-$for any]
diam $(Y)].$$\infty[$$z=\gamma_{t}$$p$$L^{p}$$p$$p$$k= \frac{8}{4p}(\frac{p-1}{p})^{p-1}$$p$$k= \frac{8}{4^{p}p^{2}}(\frac{p-1}{p})^{p-1}$$\mathbb{R}^{2}$$L^{p}$$p$$Y$$(\kappa)$$2\sqrt{\kappa}\epsilon)\tan\sqrt{\kappa}\epsilon\}$$\epsilon\in$$0,$ $R_{\kappa}/2-$

Remark 2.5. A Banach space$\delta_{Y}(\epsilon)$$:= \inf\{1-\Vert\frac{x+y}{2}\Vert$$(Y, \Vert\cdot\Vert)$$x,$ $y\in

is said to be uniformly convex if

Y,$ $\Vert x\Vert=\Vert y\Vert=1,$ $\Vert x-y\Vert\geq\epsilon\},$the modulus of convexity of , satisfies
for] $0,2]$ . For$p\geq 2,$
is said to be -uniformly convex if there exists $c>$such thatfor] $0,2]$ . It is known that for $p\geq 2,$] $0,2]$ . By Lemma 2.1 in [29],forif a Banach space
is -uniformly convex for $p\geq 2$ , then there
exists $d=d(c,p)>0$ such that
$Y$$(Y, \Vert \Vert)$$0$$\delta_{Y}(\epsilon)>0$$\epsilon\in$$p$$\delta_{Y}(\epsilon)\geq c\epsilon^{p}$$\epsilon\in$$\delta_{Lp}(\epsilon)=1-[1-(\frac{\epsilon}{2})^{p}]^{\frac{1}{p}}\geq\frac{1}{2^{p}p}\epsilon^{p}$$(Y, \Vert\cdot\Vert)$$\epsilon\in$$p$$\Vert(1-t)x+ty\Vert^{p}\leq(1-t)\Vert x\Vert^{p}+t\Vert y\Vert^{p}-d\{t(1-t)^{p}+t^{p}(1-t)\}\Vert x-y\Vert^{p}$for all$x,$ $y\in

Y$

and

$t\in$

]

$O,$

$1$

[. Actually, we can take$d= \frac{c}{p}(\frac{p-1}{p})^{p-1}$

asSince $\frac{4}{2p}\leq(1-t)^{p-l}+t^{p-1}\leq lforallt\in[0, l],p-$$uniformconvexityoftheBanachspaceimp1iesthepuniformconvexityanoptimalvalue.$of geodesic space.
The following propositions can be proved in the same way as in [28].
So we omit its proof.
Proposition 2.6 (cf. Lemma 2.3 in [28]). Let $(Y, d)$ be a -uniformly
convex space. For $x,$ $z\in Y$ , any minimal geodesic
in
$p$

$y,$

$\gamma$$:=(\gamma_{t})_{t\in[0,1]}$

26
$Y$

with

$\gamma_{0}=x,$(2.2)$\gamma_{1}=y$, and all $t\in[O, 1]$ , we have$d^{p}(z, \gamma_{t})\leq\frac{2}{k}\cdot\frac{1}{t^{p-1}+(1-t)^{p-1}}$$\cross((1-t)^{p-1}d^{p}(z, x)+t^{p-1}d^{p}(z, y)-(1-t)^{p-1}t^{p-1}d^{p}(x, y))$.

Proposition 2.7 (cf. Lemma 2.2 and Proposition 2.4 in [28]). Any
two points in a -uniformly convex space can be connected by a uniqueminimal geodesic and contmctible.
$p$

Lemma 2.8 (Projection Map to Convex Set). Let $(Y, d)$ be a complete-uniformly convex space with pammeter $k\in$ ] $0,2]$ . The the following
hold:(1) Let $F$ be a closed convex subset of $(Y, d)$ . Then for each $x\in Y,$
there exists a unique element $\pi_{F}(x)\in F$ such that $d(x, F)=$$d(\pi_{F}(x), x)$ holds.
: $Yarrow F$ the projection map to
We call$p$$\pi_{F}$$F.$(2) Let $F$ be as above. Then
(2.3)$\pi_{F}$satisfies$ff(z, \pi_{F}(z))+\frac{k}{2}\parallel(\pi_{F}(z), w)\leq ff(z, w)$,$forz\in Y,$ $w\in F,$

in particular, $( \frac{k}{2})^{1/p}d(\pi_{F}(z), w)\leq d(z, w)$ for $z\in Y,$ $w\in F.$

Definition 2.9 (Vertical Geodesics). Let $(Y, d)$ be a geodesic space.
Take a geodesic with a point on it and another geodesic throughin short) if for(write. We say that is vertical to atanyand$p_{0}$$\eta$$p_{0}$$\gamma$$\gamma 1_{p0}\eta$$Po$$\eta$$\gamma$$y\in\eta,$$x\in\gamma$$d(x,p_{0})\leq d(x, y)$holds.
Let $(Y, d)$ be a complete -uniformly convex space with parameter$k\in]0,2]$ . We consider the following conditions:$p$:(A) For any closed convex set $F$ in $(Y, d)$ , the projection map$Yarrow Y$ satisfies $d(\pi_{F}(x), y)\leq d(x, y)$ for $x\in Y,$ $y\in F.$(B) Let and be minimal geodesics among two points such that
imliesintersects at . Then
(C) Let and be minimal geodesics among two points such that. Suppose that is a minimal
andintersects at
geodesic among two points which contains . Then
implies$\pi_{F}$$\gamma$$\eta$$\gamma$$\sigma$$\sigma$$\eta$$Po$$\eta$$p_{0}$$\eta\perp_{p0}\gamma.$$\gamma\perp_{p0}\eta$$\eta$$\sigma\neq\{p_{0}\}$$\gamma$$\sigma$$\sigma\perp_{p0}\eta$$\gamma\perp_{p0}\eta.$

Lemma 2.10. (B) implies (A).

