Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in ${\text {CAT}}(1)$ spaces

Abstract

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in ${\text {CAT}}(1)$ space. Under the setting, we prove that the Korevaar–Schoen energy admits a unique minimizer.

1 Introduction

1.1 ${\text {CAT}}(0)$ target

The study of harmonic maps between singular spaces is one of the central topics in geometric analysis since the pioneering work by Gromov and Schoen [24]. In an earlier stage, the theory has been developed by Korevaar and Schoen [30], Jost [26, 27] and Lin [32], independently. Korevaar and Schoen [30] have introduced an energy for $L^2$-maps from a Riemannian domain into a complete metric space and derived its basic properties, for instance, the existence of energy density, the consistency of Sobolev functions, and its lower semicontinuity. Furthermore, they have proven that if the target space is a ${\text {CAT}}(0)$ space (i.e., non-positively curved space in the sense of A. D. Alexandrov), then the energy admits a unique minimizer for the Dirichlet problem.

One of the research directions is to generalize the theory for non-smooth source spaces. Such attempts have been done by Gregori [16], Eells and Fuglede [11], Kuwae and Shioya [31], and so on. Gregori [16] and Eells and Fuglede [11] have dealt with a domain of a Lipschitz manifold and that of a Riemannian polyhedron, respectively. Kuwae and Shioya [31] have examined a metric measure space satisfying the so-called strongly measure contraction property of Bishop-Gromov type, called ${\text {SMCPBG}}$ space. A typical example of ${\text {SMCPBG}}$ space is an Alexandrov space, which is a metric space equipped with the notion of a lower sectional curvature bound.

In recent years, the theory of metric measure spaces with a lower Ricci curvature bound has been vastly developed. In the literature, the main research object is the so-called ${\text {RCD}}(K,N)$ space, which has been introduced by Ambrosio et al. [1] and Gigli [14]. It is natural to ask whether the Korevaar and Schoen theory can be extended to ${\text {RCD}}(K,N)$ spaces. Here we notice that an ${\text {RCD}}(K,N)$ space is not necessarily a ${\text {SMCPBG}}$ space (especially, in the collapsed case) since the Bishop-type inequality is required in the ${\text {SMCPBG}}$ condition, and hence the framework of Kuwae and Shioya [31] does not cover ${\text {RCD}}(K,N)$ spaces. Gigli and Tyulenev [22] were able to work in a more general context, covering the ${\text {RCD}}$ setting. In their framework, source spaces are assumed to be locally uniformly doubling, strongly rectifiable and to satisfy a Poincaré inequality (more precisely, see Subsection 2.2). Under this setting, they have defined a Korevaar–Schoen-type energy and deduced some fundamental results such as the existence of energy density and its lower semicontinuity. They have concluded the solvability of the Dirichlet problem when the target space is a ${\text {CAT}}(0)$ space (see [22, Theorem 6.4]).

Theorem 1.1

( [22]) Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be a bounded open subset of $\textrm{X}$ with ${{\mathfrak {m}}}(\textrm{X}{\setminus } \Omega )>0$. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed ${\text {CAT}}(0)$ space. Let ${\bar{u}}\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_\textrm{o})$ be a Korevaar–Schoen-type Sobolev map which determines a boundary value. Then the Korevaar–Schoen-type energy $\textsf{E}^{\Omega }_{2,{\bar{u}}}:L^2(\Omega ,\textrm{Y}_\textrm{o})\rightarrow [0,+\infty ]$ admits a unique minimizer.

Thanks to Theorem 1.1, one can introduce the notion of harmonic map from an ${\text {RCD}}(K,N)$ space to a ${\text {CAT}}(0)$ space. Very recently, Gigli [16] has shown a quantitative Lipschitz estimate for such harmonic maps and produced a Cheng-type Liouville theorem ( [7]) based on [10, 17, 21] (see also [40, 41]). Mondino and Semola [35] have also obtained a similar result, independently.

1.2 ${\text {CAT}}(1)$ target

The purpose of this note is to yield an analog of Theorem 1.1 for the case where the target space is a regular ball in a ${\text {CAT}}(1)$ space (i.e., geodesic ball whose radius is strictly less than $\pi /2$). In the case where the source space is a Riemannian domain, the solvability of the Dirichlet problem has been established by Serbinowski [37]. Similarly to the non-positively curved case, the result in [37] has been extended to non-smooth source spaces (see Eells and Fuglede [11], Fuglede [12, 13] for Riemannian polyhedra, and Huang and Zhang [25] for Alexandrov spaces). We now aim to generalize it for ${\text {RCD}}(K,N)$ spaces. Our main result is the following:

Theorem 1.2

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be an infinitesimally Hilbertian, locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be a bounded open subset of $\textrm{X}$ with ${{\mathfrak {m}}}(\textrm{X}{\setminus } \Omega )>0$. Let $\textrm{Y}_{\textrm{o}}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed ${\text {CAT}}(1)$ space, and let ${\bar{\textrm{B}}}_{\rho }(\textrm{o})$ be a regular ball (i.e., a closed ball of radius $\rho \in (0,\pi /2)$ centered at $\textrm{o}$ $)$. Let ${\bar{u}}\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(o))$ be a Korevaar–Schoen-type Sobolev map which determines a boundary value. Then the Korevaar–Schoen-type energy $\textsf{E}^{\Omega }_{2,{\bar{u}}}:L^2(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))\rightarrow [0,+\infty ]$ admits a unique minimizer.

Theorem 1.2 enables us to introduce the notion of harmonic map from an ${\text {RCD}}(K,N)$ space into a regular ball in a ${\text {CAT}}(1)$ space. When the source space is a Riemannian domain, Zhang et al. [41] have obtained a quantitative Lipschitz estimate and a Choi-type Liouville theorem ( [8]).

2 Preliminaries

We say that $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is a metric measure space if $(\textrm{X},\textsf{d})$ is a complete separable metric space, and ${{\mathfrak {m}}}$ is a non-negative Borel measure, which is finite on bounded sets. This section is devoted to preliminaries for metric measure spaces.

2.1 Sobolev functions

We briefly recall the non-smooth differential calculus on metric measure spaces. The readers can refer to [15, 18] for the details.

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a metric measure space and let $C([0,1],\textrm{X})$ be the set of all curves in $\textrm{X}$ defined on [0, 1] with the uniform topology. For $t\in [0,1]$ the evaluation map $\textrm{e}_t:C([0,1],\textrm{X})\rightarrow \textrm{X}$ is defined as $\textrm{e}_t(\gamma ):=\gamma _t$. A curve $\gamma \in C([0,1],\textrm{X})$ is said to be absolutely continuous if there is $f\in L^1(0,1)$ such that

$$\begin{aligned} \textsf{d}(\gamma _t,\gamma _s)\le \int _s^t f(r)\,\textrm{d}r \end{aligned}$$

(2.1)

for all $t,s\in [0,1]$ with $s< t$. For an absolutely continuous $\gamma$, the minimal f satisfying (2.1) in the a.e. sense is called the metric speed and it is denoted by $|{\dot{\gamma }}_t|$. A Borel probability measure $\pi$ on $C([0,1],\textrm{X})$ is said to be a test plan if

$$\begin{aligned} \int _0^1\int |{\dot{\gamma }}_t|^2\,d\pi (\gamma )\,\textrm{d}t<+\infty ,\quad (\textrm{e}_t)_{\#}\pi \le C\,{{\mathfrak {m}}}\end{aligned}$$

for some $C>0$. The Sobolev class $S^2(\textrm{X})$ is the set of all Borel functions $f:\textrm{X}\rightarrow {\mathbb {R}}$ such that there exists a non-negative $G\in L^2({{\mathfrak {m}}})$ such that

$$\begin{aligned} \int |f(\gamma _1)-f(\gamma _0)|\,\textrm{d}\pi (\gamma ) \le \int \int _0^1G(\gamma _t)|{\dot{\gamma }}_t|\,\textrm{d}t\,\textrm{d}\pi (\gamma ) \end{aligned}$$

(2.2)

for all test plans $\pi$. For $f\in S^2(\textrm{X})$, a non-negative function $G\in L^2({{\mathfrak {m}}})$ satisfying (2.2) is called a weak upper gradient, and the minimal one in the ${{\mathfrak {m}}}$-a.e. sense is called the minimal weak upper gradient and it is denoted by |Df|. The space

$$\begin{aligned} W^{1,2}(\textrm{X}):=L^2({{\mathfrak {m}}})\cap S^2(\textrm{X}) \end{aligned}$$

equipped with the norm

is called Sobolev space and it can be proved that it is a Banach space.

It is well-known that there exists a unique couple $(L^2(T^{*}\textrm{X}),\textrm{d})$, where $L^2(T^{*}\textrm{X})$ is an $L^2({{\mathfrak {m}}})$-normed $L^{\infty }({{\mathfrak {m}}})$-module and $\textrm{d}:S^2(\textrm{X})\rightarrow L^2(T^{*}\textrm{X})$ is a linear operator such that the following hold (see e.g., [15, 18, Theorem 4.1.1]):

• (1) $L^2(T^{*}\textrm{X})$ is generated by $\{\textrm{d}f \mid f\in S^2(\textrm{X})\}$;

• (2) for every $f\in S^2(\textrm{X})$, it holds that

  $$\begin{aligned} |\textrm{d}f|=|Df|\quad {{\mathfrak {m}}}\text {-a.e.}. \end{aligned}$$

The space $L^2(T^{*}\textrm{X})$ and the operator $\textrm{d}$ are called the cotangent module and the differential, respectively. The tangent module $L^2(T\textrm{X})$ is defined as the dual module of $L^2(T^{*}\textrm{X})$. For $f\in S^2(\textrm{X})$, let ${\text {Grad}}(f)$ be the set of all $v\in L^2(T\textrm{X})$ such that

$$\begin{aligned} \textrm{d}f(v)=|\textrm{d}f|^2=|v|^2\quad {{\mathfrak {m}}}\text {-a.e.}. \end{aligned}$$

Note that ${\text {Grad}}(f)$ is non-empty (see [18, Remark 4.2.10]). The following chain rule holds (see e.g., [18, Theorem 4.2.15]):

Theorem 2.1

([18]) Let $f \in S^2(\textrm{X})$ and $v\in {\text {Grad}}(f)$. If $\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ is Lipschitz, then $\varphi ' \circ f v\in {\text {Grad}}(\varphi \circ f)$, where $\varphi '\circ f$ is arbitrarily defined on the inverse image of the non-differentiability points of $\varphi$.

A metric measure space $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is said to be infinitesimally strictly convex when for every $f\in S^2(\textrm{X})$ the set ${\text {Grad}}(f)$ is a singleton; in this case, the unique element is denoted by $\nabla f$. Also, $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is said to be infinitesimally Hilbertian if $W^{1,2}(\textrm{X})$ is a Hilbert space.

We have the following characterization (see e.g., [18, Theorem 4.3.3]):

Theorem 2.2

([18]) The following are equivalent:

• (1) $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally Hilbertian;

• (2) $L^2(T^{*}\textrm{X})$ and $L^2(T\textrm{X})$ are Hilbert modules;

• (3) $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally strictly convex, and for all $f,g\in W^{1,2}(\textrm{X})$,

  $$\begin{aligned} \nabla (f+g)=\nabla f+\nabla g \quad {{\mathfrak {m}}}\text {-a.e.}; \end{aligned}$$

• (4) $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally strictly convex, and for all $f,g\in W^{1,2}(\textrm{X}) \cap L^{\infty }({{\mathfrak {m}}})$,

  $$\begin{aligned} \nabla (f g)=f \nabla g+g \nabla f \quad {{\mathfrak {m}}}\text {-a.e.}. \end{aligned}$$

In the case where $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally Hilbertian, we denote by $\langle \cdot ,\cdot \rangle$ the pointwise scalar product on $L^2(T\textrm{X})$.

2.2 Doubling, Poincaré and strong rectifiability

Here we review a volume doubling property, a Poincaré inequality and the strongly rectifiability on metric measure spaces.

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a metric measure space. For $r>0$ and $x\in \textrm{X}$, we denote by $B_r(x)$ the open ball of radius r centered at x. A metric measure space $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is said to be locally uniformly doubling if for every $R>0$ there exists $C_R>0$ such that

$$\begin{aligned} {{\mathfrak {m}}}(B_{2r}(x)) \le C_R \,{{\mathfrak {m}}}(B_r(x)) \end{aligned}$$

for all $x\in \textrm{X}$ and $r\in (0,R)$. We also say that $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ satisfies a Poincaré inequality if for every $R>0$ there are $C_R,\lambda =\lambda _R>0$ such that

holds for every Lipschitz function $f:\textrm{X}\rightarrow {\mathbb {R}}$ and for all $x\in \textrm{X}$ and $r\in (0,R)$, where and ${\text {lip}}(f)$ is the pointwise Lipschitz constant.