Remark 2.11. Theorem 2.13 below shows that the conditions (A), (B),(C) are satisfied for any complete CAT -space with diameter strictly. For any complete -uniformly convex space $(Y, d)$ withless than$k\in$] $0,2]$ which is also a weakly -convex space in the senseparameter$(\kappa)$$R_{\kappa}/2$$prightarrow$

$L$

27
of Busemann for somesatisfying the conditions (A), (B), (C),the space $L_{h}^{p}(X, Y;m)$ of -maps fromintowith a map$XarrowY$:
is also a complete -uniformly convex space with thesame parameter $k\in$ ] $0,2]$ which is als a weakly -convex space in thesense of Busemann for the same
. and $L_{h}^{p}(X, Y;m)$ satisfies theconditions (A), (B), (C).
$(L_{1}, L_{2})$$L^{p}$$Y$$(X, \mathcal{X}, m)$

$h$

$p$

$L$

$0$

$(L_{1}, L_{2})$Lemma 2.12. Take a geodesic triangle$d_{\mathbb{M}^{n}(\kappa)}(B, C),$

$R_{\kappa}/2$$b$$:=d_{\mathbb{M}^{n}(\kappa)}(C, A),$$c$$\triangle ABC$in$\mathbb{M}^{n}(\kappa)$$:=d_{\mathbb{M}^{n}(\kappa)}(A, B)$

and $\angle BAC\geq\pi/2$ . Then for any point
holds.

$P$

and set $a:=$. Assume

on $AB,$$a,$ $b,$$c<$$d_{\mathbb{M}^{n}(\kappa)}(C, A)\leq$$d_{\mathbb{M}^{n}(\kappa)}(C, P)\leq d_{\mathbb{M}^{n}(\kappa)}(C, B)$

Theorem 2.13. Let. Any CAT -space $(Y, d)$ with diam $(Y)<$is a 2-uniformly convex space with some parameter $k\in$ ] $0,2]$ satisfying the conditions (A), (B), (C).
$\kappa\in \mathbb{R}$$(\kappa)$$R_{\kappa}/2$

3.

$L$

-CONVEX SPACES OF BUSEMANN TYPE

Definition 3.1 $(L-$ Convexity $of$ Busemann $Type, cf. Ohta [28])$ . Let$L_{2}\geq 0.$
metric space $(Y, d)$ is called an -convex space forin the sense of Busemann if $(Y, d)$ is a geodesic space, and$z\in Y$ and any minimal geodesics
for any three points:$[0,1]arrow Y$ and: $[0,1]arrow Y$ , and for all $t\in[0,1],$$A$$L_{1},$$L$$(L_{1}, L_{2})$$x,$ $y,$$\eta$(3.1)$\gamma$$:=\gamma_{xy}$$:=\gamma_{xz}$$d( \gamma_{t}, \eta_{t})\leq(1+L_{1}\frac{\min\{d(x,y)+d(x,z),2L_{2}\}}{2})td(y, z)$metric space $(Y, d)$ is called a weakly -convex space forin the sense of Busemann if $(Y, d)$ is a geodesic space, and$z\in Y$ and any minimal geodesics
for any three points:$[0,1]arrow Y$ and: $[0,1]arrow Y$ , and for all $t\in[0,1],$$d(\gamma_{t}, \eta_{t})\leq(1+L_{1}L_{2})td(y, z)$(3.2)

holds.

$A$$L$$(L_{1}, L_{2})$$x,$ $y,$$\eta$$\gamma$$:=\gamma_{xy}$$:=\gamma_{xz}$holds. metric space $(Y, d)$ is said to be quasi- -convex forin
the sense of Busemann if $(Y, d)$ is weakly -convex forin thesense of Busemann such that for any $x\in Y$ , any two minimal geodesics
and emanating from and $s\in[0,$, the limit
$A$

$L$

$L$

$\gamma$$\eta$

$x$

(3.3)

$t,$

$(L_{1}, L_{2})$$(L_{1}, L_{2})$$\infty[$$\lim_{\epsilonarrow 0}\frac{1}{\epsilon}d(\gamma_{t\epsilon}, \eta_{s\epsilon})$always exists.

Clearly, any complete separable CAT(0)-space is an -convex spaceforwith $L_{1}L_{2}=0$ in the sense of Busemann. Let $(Y, d)$ be aCAT(l)-space with diam $(Y)\leq\pi-\epsilon,$in which no triangle hasa perimeter greater than . Then by Proposition 4.1 in [28], $(Y, d)$ is
an -convex space for$L$$(L_{1}, L_{2})$$\epsilon\in]0,$$\pi[$$2\pi$$L$$(L_{1}, L_{2})=( \frac{2\{(\pi-\epsilon)-\sin\epsilon\}}{(\pi-\epsilon)\sin\epsilon}, \pi-\epsilon)$

28
By Lemma 4.1 in [28], -convexity of Busemann type implies the quasi-convexity of Busemann type.
. For $x\in Y,$
Let $(Y, d)$ be a quasi- -convex space for some
we defineas the set of unit speed minimal geodesics emanatingfrom $x\in Y$ . Then
and $s\in[0,$ , we can define the limit. Define the space of directions
at $x\in X$ by, where. Putholds if
$L$

$L$

$L$

$(L_{1}, L_{2})$$\Sigma_{x}’$$\gamma,$$t,$$\eta\in\Sigma_{x}’$$\infty[$$\lim_{\epsilonarrow 0}d(\gamma_{t\epsilon}, \eta_{s\epsilon})/\epsilon$$\Sigma_{x}$$:=\Sigma_{x}’/\sim$$\Sigma_{x}$$\lim_{\epsilonarrow 0}d(\gamma_{\epsilon}, \eta_{\epsilon})/\epsilon=0$$\gamma\sim\eta$$K_{x}’:=\Sigma_{x}\cross[0, \infty[/\sim,$

where$(\gamma, t)\sim(\eta, s)$holds if$\lim_{\epsilonarrow 0}d(\gamma_{t\epsilon}, \eta_{s\epsilon})/\epsilon=0$. Then$d_{K_{x}’}(( \gamma, t), (\eta, s)):=\lim_{\epsilonarrow 0}\frac{d(\gamma_{t\epsilon},\eta_{s\epsilon})}{\epsilon}$. Define the tangent conegives a distance function onat.
as the completion of
The following proposition can be similarly proved as for Proposition 4.2 in [28],$(K_{x}, d_{K_{x}})$$K_{x}’$$x\in X$$(K_{x}’, d_{K_{x}’})$