We say that a metric measure space $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is a d-dimensional strongly rectifiable space if for every $\varepsilon >0$ there exists an $\varepsilon$-atlas ${\mathcal {A}}^\varepsilon :=\{(U^\varepsilon _i,\varphi ^\varepsilon _i)\}_{i}$ such that the following hold:

• (1) Borel subsets $\{U^\varepsilon _i\}_i$ form a partition of $\textrm{X}$ up to an ${{\mathfrak {m}}}$-negligible set;

• (2) $\varphi ^\varepsilon _i$ is a $(1+\varepsilon )$-bi-Lipschitz map from $U^\varepsilon _i$ to $\varphi ^\varepsilon _i(U^\varepsilon _i)\subset {\mathbb {R}}^d$;

• (3) for some $c_i>0$, we have

  $$\begin{aligned} c_i\,{\mathcal {L}}^d|_{\varphi ^\varepsilon _i(U^\varepsilon _i)}\le (\varphi ^\varepsilon _i)_{\#}{{\mathfrak {m}}}|_{U^\varepsilon _i}\le (1+\varepsilon )c_i\,{\mathcal {L}}^d|_{\varphi ^\varepsilon _i(U^\varepsilon _i)}. \end{aligned}$$

Remark 2.3

For $K\in {\mathbb {R}}$ and $N\in [1,\infty )$, every ${\text {RCD}}(K,N)$ space is a locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality. Actually, the doubling property is due to [33, 38, 39]; the Poincaré inequality is due to [36]. Moreover, as stated in [22, Theorem 2.19], the strong rectifiability is a consequence of the results by Brué and Semola [6], Gigli and Pasqualetto [19], Kell and Mondino [28], Mondino and Naber [34], De Philippis et al. [9].

Remark 2.4

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a strongly rectifiable space, and let $\{\varepsilon _n\}_n$ be a sequence with $\varepsilon _n\downarrow 0$. A family $\{{\mathcal {A}}^{\varepsilon _n}\}_{n}$ of atlases is said to be aligned if for all n, m and $(U^{\varepsilon _n}_i,\varphi ^{\varepsilon _n}_i)\in {\mathcal {A}}^{\varepsilon _n}$, $(U^{\varepsilon _m}_j,\varphi ^{\varepsilon _m}_j)\in {\mathcal {A}}^{\varepsilon _m}$, the map $\varphi ^{\varepsilon _n}_i-\varphi ^{\varepsilon _m}_j$ is $(\varepsilon _n+\varepsilon _m)$-Lipschitz on $U^{\varepsilon _n}_i\cap U^{\varepsilon _m}_j$. It has been observed in [20, Theorem 3.9], [22, Subsection 2.4] that for any sequence $\{\varepsilon _n\}_n$, an aligned family of atlases $\{{\mathcal {A}}^{\varepsilon _n}\}_{n}$ exists.

2.3 Approximate metric differentiability

We further review the notion of approximate metric differentiability of maps introduced by Kirchheim [29] and Gigli and Tyulenev [22].

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a metric measure space, and let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a metric space. Let $u:\textrm{X}\rightarrow \textrm{Y}$ be a Borel map. The pointwise Lipschitz constant of u at $x\in \textrm{X}$ is defined as

$$\begin{aligned} {\text {lip}}(u)(x):=\varlimsup _{y\rightarrow x}\frac{\textsf{d}_\textrm{Y}(u(x),u(y))}{\textsf{d}(x,y)} \end{aligned}$$

if x is not an isolated point, and ${\text {lip}}(u)(x):=0$ otherwise. For a Borel subset $U\subset \textrm{X}$, a point $x\in U$ is said to be a density point of U if

$$\begin{aligned} \lim _{r\downarrow 0} \frac{{{\mathfrak {m}}}(B_r(x)\cap U)}{{{\mathfrak {m}}}(B_r(x))}=1. \end{aligned}$$

The approximate Lipschitz constant is defined by

$$\begin{aligned} {{\,\mathrm{\mathrm{ap\,-lip}}\,}}(u)(x):=\mathop {\mathrm {\mathrm{ap\,-}\varlimsup }}\limits _{y\rightarrow x}\frac{\textsf{d}_\textrm{Y}(u(x),u(y))}{\textsf{d}(x,y)}, \end{aligned}$$

where the right-hand side means the approximate upper limit (see e.g., [22, Subsection 2.1] for the precise definition). We have the following (see [22, Proposition 2.5]):

Proposition 2.5

([22]) Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a uniformly locally doubling space, and let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a complete metric space. Let U be a Borel subset of $\textrm{X}$, and let $u:U\rightarrow \textrm{X}$ be Lipschitz. Then for every density point $x\in \textrm{X}$ of U we have

$$\begin{aligned} {\text {lip}}(u)(x)={{\,\mathrm{\mathrm{ap\,-lip}}\,}}(u)(x). \end{aligned}$$

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a d-dimensional strongly rectifiable space, and let $\{{\mathcal {A}}^{\varepsilon _n}\}_n$ be an aligned family of atlases for a sequence $\{\varepsilon _n\}_n$ with $\varepsilon _n\downarrow 0$. Let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a metric space. We denote by $\textsf{sn}^d$ the set of all semi-norms on ${\mathbb {R}}^d$ equipped with the complete separable metric

$$\begin{aligned} \textsf{D}({{\mathfrak {n}}}_1,{{\mathfrak {n}}}_2):=\sup _{z}|{{\mathfrak {n}}}_1(z)-{{\mathfrak {n}}}_2(z)|, \end{aligned}$$

where the supremum is taken over all $z\in {\mathbb {R}}^d$ with $|z|\le 1$. We write $\Vert {{\mathfrak {n}}}\Vert :=\textsf{D}({{\mathfrak {n}}},0)$. We say that a Borel map $u:\textrm{X}\rightarrow \textrm{Y}$ is approximately metrically differentiable at $x\in \textrm{X}$ relatively to $\{{\mathcal {A}}^{\varepsilon _n}\}_n$ if the following hold:

• (1) For every n, there exists $i=i_{x,n}$ such that x belongs to $U_{i}^{\varepsilon _n}$, it is a density point of $U_{i}^{\varepsilon _n}$ and $\varphi ^{\varepsilon _n}_{i}(x)$ is a density point of $\varphi ^{\varepsilon _n}_{i}(U^{\varepsilon _n}_{i})$;

• (2) there exists $\textsf{md}_x(u)\in \textsf{sn}^d$, called the metric differential of u at x, such that

  $$\begin{aligned} \varlimsup _{n\rightarrow \infty }\mathop {\mathrm {\mathrm{ap\,-}\varlimsup }}\limits _{y\rightarrow x,\,y\in U_{i}^{\varepsilon _n}}\frac{\big |\textsf{d}_\textrm{Y}(u(y),u(x))-\textsf{md}_x(u)(\varphi ^{\varepsilon _n}_{i}(y)-\varphi ^{\varepsilon _n}_{i}(x))\big |}{\textsf{d}(y,x)}=0. \end{aligned}$$

Moreover, Gigli and Tyulenev [22] have shown the following (see [22, Lemma 3.4]):

Lemma 2.6

([22]) Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a strongly rectifiable space, and let $\{{\mathcal {A}}^{\varepsilon _n}\}_n$ be an aligned family of atlases for a sequence $\{\varepsilon _n\}_n$ with $\varepsilon _n\downarrow 0$. Let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a complete metric space. If $u:\textrm{X}\rightarrow \textrm{Y}$ is approximately metrically differentiable at $x\in \textrm{X}$ relatively to $\{{\mathcal {A}}^{\varepsilon _n}\}_n$, then

$$\begin{aligned} {{\,\mathrm{\mathrm{ap\,-lip}}\,}}(u)(x)=\Vert \textsf{md}_x(u) \Vert . \end{aligned}$$

Gigli and Tyulenev [22] have further concluded the ${{\mathfrak {m}}}$-a.e. approximately metrical differentiability of Borel maps satisfying the Lusin–Lipschitz property (see [22, Proposition 3.6]).

2.4 Korevaar–Schoen space

In this section we recall the formulation and basic results concerning the Korevaar–Schoen-type energy introduced in Gigli and Tyulenev [22]. Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a metric measure space, and let $\Omega$ be an open subset of $\textrm{X}$. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed complete metric space.

We denote by $L^0(\Omega ,\textrm{Y})$ the set of all Borel maps (up to ${{\mathfrak {m}}}$-a.e. equality) from $\Omega$ to $\textrm{Y}$ with separable range. Let $L^2(\Omega ,\textrm{Y}_\textrm{o})$ be the set of all $u\in L^0(\Omega ,\textrm{Y})$ such that

$$\begin{aligned} \int _{\Omega } \textsf{d}_{\textrm{Y}}^2(u(x),\textrm{o})\,\textrm{d}{{\mathfrak {m}}}(x)<+\infty , \end{aligned}$$

which is endowed with the metric

$$\begin{aligned} \textsf{d}_{L^2}(u,v):=\left| \int _{\Omega } \textsf{d}_\textrm{Y}^2\big (u(x),v(x)\big )\,\textrm{d}{{\mathfrak {m}}}(x)\right| ^{1/2}. \end{aligned}$$

Since $\textrm{Y}$ is complete, it can be proved that so is $L^2(\Omega ,\textrm{Y}_{\textrm{o}})$.

Let $u\in L^2(\Omega ,\textrm{Y}_{\textrm{o}})$. For $r>0$ and $x\in \Omega$, the approximate energy is defined by

The Korevaar–Schoen space $\textsf{KS}^{1,2}(\Omega ,\textrm{Y}_\textrm{o})$ is defined by the set of all $u\in L^2(\Omega ,\textrm{Y}_\textrm{o})$ such that

$$\begin{aligned} \textsf{E}_2^\Omega (u):=\varliminf _{r\downarrow 0}\,\int _\Omega \,\textsf{ks}^2_{2,r}[u,\Omega ]\,\textrm{d}{{\mathfrak {m}}}<+\infty . \end{aligned}$$

(2.3)

Remark 2.7

Let $W^{1,2}(\Omega )$ denote the Sobolev space provided by means of [22, Definition 5.2]. One can also introduce the associated Sobolev space $W^{1,2}(\Omega ,\textrm{Y}_{\textrm{o}})$ as the set of all $u\in L^2(\Omega ,\textrm{Y}_{\textrm{o}})$ such that there is $G\in L^2(\Omega )$ such that for all 1-Lipschitz functions $f:\textrm{Y}\rightarrow {\mathbb {R}}$ we have $f\circ u\in W^{1,2}(\Omega )$ and $|D(f\circ u)|\le G$ ${{\mathfrak {m}}}$-a.e. on $\Omega$ (see [22, Definition 5.3]).

We now present some basic properties of the Korevaar–Schoen space $\textsf{KS}^{1,2}(\Omega ,\textrm{Y}_\textrm{o})$ (see [22, Theorem 5.7]):

Theorem 2.8

([22]) Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be an open subset. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed complete metric space. Then the following hold:

• (1) $\textsf{KS}^{1,2}(\Omega ,\textrm{Y}_\textrm{o})=W^{1,2}(\Omega ,\textrm{Y}_\textrm{o})$;

• (2) for every $u\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_{\textrm{o}})$ there exists a function $\textrm{e}_2[u]\in L^2(\Omega )$, called the energy density, such that

  $$\begin{aligned} \textsf{ks}_{2,r}[u,\Omega ] \rightarrow \textrm{e}_2[u]\quad {{\mathfrak {m}}}\hbox { -a.e. on } \Omega \hbox { and in } L^2(\Omega ) \hbox { as } r\downarrow 0. \end{aligned}$$

  In particular $\liminf$ in (2.3) is a limit and $\textsf{E}_2^\Omega (u)$ can be written as

  $$\begin{aligned} \textsf{E}_2^\Omega (u)= {\left\{ \begin{array}{ll} \displaystyle {\int _\Omega \textrm{e}_2^2[u]\, \textrm{d}{{\mathfrak {m}}}} &{} \text {if}\ u\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_{\textrm{o}}),\\ +\infty &{} \text {otherwise}; \end{array}\right. } \end{aligned}$$

• (3) $\textsf{E}^\Omega _2:L^2(\Omega ,\textrm{Y}_{\textrm{o}})\rightarrow [0,+\infty ]$ is lower semicontinuous;

• (4) any $u\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_\textrm{o})$ is approximately metrically differentiable ${{\mathfrak {m}}}$-a.e. in $\Omega$, where we extend u to the whole $\textrm{X}$ by setting it to be constant outside of $\Omega$.

We also need the following (see [22, Propositions 4.6 and 4.19]):

Proposition 2.9

([22]) Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a d-dimensional, locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be an open subset. Then there exists $c_d>0$ depending only on d such that for every $u\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$,

$$\begin{aligned} \textrm{e}_2[u]=c_d\,|Du| \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega . \end{aligned}$$

(2.4)

Let $W^{1,2}_0(\Omega )$ be the $W^{1,2}(\Omega )$-closure of the set of all functions in $W^{1,2}(\Omega )$ whose support is contained in $\Omega$. For a fixed ${{\bar{u}}}\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_{\textrm{o}})$, we define

$$\begin{aligned} \textsf{KS}^{1,2}_{{{\bar{u}}}}(\Omega ,\textrm{Y}_{\textrm{o}}):=\big \{u\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_{\textrm{o}}) \mid \textsf{d}_\textrm{Y}(u,{{\bar{u}}})\in W^{1,2}_0(\Omega )\big \}, \end{aligned}$$

and the associated energy functional $\textsf{E}^\Omega _{2,{{\bar{u}}}}:L^2(\Omega ,\textrm{Y}_\textrm{o})\rightarrow [0,+\infty ]$ as

$$\begin{aligned} \textsf{E}^\Omega _{2,{{\bar{u}}}}(u):= {\left\{ \begin{array}{ll} \displaystyle {\textsf{E}_2^\Omega (u)} &{} \text {if}\ u\in \textsf{KS}^{1,2}_{{{\bar{u}}}}(\Omega ,\textrm{Y}_{\textrm{o}}),\\ +\infty &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

We close this section with the following (see [22, Proposition 5.10]):

Proposition 2.10

([22]) Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be an open subset. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed complete metric space. Then for a fixed ${{\bar{u}}}\in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_{\textrm{o}})$ the following hold:

• (1) $\textsf{E}^\Omega _{2,{{\bar{u}}}}$ is lower semicontinuous;

• (2) for any $u,v\in \textsf{KS}^{1,2}_{{{\bar{u}}}}(\Omega ,\textrm{Y}_\textrm{o})$ we have $\textsf{d}_\textrm{Y}(u,v)\in W^{1,2}_0(\Omega )$.