Proposition 3.2 (cf. Proposition 4.2 in [28]). For a -uniformly convex
space $(Y, d)$ having the quasi- -convexity of Busemann type for someand $x\in Y$ , the tangent cone
is a geodesic space.
Moreover, it is weakly -convex in the sense of Busemann with $L_{1}L_{2}=$, that is, a Busemann’s $NPC$ space.
$p$

$L$

$(L_{1}, L_{2})$$(K_{x}, d_{K_{x}})$

$L$

$0$

4. WEAK CONVERGENCE OVER $p-$-UNIFORMLY CONVEX SPACES
Throughout this section, we denote byrected set. We need the following:$i$any element of a given di-$\{i\}$

Proposition 4.1. Let
be a net of complete -uniformly
convex spaces with common pammeter $k\in$ ] $0,2]$ and all
have
the weak -convexity of Busemann type for some common. Let
be a net and: $[0,1]arrow H_{i}$ a net of minimal segments. Set$\{(H_{i}, d_{H_{t}})\}$$p$$(H_{i}, d_{H}.)$$L$$x_{i}\in H_{i}$$(L_{1}, L_{2})$$\gamma^{i},$$\eta^{i}$$\alpha_{0}:=\varlimsup_{i}d_{H_{i}}(\gamma_{0}^{i}, \eta_{0}^{i}) , \alpha_{1}:=\varlimsup_{i}d_{H_{i}}(\gamma_{1}^{i}, \eta_{1}^{i})$and $A:=(1+L_{1}L_{2})(\alpha_{0}+\alpha_{1})$ . Then$\varlimsup_{i}d_{H_{i}}(\pi_{\gamma^{t}}(x_{i}), \pi_{\eta^{t}}(x_{i}))\leq A+(\frac{2p}{k})^{1/p}(\sup_{j}d_{j}(x_{j}, y_{j})+2A)^{\frac{p-1}{p}}\cdot(2A)^{\frac{1}{p}}$$or$$\varlimsup_{i}d_{H_{i}}(\pi_{\gamma^{i}}(x_{i}), \pi_{\eta}\cdot(x_{i}))\leq A+(\frac{2p}{k})^{1/p}(\sup_{j}d_{j}(x_{j}, y_{j})+2A)^{L^{-\underline{1}}}p.$$(2A)^{\frac{1}{p}}$holds.

Corollary 4.2. Let
be a net of complete -uniformly convex
spaces with common pammeter $k\in$ ] $0,2]$ and all
have the weak$\{(H_{i}, d_{H}.)\}$$p$$(H_{i}, d_{H_{i}})$

29
. Let
-convexity of Busemann type for some common
be a net and: $[0,1]arrow H_{i}$ a net of minimal segments. If$L$$(L_{1}, L_{2})$$\gamma^{i},$$x_{i}\in H_{i}$$\eta^{i}$li$imd_{H_{i}}(\gamma_{0}^{i}, \eta_{0}^{i})=$li$imd_{H_{i}}(\gamma_{1}^{i}, \eta_{1}^{i})=0$

holds, thenliLetDefine$\{(H_{i}, d_{H_{i}})\}$$\mathcal{H}$$imd_{H_{i}}(\pi_{\gamma^{i}}(x_{i}), \pi_{\eta^{i}}(x_{i}))=0.$

be a net of metric spaces and $(H, d_{H})$ a metric space.
(disjoint union).

$:=(\sqcup_{i}H_{i})\sqcup H$

Definition 4.3 (Asymptotic Relation on ). We call a topology on
satisfying the following $(A1)-(A4)$ an asymptotic relation between
and $(H, d_{H})$ .
(Al)
and $H$ are all closed inand the restricted topology ofon each of
and $H$ coincides with its original topology.
(A2) For any $x\in H$ there exists a net
converging to in$H_{i}\niy_{i}arrowy\inH$$H_{i}\nix_{i}arrowx\inH$(A3) If
in , then we haveand.
(A4) If $H_{i}\ni x_{i}arrow x\in H$ in
and if $y_{i}\in H_{i}$ is a net with$\mathcal{H}$$\mathcal{H}$$\{(H_{i}, d_{H_{i}})\}$$H_{i}$$\mathcal{H}$$\mathcal{H}$$H_{i}$$x_{i}\in H_{i}$$x$$\mathcal{H}.$$\mathcal{H}$$d_{H_{i}}(x_{i}, y_{i})arrow d_{H}(x, y)$$\mathcal{H}$$d_{H_{i}}(x_{i}, y_{i})arrow 0$, then $y_{i}arrow x$ in$\mathcal{H}.$Definition 4.4 (Asymptotic Compactness of Asymptotic Relation).
Assume thatand $(H, d_{H})$ have an asymptotic relation. We
say that a net
is bounded for someis bounded if
convergent net. The asymptotic relation is said to be asymptotically compact if any bounded net
has a convergent subnet
in with respect to the asymptotic relation.
Hereafter, strong convergence on means the convergence with respect to a given asymptotic relation over . Assume that an asymptotic relation between metric spaces
and $H$ given. Consider thefollowing condition:(G) If : $[0,1]arrow H_{i}$ and : $[0,1]arrow H$ are minimal geodesics suchthatfor any $t\in[O, 1].$, thenand
Proposition 4.5.
(1) If (G) is satisfied and if eachis a geodesic space, then $H$ is so.
(2) If (G) is satisfied and if eachis -uniformly convex with$H$
$k\in$$0,2]$, thenis so.
]
common pammeter(3) If eachis -uniformly convex with common pammeter $k\in$$]0,2]$ and $H$ is a geodesic space, then (G) is satisfied and $H$ is-uniformly convex with pammeter $k\in$ ] $0,2].$In the proof of Proposition 4.5, we use Proposition 2.6.
We now define the weak convergence over , which generalize thenotions introduced in [8, 6, 20].
$\{(H_{i}, d_{H_{i}})\}$$x_{i}\in H_{i}$$d_{H_{i}}(x_{i}, 0_{i})$$0_{1}\in H_{i}$$x_{i}\in H_{i}$$\mathcal{H}$$\mathcal{H}$$\mathcal{H}$$\{H_{i}\}$$\gamma^{i}$$\gamma$$\gamma_{0}^{i}arrow\gamma_{0}$$\gamma_{1}^{i}arrow\gamma_{1}$$\gamma_{t}^{i}arrow\gamma_{t}$$H_{i}$$H_{i}$$H_{i}$

$p$

$p$

$p$

$\mathcal{H}$30
Definition 4.6 (Weak Convergence on ). Let
be a netof complete p–uniformly convex spaces with common parameter $k\in$$]0,2]$ and $(H, d_{H})$ a complete p–uniformly convex space with the same
parameter . We say that a net
weakly converges to a point$x\in H$ if for any net of geodesic segments
instrongly converging
to a geodesic segment in $H$ withstrongly converges
to . Here the strong convergence ofto means that for any$t\in[O, 1],$strongly converges to . It is easy to prove that a strong
convergence implies a weak convergence and that a weakly convergent
net always has a unique weak limit.
$\{(H_{i}, d_{H_{i}})\}$$\mathcal{H}$$k$$x_{i}\in H_{i}$$\gamma^{i}$$\gamma_{0}=x,$$\gamma$$x$$\{\gamma^{i}\}$$H_{i}$$\pi_{\gamma^{i}}(x_{i})$$\gamma$$\gamma_{t}^{i}$$\gamma_{t}$The following proposition is omitted in [20]. We shall give it forcompleteness.
Proposition 4.7 (Weak Topology on ). The weak convergence overof complete -uniformly convex spaces with pammeter $k\in$ ] $0,2]$ induces a Hausdorff topology on it. We call it weak topology of $(H, d_{H})$ .
$\mathcal{H}$$\mathcal{H}$$p$Remark 4.8. The notion of weak convergence over a fixed CAT(0)-space
is proposed by Jost [8]. In [20], we extend it over of CAT(0)-spaces.
In Kirk-Panyanak [14], they give a different approach on the weak convergence, so-called -convergence, and Esp\’inola and Fern\’andez-Le\’on[6] proved the equivalence between the weak convergence and theconvergence over a fixed CAT(0)-space or CAT(l)-space whose diameter strictly less than(see Proposition 5.2 in [6]). Such an equivalence is also valid for a fixed p–uniformly convex space in the same way
as in the proof of Proposition 5.2 in [6].
$\mathcal{H}$$\Delta$$\triangle-$$\pi/2$

Lemma 4.9. Let
be a net of complete -uniformly convex space with common pammeter $k\in$ ] $0,2]$ and $(H, d_{H})$ a complete-uniformly convex space with the same pammeter . Suppose that anet
is weakly convergent to $x\in H$ and a net
is strongly$y\inH$
convergent to. Then we have the following:(1) Under (A) for all.
$\lim_{i}d_{H_{i}}(x_{i},y_{i})=d_{H}(x,y)$(2) Under (B) for all
if andonly ifstrongly converges to $x\in H.$$\{(H_{i}, d_{H_{:}})\}$

$p$

$k$

$p$

$x_{i}\in H_{i}$$y_{i}\subset H_{i}$$(H_{i}, d_{H_{i}}),$$d_{H}(x, y)\leq\varliminf_{i}d_{H_{i}}(x_{i}, y_{i})$$(H_{i}, d_{H_{i}}),$$x_{i}\in H_{i}$The main result of this section is the following theorem:
Theorem 4.10 (Banach-Alaoglu Type Theorem). Letbea net of complete -uniformly convex spaces with common pammeter $k\in]0,2]$ and $(H, d_{H})$ a complete -uniformly convex space withthe same pammeter and all
and $(H, d_{H})$ have the weak Lconvexity of Busemann type for some common. Suppose oneof the following:(1) (B) and (C) hold for $(H, d_{H})$ and $(H_{i}, d_{H_{i}})=(H, d_{H})$ holds
for all(2) $(H, d_{H})$ is sepamble.
$\{(H_{i}, d_{H_{1}})\}$

$p$

$p$

$k$

$(H_{i}, d_{H_{i}})$$(L_{1}, L_{2})$

$i.$

31
Then every bounded net$\{x_{i}\}\subset \mathcal{H}$

has a weakly convergent subsequence.

Combining Theorems 2.13 and 4.10, we obtain the following:
Corollary 4.11 (Banach-Alaoglu Type Theorem over CAT -Spaces).
Letbe a net of complete $CAT(\kappa)$ -spaces with diam $(H_{i})<$with, and $(H, d_{H})$ a complete $CAT(\kappa)$ -space with]diam $(H)<R_{\kappa}/2-\epsilon$ with][. Assume that $(H_{i}, d_{H_{i}})=$$(H, d_{H})$for all or $(H, d_{H})$ is sepamble. Then every bounded net
has a weakly convergent subsequence.
$(\kappa)$$\{(H_{i}, d_{H_{i}})\}$$R_{\kappa}/2-\epsilon$$\epsilon\in$$0,$ $R_{\kappa}/2[$$\epsilon\in$$0,$$R_{\kappa}/2$$i$$\{x_{i}\}\subset \mathcal{H}$Remark 4.12. The assertion of Theorem 4.10 was proved by Theorem 2.1 in Jost [8] over a fixed complete CAT(0)-space without assuming the separability. In the framework of convergence over CAT $(O)-$spaces, Lemma 5.5 in [20] extends Theorem 2.1 in [8]. For a fixed]
CAT -space $(H, d_{H})$ with diam $(H)<R_{\kappa}/2-\epsilon$ withthe assertion of Corollary 4.11 is essentially shown by combining Corollary 4.4 and Remark 5.3 of [6]. Corollary 4.11 also extends the resultin [6].
$(\kappa)$$\epsilon\in$

5. VARIATIONAL CONVERGENCE OVER

$p$$0,$ $R_{\kappa}/2[,$-UNIFORMLY CONVEXSPACESIn this section we fix $p\geq 2.$

5.1. Resolvents. Throughout this subsection, we fix a complete puniformly convex space $(H, d_{H})$ with parameter $k\in$ ] $0,2]$ . Consider afunction $E:Harrow[O, \infty]$ and set $D(E):=\{x\in H|E(x)<\infty\}.$Definition 5.1 (Moreau-Yosida Approximation, [9]). For $E:Harrow$$[0, +\infty]$ we define: $Harrow[O, +\infty]$ by$E^{\lambda}$$E^{\lambda}(x);= \inf_{y\in H}(\lambda^{p-1}E(y)+d_{H}^{p}(y, x)) , x\in H, \lambda>0,$

and call it the Moreau-Yosida approximation or the Hopf-Lax formulafor $E.$-continuous,
Theorem 5.2 (Existence of Resolvent). If $E$ is lower
convex and, then for any $x\in H$ there exists a unique point,say, such that$\mathcal{S}emi$$E\not\equiv+\infty$$J_{\lambda}(x)\in H$$E^{\lambda}(x)=\lambda^{p-1}E(J_{\lambda}(x))+d_{H}^{p}(x, J_{\lambda}(x))$

This defines a map$J_{\lambda}$.