3 Dirichlet problem

In the present section, we give a proof of Theorem 1.2. We proceed along the line of the proof of [37, Theorem 1.16], [22, Theorem 6.4].

3.1 Technical lemma

This subsection is devoted to the proof of a technical lemma. Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a metric measure space, and let $\Omega$ be an open subset of $\textrm{X}$. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed complete metric space. Let $u,v,w \in L^2(\Omega ,\textrm{Y}_{\textrm{o}})$. For $\alpha >0$ and $x\in \Omega$, we define the following (Borel) sets

$$\begin{aligned} \Omega _{v,\alpha ,x}&:=\{y\in \Omega \mid \textsf{d}_\textrm{Y}(v(x),v(y))\ge \alpha \},\nonumber \\ \Omega _{w,\alpha ,x}&:=\{y\in \Omega \mid \textsf{d}_\textrm{Y}(w(x),w(y))\ge \alpha \},\nonumber \\ \Omega _{v,w,\alpha ,x}&:=\Omega _{v,\alpha ,x}\cup \Omega _{w,\alpha ,x}. \end{aligned}$$

(3.1)

We also define a modified approximate energy by

$$\begin{aligned} \textsf{ks}_{2,r}[u,\Omega ;v,w,\alpha ](x):= {\left\{ \begin{array}{ll} \displaystyle {\left| \frac{1}{{{\mathfrak {m}}}(B_r(x))}\int _{B_r(x)\setminus \Omega _{v,w,\alpha ,x}} \frac{\textsf{d}^2_\textrm{Y}(u(x),u(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y) \right| ^{1/2}} &{} \text {if}\ B_{r}(x)\subset \Omega ,\\ 0 &{} \text {otherwise}. \end{array}\right. }\nonumber \\ \end{aligned}$$

(3.2)

We now state and prove our key lemma (cf. [37, Lemma 1.12], [25, Lemma 3.1]):

Lemma 3.1

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be a locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be an open subset. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed complete metric space. Let $u,v,w \in \textsf{KS}^{1,2}(\Omega ,\textrm{Y}_\textrm{o})$ and $\alpha >0$ fixed. Then we have

$$\begin{aligned} \textsf{ks}_{2,r}[u,\Omega ;v,w,\alpha ] \rightarrow \textrm{e}_2[u]\quad {{\mathfrak {m}}}\hbox { -a.e. on } \Omega \hbox { and in } L^2(\Omega )\hbox { as }r\downarrow 0. \end{aligned}$$

(3.3)

Proof

We first prove the ${{\mathfrak {m}}}$-a.e. convergence. Note that for ${{\mathfrak {m}}}$-a.e. $x\in \Omega$ the following hold:

• (1) $\textsf{ks}_{2,r}[u,\Omega ](x)\rightarrow \textrm{e}_2[u](x), \textsf{ks}_{2,r}[v,\Omega ](x)\rightarrow \textrm{e}_2[v](x),\textsf{ks}_{2,r}[w,\Omega ](x)\rightarrow \textrm{e}_2[w](x)$ as $r\downarrow 0$;

• (2) ${\text {lip}}u(x)<+\infty$.

Indeed, the first property is a consequence of Theorem 2.8. Concerning the second one, Theorem 2.8 implies that u is approximately metrically differentiable ${{\mathfrak {m}}}$-a.e. $x\in \Omega$ relatively to an aligned family $\{{\mathcal {A}}^{\varepsilon _n}\}_n$ of atlases. For such an $x\in \Omega$, and for each n, there is $i=i_{x,n}$ such that x belongs to a chart $U_{i}^{\varepsilon _n}$ of ${\mathcal {A}}^{\varepsilon _n}$, and it is a density point of $U_{i}^{\varepsilon _n}$. Thanks to [22, (2.15), Corollary 3.10], we may assume that $u|_{U_{i}^{\varepsilon _n}}$ is Lipschitz. By Proposition 2.5 and Lemma 2.6, ${\text {lip}}u(x)$ coincides with $\Vert \textsf{md}_x(u)\Vert$, which is finite.

Let $x\in \Omega$ be such that two properties above hold: we shall prove (3.3) for such an x. Since

$$\begin{aligned} \textsf{ks}^2_{2,r}[u,\Omega ;v,w,\alpha ](x)=\textsf{ks}^2_{2,r}[u,\Omega ](x)-\frac{1}{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x)\cap \Omega _{v,w,\alpha ,x}} \frac{\textsf{d}^2_\textrm{Y}(u(x),u(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y), \end{aligned}$$

it suffices to show that

$$\begin{aligned} \frac{1}{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x)\cap \Omega _{v,w,\alpha ,x}} \frac{\textsf{d}^2_\textrm{Y}(u(x),u(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\rightarrow 0 \end{aligned}$$

(3.4)

as $r\downarrow 0$. We have

$$\begin{aligned}&\frac{1}{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x)\cap \Omega _{v,w,\alpha ,x}} \frac{\textsf{d}^2_\textrm{Y}(u(x),u(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\\&\quad \le \frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{v,w,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))} \sup _{y\in B_r(x)\setminus \{x\}}\frac{\textsf{d}^2_{\textrm{Y}}(u(x),u(y))}{\textsf{d}^2(x,y)}\\&\quad \le \left( \frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{v,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}+\frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{w,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))} \right) \sup _{y\in B_r(x)\setminus \{x\}}\frac{\textsf{d}^2_{\textrm{Y}}(u(x),u(y))}{\textsf{d}^2(x,y)}\\&\quad \le \frac{r^2}{\alpha ^2} \left( \textsf{ks}^2_{2,r}[v,\Omega ](x)+\textsf{ks}^2_{2,r}[w,\Omega ](x) \right) \sup _{y\in B_r(x)\setminus \{x\}}\frac{\textsf{d}^2_{\textrm{Y}}(u(x),u(y))}{\textsf{d}^2(x,y)}, \end{aligned}$$

where the last inequality follows from

$$\begin{aligned} \frac{\alpha ^2}{r^2} \frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{v,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}&\le \frac{1}{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x)\cap \Omega _{v,\alpha ,x}} \frac{\textsf{d}^2_\textrm{Y}(v(x),v(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\le \textsf{ks}^2_{2,r}[v,\Omega ](x),\\ \frac{\alpha ^2}{r^2} \frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{w,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}&\le \frac{1}{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x)\cap \Omega _{w,\alpha ,x}} \frac{\textsf{d}^2_\textrm{Y}(w(x),w(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\le \textsf{ks}^2_{2,r}[w,\Omega ](x). \end{aligned}$$