: $Harrow H$ , called the resolvent of $E.$

Note that if $H$ is a Hilbert space and $p=2$ , and if $E$ is a closed
densely defined non-negative quadratic form on $H$ , then we have
. Here, is the identity operator, the infinitesimal
generator associated with $E$ , i.e., the non-negative self-adjoint operator
on $H$ such that $D(E)=\sqrt{A}$ and $E(x)=(\sqrt{A}x, \sqrt{A}x)_{H}$ for any $x\in$$D(E)$ , whereis the Hilbert inner product on $H$ , and$(\alpha+A)^{-1},$ $\alpha>0$ is the resolvent operator associated with $A.$
$J_{\lambda}=$$(I+ \lambda A)^{-1}=\frac{1}{\lambda}G_{\frac{1}{\lambda}}$$(\cdot, \cdot)_{H}$

$I$

$A$

$G_{\alpha}=$32
To the end of this subsection, we always assume the convexity of $E.$
We have the following lemmas and theorems which are known for thecase that $(H, d_{H})$ is a CAT(0)-space. The proofs are omitted.
Lemma 5.3. For$\lambda,$$\mu>0$ ,we have$\frac{1}{\mu^{p-1}}(\frac{1}{\lambda^{p-1}}E^{\lambda})^{\mu}=\frac{1}{(\lambda+\mu)^{p-1}}E^{\lambda+\mu}.$

Lemma 5.4. Let $E:Harrow[O, \infty]$ be a lower semi-continuous function. For $x\in H$ and $s\in[O, 1]$ , we havewith
$E\not\equiv\infty$,$J_{\lambda}(x)=J_{(1-s)\lambda}((1-s)x+sJ_{\lambda}(x))$

where $(1-s)x+sJ_{\lambda}(x)$ is the point on the geodesic joining
such that $d_{H}(x, (1-s)x+sJ_{\lambda}(x))=sd_{H}(x, J_{\lambda}(x))$ .

$x$

to

$J_{\lambda}(x)$Lemma 5.5. Let: $Harrow H,$
be the resolvent associated witha lower semi-continuous convex function $E:Harrow[O, \infty]$ with
$Forx\in\overline{D(E)}$, then$\lambda>0$$J_{\lambda}$$E\not\equiv\infty.$$\lim_{\lambdaarrow 0}d_{H}(J_{\lambda}(x), x)=0.$

Theorem 5.6. Let $E:Harrow[O, \infty]$ be a lower semi-continuous convex. Take $x\in H$ and assume thatisfunction with. Thenconverges to abounded for some equence$E\not\equiv\infty$$(J_{\lambda_{n}}(x))_{n\in \mathbb{N}}$$\lambda_{n}arrow\infty$$\mathcal{S}$$(J_{\lambda}(x))_{\lambda>0}$

minimizer of $E.$

5.2. Variational Convergence. Throughout this subsection, we fix
a netof complete p–uniformly convex spaces with common$k\in$
parameter] $0,2]$ and a complete -uniformly convex space $(H, d)$of functions
with the same parameter $k\in$ ] $0,2]$ . Consider a net$E_{i}:H_{i}arrow[0, \infty]$ and a function $E:Harrow[O, \infty].$$\{(H_{i}, d_{H_{i}})\}$$p$$\{E_{i}\}$of
Definition 5.7 (Asymptotic Compactness, [24],[20]). The net
functions is said to be asymptotically compact if for any bounded net$x_{i}\in H$ withthere exists a convergent subnet of$\{E_{i}\}$$\varlimsup_{i}E_{i}(x_{i})<+\infty$$\{x_{i}\}.$-converges to $E$ if
Definition 5.8 ( -convergence). We say thatthe following
andare satisfied:and
For any $x\in H$ there exists a net
such that$E_{i}(x_{i})arrow E(x)$ .
If $H_{i}\ni x_{i}arrow x\in H$ then $E(x)\leq\varliminf_{i}E_{i}(x_{i})$ .
$E_{i}\Gamma$$\Gamma$$(\Gamma 1)$$(\Gamma 2)$$x_{i}\in H_{i}$$(\Gamma 1)$$x_{i}arrow x$$(\Gamma 2)$

converges to $E$
Definition 5.9 (Mosco convergence). We say thatin Definition 5.8 and the following
in the Mosco sense if bothhold.
.
If $H_{i}\ni x_{i}arrow x\in H$ weakly, then$E_{i}$$(\Gamma 2’)$$(\Gamma 1)$$E(x)\leq\varliminf_{i}E_{i}(x_{i})$$(\Gamma 2’)$, so that a Mosco
Note thatis a stronger condition thanconvergence implies a -convergence.
It is easy to prove the following proposition. The proof is omitted.
$(\Gamma 2’)$$(\Gamma 2)$$\Gamma$33
Proposition 5.10. Assume thatis ymptotically compact. Thenthe following (1) $-(3)$ are all equivalent to each other.
(1)converges to $E$ in the Mosco sense.
(2)-converges to $E.$
(3)
compactly converges to $E.$$\{E_{i}\}$$a\mathcal{S}$$E_{i}$$E_{i}\Gamma$$E_{i}$In what follows, we assume that all
and $H$ are -uniformly convex
spaces with a common parameter $k\in$ ] $0,2]$ having the weak -convexity: $H_{i}arrow[0, +\infty]$ and $E:Harrow$of Busemann type, and all functions$[0, +\infty]$ are all lower semi-continuous, convex, and are not identicallyequal $to+\infty$ . Let
and $E$ respectively.
andbe the resolvents of$H_{i}$$p$$L$$E_{i}$$E_{i}$$J_{\lambda}$$J_{\lambda}^{i}$