Therefore, by letting $r\downarrow 0$ we obtain (3.4) and the sought ${{\mathfrak {m}}}$-a.e. convergence. The $L^2$-convergence follows from $\textsf{ks}^2_{2,r}[u,\Omega ;v,w,\alpha ]\le \textsf{ks}^2_{2,r}[u,\Omega ]$ and [22, Lemma 3.14]. $\square$

3.2 ${\text {CAT}}(\kappa )$ spaces

In this subsection we shall review some basic properties of ${\text {CAT}}(\kappa )$ spaces. We refer to [3, 5]. For $\kappa \in {\mathbb {R}}$, let $M_\kappa$ be the 2-dimensional space form of constant curvature $\kappa$. Let $D_{\kappa }$ be the diameter of $M_{\kappa }$; more precisely,

$$\begin{aligned} D_\kappa := {\left\{ \begin{array}{ll} \pi /\sqrt{\kappa } &{} \text { if }\ \kappa >0,\\ +\infty &{} \text { otherwise }. \end{array}\right. } \end{aligned}$$

Let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a metric space. A curve $\gamma :[0,1]\rightarrow \textrm{Y}$ is said to be a geodesic if

$$\begin{aligned} \textsf{d}_\textrm{Y}(\gamma _t,\gamma _s)=|s-t|\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1) \end{aligned}$$

for all $t,s\in [0,1]$. Further, $(\textrm{Y},\textsf{d})$ is called a geodesic space if for any pair of points $x,y\in \textrm{Y}$, there is a geodesic $\gamma :[0,1]\rightarrow \textrm{Y}$ with $\gamma _0=x$ and $\gamma _1=y$. For $\kappa \in {\mathbb {R}}$, a complete geodesic space $(\textrm{Y},\textsf{d}_\textrm{Y})$ is said to be a ${\text {CAT}}(\kappa )$-space if it satisfies the following $\kappa$-triangle comparison principle: for every geodesic triangle $\triangle _{pqr}$ in $\textrm{Y}$ whose perimeter is less than $2 D_{\kappa }$, and for every point x on the segment between q and r, it holds that

$$\begin{aligned} \textsf{d}_{\textrm{Y}}(p,x)\le \textsf{d}_{\kappa }({\bar{p}},{\bar{x}}), \end{aligned}$$

where $\textsf{d}_{\kappa }$ is the Riemannian distance on $M_\kappa$, and ${\bar{p}}$ and ${\bar{x}}$ are comparison points in $M_\kappa$.

Let us recall the following basic properties (see e.g., [5, Proposition 1.4]):

Proposition 3.2

([5]) For $\kappa \in {\mathbb {R}}$, let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a ${\text {CAT}}(\kappa )$ space. Then we have:

• (1) For each pair of points $x,y\in \textrm{Y}$ with $\textsf{d}_\textrm{Y}(x,y)<D_{\kappa }$, there exists a unique geodesic from x to y; moreover, it varies continuously with its end points;

• (2) any ball in $\textrm{Y}$ whose radius is less than $D_{\kappa }/2$ is convex; namely, for each pair of points in such a ball, the unique geodesic joining them is contained the ball.

Remark 3.3

In [5], Proposition 3.2 has been deduced from the following: let $\gamma ,\eta :[0,1]\rightarrow \textrm{Y}$ be geodesics such that

$$\begin{aligned} \gamma _0=\eta _0,\quad \textsf{d}_\textrm{Y}(\gamma _0,\gamma _1)+\textsf{d}_\textrm{Y}(\gamma _1,\eta _1)+\textsf{d}_\textrm{Y}(\eta _1,\eta _0)<2D_{\kappa }. \end{aligned}$$

For $l\in (0,D_{\kappa })$, we assume $\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1)\le l$ and $\textsf{d}_\textrm{Y}(\eta _0,\eta _1)\le l$. Then there is a constant $c_{\kappa ,l}>0$ depending only on $\kappa ,l$ such that for all $t\in [0,1]$,

$$\begin{aligned} \textsf{d}_\textrm{Y}(\gamma _t,\eta _t)\le c_{\kappa ,l}\,\textsf{d}_\textrm{Y}(\gamma _1,\eta _1). \end{aligned}$$

(3.5)

We now collect some useful estimates on ${\text {CAT}}(1)$ spaces, which have been stated in [37] without proof. The detailed proofs can be found in [4, Appendix A]. The first one is the following (see [37, p. 11, ESTIMATE I], and also [4, Lemma A.2]):

Lemma 3.4

([37]) Let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a ${\text {CAT}}(1)$ space, and let $\gamma ,\eta :[0,1]\rightarrow \textrm{Y}$ be geodesics with

$$\begin{aligned} \textsf{d}_\textrm{Y}(\gamma _0,\eta _0)+\textsf{d}_\textrm{Y}(\eta _0,\eta _1)+\textsf{d}_\textrm{Y}(\eta _1,\gamma _1)+\textsf{d}_\textrm{Y}(\gamma _1,\gamma _0)<2\pi . \end{aligned}$$

Then we have

$$\begin{aligned} \cos ^2 \frac{\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1)}{2}&\,\textsf{d}^2_\textrm{Y}(\gamma _{1/2},\eta _{1/2})+\frac{1}{4}(\textsf{d}_\textrm{Y}(\eta _0,\eta _1)-\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1))^2\\&\le \frac{1}{2}\left( \textsf{d}^2_\textrm{Y}(\gamma _0,\eta _0)+\textsf{d}^2_\textrm{Y}(\gamma _1,\eta _1) \right) \\&\quad +{\text {Cub}}(\textsf{d}_\textrm{Y}(\gamma _0,\eta _0),\textsf{d}_\textrm{Y}(\gamma _1,\eta _1),\textsf{d}_\textrm{Y}(\eta _0,\eta _1)-\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1),\textsf{d}_\textrm{Y}(\gamma _{1/2},\eta _{1/2})), \end{aligned}$$

where ${\text {Cub}}$ denotes cubic terms in the indicated variables.