satisfy the condition (B).
Theorem 5.11. Suppose that all$(H_{i},d_{H_{i}})=(H,d_{H})$
Assume thatfor all and $(H, d_{H})$ satisfies (C), $or$$(H, d_{H})$ is sepamble.
If converges to $E$ in the Mosco sense, then forany
we have the following (1) and (2).
(1)
strongly converges to(2)stmngly converges to$(H_{i}, d_{H_{i}})$$i$$E_{i}$$\lambda>0$$E^{\lambda}.$$E_{i}^{\lambda}$$J_{\lambda}.$$J_{\lambda}^{i}$

Proposition 5.12. If$E_{i}\Gamma$-converges to$E_{i}^{\lambda}$

strongly converges to$E^{\lambda}$

for any$\lambda>0$, then$E.$

Propositions 5.10, 5.12 and Theorem 5.11 together imply the following

Corollary 5.13. Assume thatis asymptotically compact and all
atisfies the condition (A). Then, the following (1) and (2)are equivalent.
(1)compactly converges to $E.$(2)strongly converges tofor any$\{E_{i}\}$$(H_{i}, d_{H_{i}})\mathcal{S}$$E_{i}$$\lambda>0.$$E^{\lambda}$$E_{i}^{\lambda}$

6. CHEEGER TYPE SOBOLEV SPACE OVER

$L^{p}$-MAPS

In this section, we prepare several notions for our main Theorem 1.1.
be a -finite measurespace. Denote bywith respect to $m$ . In
the completion of-measumble) for
what follows, we simply say measumble (resp.
-measurable).
measurable (resp.
numerical function on $X$ is amap : $Xarrow[-\infty, \infty]$ . For a measurable numerical function on $X,$$:= \inf\{\lambda>0||f(x)|\leq$
we set $\Vert f\Vert_{p};=(\int_{X}|f(x)|^{p}m(dx))^{1/p},$, denote by $U(X;m)$ the family of
m-a.e. $x\in X\}$ . For $p\in$ ]$m$ -equivalence classes of-measurable functions finite with respectto. Denote by $L^{0}(X;m)$ the family of $m$-equivalence classes of-measurable numerical functions : $Xarrow[-\infty, \infty]$ with $|f|<\infty$m-a.

6.1. The space of $U$ -maps. Let$(X, \mathcal{X}, m)$$\mathcal{X}^{m}$$\sigma$$\mathcal{X}$$\mathcal{X}^{m}$$A$$\mathcal{X}^{m}$

$f$

$f$

$f$

$\Vert f\Vert_{\infty}$$0,$$\lambda$$\infty]$$\mathcal{X}^{m}$$\Vert_{p}$$\Vert$$f$$\mathcal{X}^{m}$$e.$$\mathcal{X}-$34
Let $(Y, d)$ be a metric space. For $p\in$ ]and measurable maps$f,$
: $Xarrow Y$ , define a pseudo distance $d_{p}(f, g)$ by $d_{p}(f, g)$
If $p<\infty$ , then$0,$$\infty]$$:=\Vert d(f, g)\Vert_{p}.$$g$$d_{p}(f, g) :=( \int_{X}d^{p}(f(x), g(x))m(dx))^{1/p}$

If $p=\infty$ , thenWe say that andis the $m$-essentially supremum of $x\mapsto d(f(x), g(x))$ .
are $m$-equivalent if$d_{\infty}(f, g)$

$f$

$g$

$f(x)=g(x)$ m-a.e. $x\in X$

and write $f\sim mg$ . For a fixed measurable map $h:Xarrow Y$ , we set$L_{h}^{p}(X, Y;m) :=\{f\in \mathcal{X}/\mathcal{B}(Y)|d(f, h)\in L^{p}(X;m)\}/\sim m$

The map : $Xarrow Y$ is called a base map. If $m(X)<\infty$ and : $Xarrow Y$
is bounded, then $L_{h}^{p}(X, Y;m)$ is independent of the choice of such
$h$

$h$

$h.$

Lemma 6.1. Let $(Y, d)$ be a metric space. For a fixed measumble map$h:Xarrow Y$ and $p\in[1, \infty]$ , we have the following;(1) If $(Y, d)$ is complete (resp. sepamble), then $(L_{h}^{p}(X, Y;m), d_{p})$ is$so.$(2) Suppose that $(Y, d)$ is a geodesic space and any two points can beconnected by a unique minimal geodesic. For given
be the -point in a unique minimaland each $t\in[0,1]$ , let$[0,1]arrowY$joiningto . Assume that for eachgeodesic :$t\in[0,1],$. Then foris continuous with respect to$f_{1}\in L_{h}^{p}(X, Y;m)$ , the map: $Xarrow Y$ defined bygiven$f_{t}(x);=(f_{0}(x)f_{1}(x))_{t}$ belongs to $L_{h}^{p}(X, Y;m)$ andforms a $\min-$toin $L_{h}^{p}(X, Y;m)$ . In particular,
imal geodesic joining$\gamma_{0},$$\gamma_{1}\in Y$$t$$\gamma_{t}$$\gamma_{0}$$\gamma$$\gamma_{1}$$(\gamma_{0}, \gamma_{1})$$\gamma_{t}$$f_{0},$$f_{t}$$f_{0}$$(L_{h}^{p}(X, Y;m), d_{p})$$f_{1}$

is a geodesic space.

Theorem 6.2. Let $(Y, d)$ be a complete -uniformly convex space having the weak -convexity of Busemann type. Fix a measumble map$h:Xarrow Y$ . Then we have the following:(1) $(L_{h}^{p}(X, Y;m), d_{p})$ is a complete -uniformly convex space having
the weak -convexity of Busemann type.
(2) Let : [ $\infty[arrow L_{h}^{p}(X, Y;m)$ be a minimal geodesic. Then foreach $x\in X$ and $L\in[0,$ , there exists a minimal segment: $[0, L]arrow Y$ such thatfor all $t\in[0, L],$$[0,L]arrow
L_{h}^{p}(X,
Y;m)$
:
where
is a minimal segment definedby.
(3) Assume that $(Y, d)$ satisfies the quasi- -convexity of Busemann. $Then(L_{h}^{p}(X, Y;m), d_{p})is$ so.
type for some
$p$

$L$

$p$

$L$

$0,$

$\gamma$$\infty[$$d_{p}(\gamma_{t},\tilde{\gamma}_{t}^{(L)})=0$$\tilde{\gamma}^{(L)}(x)$$\tilde{\gamma}^{(L)}$$\tilde{\gamma}^{(L)}(x)$$L$$(L_{1}, L_{2})$Lemma 6.3. Let $(Y, d)$ be a complete -uniformly convex space having
the weak -convexity of Busemann type such that $(Y, d)$ satisfies (A).
Let $F$ be a closed convex subset of $(L_{h}^{p}(X, Y;m), d_{p})$ . For each $x\in X,$
set $F(x)$ $:=\{f(x)|f\in F\}.$
$p$

$L$

35
(1) For each $x\in X,$ $F(x)$ is convex in $(Y, d)$ .
(2) Take an $f\in L_{h}^{p}(X, Y;m)$ . Then$\pi_{F}(f)=(\pi_{\overline{F(x)}}(f(x)))_{x\in X}$$L_{h}^{p}(X, Y;m)$in.

Theorem 6.4. Let $(Y, d)$ be a complete -uniformly convex space hav$ing$ the weak
-convexity of Busemann type. The following hold:
$p$

$L$

(1) If $(Y, d)$ satisfies (A), then $(L_{h}^{p}(X, Y;m), d_{p})$ does so.
(2) If $(Y, d)$ satisfies (B), then $(L_{h}^{p}(X, Y;m), d_{p})$ does so.
(3) If $(Y, d)$ satisfies (C), then $(L_{h}^{p}(X, Y;m), d_{p})$ does so.

Corollary 6.5. For $p\geq 2,$ $U(X;m)$ satisfies (A), (B), (C).
Corollary 6.6. Let $(Y, d)$ be a complete $CAT(\kappa)$ -space with a diameter
strictly less than
. Then we have the following:$R_{\kappa}/2$(1) $(L_{h}^{2}(X, Y;m), d_{2})$ is a 2-uniformly convex space with the same
pammeter $k\in$ ] $0,2]$ having the weak -convexity of Busemann
type.
(2) $(L_{h}^{2}(X, Y;m), d_{2})$ satisfies (A), (B) and (C).
$L$

Hereafter, we focus only on the case that $X$ is a locally compact
separable metric space and $h\equiv 0$ , where $0\in Y$ is a fixed base point.
We write $L_{o}^{r}(X, Y;m)$ instead of $L_{h}^{r}(X, Y;m)$ in such a case.
Definition 6.7 (Lipschitz Maps with Compact Support). The support$supp[u]$ ‘for a measurable map $u:Xarrow Y$ is defined to be the subset
of $X$ satisfying the condition that $x\in X\backslash supp[u]$ if and only if thereof such that $u=0$ on . Denote
exists an open neighborhood
by $C_{o}^{Lip}(X, Y)$ the set of Lipschitz continuous maps : $Xarrow Y$ withcompact support $supp[u].$
$U$

$U$

$x$

$u$

Theorem 6.8. Suppose that $(Y, d)$ is a separable geodesic space. Let. Then $C_{o}^{Lip}(X, Y)is$ adense subset of $(L_{o}^{r}(X, Y;m), d_{r})$ .

$r\geq 1$

6.2. Upper gradient and Cheeger’s Sobolev spaces. In what $fo1-$
lows, let $(X, d_{X})$ be a metric space, and $U\subset X$ be an open set, and $m$
be a Borel regular measure on $X$ such that any ball with finite positive
radius is of finite positive measure. Let $(Y, d)$ be a complete geodesic
space.
Definition 6.9 (Upper Gradient). A Borel function $g:Uarrow[O, \infty]$ iscalled an upper gmdient for a map : $Uarrow Y$ if, for any unit speed
curve $c:[0, \ell]arrow U$ , we have$u$$\Phi(u(c(0)), u(c(\ell)))\leq\int_{0}^{\ell}g(c(s))ds.$

36
Definition 6.10 (Upper Pointwise Lipschitz Constant Function). Fora map $u:Uarrow Y$ and a point $z\in U$ , we defineLip$u(z):= \varliminf_{rarrow 0}\sup_{d_{X}(z,w)=r}\frac{d(u(z),u(w))}{r},$Lip $u(z):=$ lim$d(u(z), u(w))$

sup$rarrow 0_{0<d_{X}(z,w)<r} d_{X}(z, w)$

and we put Lip $u(z)=$ Lip $u(z)=0$ if is an isolated point. Clearly
Lip $u\leq$ Lip on $X$ . We call Lip the upper pointwise Lipschitz constant function for
$z$

$u$

$u$

$u.$

Cheeger [4] proved that for a locally Lipschitz function $u:Uarrow \mathbb{R},$then Lipu, hence Lip , is an upper gradient for . We next define
the Cheeger type Sobolev spaces. Fix a point $0\in Y$ as a base point. Let $U_{o}(U, Y;m)$ be the space of -maps as defined inand $p\in[1,$the previous section. We write $L^{p}(U, Y;m)$ instead of $U_{o}(U, Y;m)$ for
simplicity.
$u$

$u$

$L^{p}$$\infty[$

Definition 6.11 (Cheeger Type Sobolev Space). For $u\in L^{p}(U, Y;m)$ ,we define the Cheeger type -energy of as
$u$

$p$

$E_{p}(u) := inf\varliminf\Vert g_{i}\Vert_{L^{p}(U;m)}^{p},$$\{(u_{i},g_{i)\}_{i=1}^{\infty}}iarrow\infty$

such that
where the infimum is taken over all sequences
and is an upper gradient for forin If $(U, Y;m)$ as
$(1,
p)$
.
-Sobolev space is defined byeach The Cheeger type$\{(u_{i}, g_{i})\}_{i=1}^{\infty}$$iarrow\infty$$u_{i}arrow u$$u_{i}$$g_{i}$$i$$H^{1,p}(U, Y;m) :=\{u\in L^{p}(U, Y;m)|E_{p}(u)<\infty\}.$By definition, if $u=v$ m-a.e. on$U$, then $E_{p}(u)=E_{p}(v)$ .