The second one is the following (see [37, p. 13, ESTIMATE II], and also [4, Lemma A.4]):

Lemma 3.5

([37]) Let $(\textrm{Y},\textsf{d}_\textrm{Y})$ be a ${\text {CAT}}(1)$ space, and let $\gamma ,\eta :[0,1]\rightarrow \textrm{Y}$ be geodesics with $\gamma _1=\eta _1$ and

$$\begin{aligned} \textsf{d}_\textrm{Y}(\gamma _0,\eta _0)+\textsf{d}_\textrm{Y}(\eta _0,\eta _1)+\textsf{d}_\textrm{Y}(\eta _1,\gamma _0)<2\pi . \end{aligned}$$

Then for every $t,s\in [0,1]$ we have

$$\begin{aligned} \textsf{d}^2_\textrm{Y}(\gamma _t,\eta _s)&\le \frac{\sin ^2(1-t)\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1)}{\sin ^2 \textsf{d}_\textrm{Y}(\gamma _0,\gamma _1) }\left( \textsf{d}^2_{\textrm{Y}}(\gamma _0,\eta _0)-(\textsf{d}_\textrm{Y}(\eta _0,\eta _1)-\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1))^2 \right) \\&+(1-t)^2(\textsf{d}_\textrm{Y}(\eta _0,\eta _1)-\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1))^2+\textsf{d}^2_\textrm{Y}(\gamma _0,\gamma _1)(s-t)^2\\&-2(1-t)(s-t)\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1)(\textsf{d}_\textrm{Y}(\eta _0,\eta _1)-\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1))\\&+{\text {Cub}}(s-t,\textsf{d}_{\textrm{Y}}(\gamma _0,\eta _0),\textsf{d}_\textrm{Y}(\eta _0,\eta _1)-\textsf{d}_\textrm{Y}(\gamma _0,\gamma _1),\textsf{d}_\textrm{Y}(\gamma _t,\eta _s)). \end{aligned}$$

Remark 3.6

In Lemmas 3.4 and 3.5, we further see that in each term of ${\text {Cub}}$, at least one of the variables has an exponent which is greater than or equal to two.

3.3 Energy estimates

In this subsection we are going to obtain some energy estimates which will be exploited in the proof of our main theorem. Let $(\textrm{X},\textrm{d},{{\mathfrak {m}}})$ be a locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be an open subset of X. Let $\textrm{Y}_\textrm{o}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed ${\text {CAT}}(1)$ space. For $\rho \in (0,\pi /2)$, we denote by ${\bar{\textrm{B}}}_{\rho }(\textrm{o})$ the closed ball in $\textrm{Y}$ of radius $\rho$ centered at $\textrm{o}$. In view of Proposition 3.2, ${\bar{\textrm{B}}}_{\rho }(\textrm{o})=({\bar{\textrm{B}}}_{\rho }(\textrm{o}),\textsf{d}_\textrm{Y},\textrm{o})$ is a pointed ${\text {CAT}}(1)$ space with itself. We call ${\bar{\textrm{B}}}_{\rho }(\textrm{o})$ a regular ball.

For $u,v\in L^0(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$ and $t\in [0,1]$, we define a map $\textsf{G}^{u,v}_t:\Omega \rightarrow {\bar{\textrm{B}}}_{r}(\textrm{o})$ by $x\mapsto \textsf{G}^{u(x),v(x)}_t$, where $\textsf{G}^{u(x),v(x)}$ denotes the unique geodesic from u(x) to v(x). By virtue of Proposition 3.2, $\textsf{G}^{u,v}_t$ belongs to $L^0(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$. By the same argument as in [22, Section 6], we see that if $u,v\in L^2(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$, then $\textsf{G}^{u,v}_t\in L^2(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$, and it is the unique geodesic from u to v.

We begin with the following energy density estimate (cf. [37, Lemma 1.13]):

Lemma 3.7

Let $u,v\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o})), m:=\textsf{G}^{u,v}_{1/2}$ and $\textsf{d}:=\textsf{d}_\textrm{Y}(u,v)$. Then we have $m \in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o})), \textsf{d}\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$ and

$$\begin{aligned} \cos ^2 \frac{\textsf{d}}{2}\, \textrm{e}^2_2[m]+\frac{1}{4}\textrm{e}^2_2[\textsf{d}]\le \frac{1}{2}\left( \textrm{e}^2_2[u]+\textrm{e}^2_{2}[v] \right) \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega . \end{aligned}$$

(3.6)

Proof

We check $m \in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o})), \textsf{d}\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$. The claim for $\textsf{d}$ immediately follows from the triangle inequality. For what concerns m let us set $\alpha :=2(\pi -2\rho )>0$. For $x\in \Omega$, we define $\Omega _{u,\alpha ,x},\Omega _{v,\alpha ,x},\Omega _{u,v,\alpha ,x}$ as in (3.1). For all $y\in \Omega {\setminus } \Omega _{u,v,\alpha ,x}$, we have

$$\begin{aligned} \textsf{d}_\textrm{Y}(u(x),v(x))+\textsf{d}_\textrm{Y}(v(x),u(y))+\textsf{d}_\textrm{Y}(u(y),u(x))< 4\rho +\alpha =2\pi ,\\ \textsf{d}_\textrm{Y}(v(y),u(y))+\textsf{d}_\textrm{Y}(u(y),v(x))+\textsf{d}_\textrm{Y}(v(x),v(y))< 4\rho +\alpha =2\pi . \end{aligned}$$

Therefore, by using (3.5) for $l=2\rho$ twice via the midpoint of u(y) and v(x), for all $y\in \Omega {\setminus } \Omega _{u,v,\alpha ,x}$,

$$\begin{aligned} \textsf{d}_\textrm{Y}(m(x),m(y))\le c_{\rho }(\textsf{d}_\textrm{Y}(u(x),u(y))+\textsf{d}_\textrm{Y}(v(x),v(y)) \end{aligned}$$

for some constant $c_\rho >0$ depending only on $\rho$. This implies

$$\begin{aligned}&\textsf{ks}^2_{2,r}[m,\Omega ](x)\le \frac{2c^2_\rho }{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x) \setminus \Omega _{u,v,\alpha ,x}}\, \left( \frac{\textsf{d}^2_\textrm{Y}(u(x),u(y))}{r^2}+\frac{\textsf{d}^2_\textrm{Y}(v(x),v(y))}{r^2}\right) \,\textrm{d}{{\mathfrak {m}}}(y)\nonumber \\&\quad + \frac{1}{{{\mathfrak {m}}}(B_r(x))} \int _{B_r(x) \cap \Omega _{u,v,\alpha ,x}}\, \frac{\textsf{d}^2_\textrm{Y}(m(x),m(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\nonumber \\&\le 2c^2_\rho \,\left( \textsf{ks}^2_{2,r}[u,\Omega ](x)+\textsf{ks}^2_{2,r}[v,\Omega ](x)\right) + \frac{4\rho ^2}{r^2} \frac{{{\mathfrak {m}}}(B_r(x) \cap \Omega _{u,v,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}\nonumber \\&\le \left( 2c^2_\rho +\frac{4\rho ^2}{\alpha ^2}\right) \,\left( \textsf{ks}^2_{2,r}[u,\Omega ](x)+\textsf{ks}^2_{2,r}[v,\Omega ](x)\right) , \end{aligned}$$

(3.7)

where we used

$$\begin{aligned} \frac{\alpha ^2}{r^2} \frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{u,v,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}&\le \frac{\alpha ^2}{r^2} \left( \frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{u,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}+\frac{{{\mathfrak {m}}}(B_r(x)\cap \Omega _{v,\alpha ,x})}{{{\mathfrak {m}}}(B_r(x))}\right) \\&\le \textsf{ks}^2_{2,r}[u,\Omega ](x)+\textsf{ks}^2_{2,r}[v,\Omega ](x). \end{aligned}$$

Integrating (3.7) over $\Omega$, and letting $r\downarrow 0$, we deduce $m \in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$.