The following is proved in [26].
Theorem 6.12 (Lower Semi Continuity of Energy, see Theorem 2.8 inconverges to in $U(U, Y;m)$ , then $E_{p}(u)\leq$[26] . If a sequence$u$$\{u_{i}\}_{i=1}^{\infty}$$)$$\varliminf_{iarrow\infty}E_{p}(u_{i})$.

Definition 6.13 (Generalized Upper Gradient). function $g\in L^{p}(U;m)$is called a genemlized upper gmdient for $u\in H^{1,p}(U, Y;m)$ if there exsuch that is an upper gradient for andists a sequence
in $L^{p}(U, Y;m),$ $L^{p}(U;m)$ respectively as$A$$\{(u_{i}, g_{i})\}_{i=1}^{\infty}$$u_{i}$$g_{i}$$iarrow\infty.$$u_{i}arrow u,$ $g_{i}arrow g$

for any genFrom the definition of the p–energy,
eralized upper gradient $g\in L^{p}(U;m)$ for $u\in H^{1,p}(U, Y;m)$ .
$E_{p}(u)\leq\Vert g\Vert_{L^{p}(U;m)}^{p}$generalDefinition 6.14 (Minimal Generalized Upper Gradient).
ized upper gradient $g\in L^{p}(U;m)$ for a map $u\in H^{1,p}(U, Y;m)$ is said
to be minimal if it satisfies$A$$E_{p}(u)=\Vert g\Vert_{L^{p}(U;m)}^{p}.$37Hereafter, we assume that $(Y, d)$ is weakly -convex with $L_{1}L_{2}=0,$
that is, $(Y, d)$ is a Busemann’s NPC space. Then the distance function$d:Y\cross Yarrow[O,$is convex. We know the following results:$L$$\infty[$

Lemma 6.15 (See, Lemma 3.1 in [28]). Suppose that $(Y, d)$ is weakly-convex with $L_{1}L_{2}=0$ . Let: $Uarrow Y$ be. For any upper$0\leq\lambda\leq
1$
gmdient
respectively and. The functionfor$g:=(1-\lambda)g_{1}+\lambda g_{2}$ isan upper gmdient for the map $v:=(1-\lambda)u_{1}+\lambda u_{2}.$$u_{2}\in H^{1,p}(U, Y;m)$ with $1\leq p<\infty$ andIn particular, for any
forany, we have$L$$u_{1},$$g_{1},$ $g_{2}$$u_{1},$$u_{2}$$map_{\mathcal{S}}$$u_{2}$$u_{1},$$0\leq\lambda\leq 1$$E_{p}((1-\lambda)u_{1}+\lambda u_{2})^{1/p}\leq(1-\lambda)E_{p}(u_{1})^{1/p}+\lambda E_{p}(u_{2})^{1/p}.$

Theorem 6.16 (See, Theorem 3.2 in [26]). Let $p\in$ ]. Supposethat $(Y, d)$ is weakly -convex with $L_{1}L_{2}=0$ . Then for any $u\in$$H^{1,p}(U, Y;m)$ , there exists a unique minimal genemlized upper gmdient$1,$$\infty[$

$L$

for

$g_{u}$$u.$

For $p\in]1,$$\infty[$, we define a distance$d_{H^{1,p}}$on $H^{1,p}(U, Y;m)$ : for$u,$ $v\in$$H^{1,p}(U, Y;m)$ ,

(6.1)

where

$d_{H^{1,p}}(u, v) :=d_{p}(u, v)+\Vert g_{u}-g_{v}\Vert_{L^{p}(U;m)},$$v\in$
is the minimal generalized upper gradient for$H^{1,p}(U, Y;m)$ , respectively. Let
be the completion of $(H^{1,p}(U, Y;m), d_{H^{1,p}})$ .
The following assertion is not declared clearly in [26]. We provide
its proof for completeness.
$g_{u},$$u,$$g_{v}$$(\overline{H}^{1,p}(U, Y;m), d_{\overline{H}^{1,p}})$

Theorem 6.17. Let $p\in$ ]$1,$$\infty[$. We have $\overline{H}^{1,p}(U, Y;m)=H^{1,p}(U, Y;m)$ .

Remark 6.18. Theorem 6.17 does not necessarily imply the-comple$H^{1,p}(U,Y;m)$
teness of $H^{1,p}(U, Y;m)$ , that is,on.
$d_{H^{1,p}}$$d_{\overline{H}^{1,p}}=d_{H^{1,p}}$6.3.

-harmonic maps. In this subsection, we still assume that $(Y, d)$is weakly -convex with $L_{1}L_{2}=0.$
$p$

$L$

Definition 6.19 ( -Harmonic Map).
$H_{v}^{1,p}(U, Y;m)$ be the-closure of$p$

For $v\in H^{1,p}(U, Y;m)$ , let$d_{H^{1,p}}$$\{u\in H^{1,p}(U, Y;m)|suppd(u, v)\Subset U\}.$$v$

is said to be -harmonic if and only if $E_{p}(v)= \inf_{u\in H_{v}^{1,p}(U,Y;m)}E_{p}(u)$ .
$p$

Theorem 6.20. Suppose $p\geq 2$ .
any $f\in H_{0}^{1,p}(U)$ ,If there exists $C>0$ such that for$\int_{U}|f|^{p}dm\leq C\int_{U}|g_{f}|^{p}dm$,(Poincar\’e Inequality)

then there exists a -harmonic map in $H_{v}^{1,p}(U, Y;m)$ for given $v\in$$H^{1,p}(U, Y;m)$ .
$p$

38
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DEPARTMENT OF MATHEMATICS AND ENGINEERINGGRADUATE SCHOOL OF SCIENCE AND TECHNOLOGYKUMAMOTO UNIVERSITYKUMAMOTO, 860-S555
JAPAN-mail address: kuwae@gpo.kumamoto-u.ac. jp$E$