We prove (3.6). Now let us set $\beta :=\pi -2\rho >0$ and for $x\in \Omega$ define $\Omega _{u,\beta ,x},\Omega _{v,\beta ,x},\Omega _{u,v,\beta ,x}$ as in (3.1) again. For all $y\in \Omega {\setminus } \Omega _{u,v,\beta ,x}$, there holds

$$\begin{aligned} \textsf{d}_\textrm{Y}(u(x),v(x))+\textsf{d}_\textrm{Y}(v(x),v(y))+\textsf{d}_\textrm{Y}(v(y),u(y))+\textsf{d}_\textrm{Y}(u(y),u(x))< 2\beta +4\rho =2\pi . \end{aligned}$$

Lemma 3.4 tells us that for all $y\in \Omega {\setminus } \Omega _{u,v,\beta ,x}$,

$$\begin{aligned} \cos ^2\frac{\textsf{d}(x)}{2}&\,\textsf{d}^2_\textrm{Y}(m(x),m(y))+\frac{1}{4}(\textsf{d}(x)-\textsf{d}(y))^2\\&\le \frac{1}{2} \left( \textsf{d}^2_\textrm{Y}(u(x),u(y))+\textsf{d}^2_\textrm{Y}(v(x),v(y)) \right) \\&\quad +{\text {Cub}}(\textsf{d}_\textrm{Y}(u(x),u(y)),\textsf{d}_\textrm{Y}(v(x),v(y)),\textsf{d}(x)-\textsf{d}(y),\textsf{d}_\textrm{Y}(m(x),m(y))). \end{aligned}$$

We shall write the last term as ${\text {Cub}}(x,y)$ for short. It follows that

$$\begin{aligned}&\cos ^2\frac{\textsf{d}(x)}{2}\,\textsf{ks}^2_{2,r}[m,\Omega ;u,v,\beta ](x)+\frac{1}{4}\textsf{ks}^2_{2,r}[\textsf{d},\Omega ;u,v,\beta ](x)\nonumber \\&\quad \le \frac{1}{2}\left( \textsf{ks}^2_{2,r}[u,\Omega ](x)+\textsf{ks}^2_{2,r}[v,\Omega ](x) \right) +\frac{1}{{{\mathfrak {m}}}(B_r(x))}\int _{B_r(x) \setminus \Omega _{u,v,\beta ,x}}\,\frac{{\text {Cub}}(x,y)}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y), \end{aligned}$$

(3.8)

where $\textsf{ks}^2_{2,r}[m,\Omega ;u,v,\beta ](x)$ and $\textsf{ks}^2_{2,r}[\textsf{d},\Omega ;u,v,\beta ](x)$ are defined as (3.2).

By Theorem 2.8 and Lemma 3.1, and by the same argument in the proof of Lemma 3.1, ${{\mathfrak {m}}}$-a.e. $x\in \Omega$ satisfies the following properties:

• (1) $\textsf{ks}_{2,r}[m,\Omega ;u,v,\beta ](x)\rightarrow \textrm{e}_2[m](x), \textsf{ks}_{2,r}[\textsf{d},\Omega ;u,v,\beta ](x) \rightarrow \textrm{e}_2[\textsf{d}](x)$ as $r\downarrow 0$;

• (2) $\textsf{ks}_{2,r}[u,\Omega ](x)\rightarrow \textrm{e}_2[u](x), \textsf{ks}_{2,r}[v,\Omega ](x)\rightarrow \textrm{e}_2[v](x)$ as $r\downarrow 0$;

• (3) ${\text {lip}}u(x),{\text {lip}}v(x),{\text {lip}}\textsf{d}(x),{\text {lip}}m(x)<+\infty$.

We now fix such a point $x\in \Omega$. Let us verify

$$\begin{aligned} \frac{1}{{{\mathfrak {m}}}(B_r(x))}\int _{B_r(x) \setminus \Omega _{u,v,\beta ,x}}\,\frac{{\text {Cub}}(x,y)}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\rightarrow 0 \end{aligned}$$

(3.9)

as $r\downarrow 0$. Each term in (3.9) can be written as

$$\begin{aligned} \frac{f(x)}{{{\mathfrak {m}}}(B_r(x))}\int _{B_r(x) \setminus \Omega _{u,v,\beta ,x}}\,\frac{\textsf{d}^{\theta _1}_\textrm{Y}(u(x),u(y))\textsf{d}^{\theta _2}_\textrm{Y}(v(x),v(y))(\textsf{d}(x)-\textsf{d}(y))^{\theta _3}\textsf{d}^{\theta _4}_\textrm{Y}(m(x),m(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\nonumber \\ \end{aligned}$$

(3.10)

for $\theta _1,\theta _2,\theta _3,\theta _4\ge 0$ such that $\theta _1+\theta _2+\theta _3+\theta _4\ge 3$ and at least one of $i=1,\dots ,4$ has $\theta _i\ge 2$ (see Remark 3.6). Let us assume $\theta _3\ge 2$ since the other cases can be handled in a similar way. For the absolute value of (3.10) we have

$$\begin{aligned}&\frac{|f(x)|}{{{\mathfrak {m}}}(B_r(x))}\int _{B_r(x)\setminus \Omega _{u,v,\beta ,x}}\,\frac{\textsf{d}^{\theta _1}_\textrm{Y}(u(x),u(y))\textsf{d}^{\theta _2}_\textrm{Y}(v(x),v(y)) |\textsf{d}(x)-\textsf{d}(y)|^{\theta _3}\textsf{d}^{\theta _4}_\textrm{Y}(m(x),m(y))}{r^2}\,\textrm{d}{{\mathfrak {m}}}(y)\nonumber \\&\quad \le |f(x)|\,r^{\theta _1+\theta _2+\theta _3+\theta _4-2}\,\textsf{ks}^2_{2,r}[\textsf{d},\Omega ;u,v,\beta ](x) \,I, \end{aligned}$$

(3.11)

where

$$\begin{aligned} I:=\sup _{y\in B_r(x)\setminus \{x\}} \left( \frac{\textsf{d}^{\theta _1}_\textrm{Y}(u(x),u(y))}{\textsf{d}^{\theta _1}(x,y)}\frac{\textsf{d}^{\theta _2}_\textrm{Y}(u(x),u(y))}{\textsf{d}^{\theta _2}(x,y)}\frac{|\textsf{d}(x)-\textsf{d}(y)|^{\theta _{3}-2}}{\textsf{d}^{\theta _3-2}(x,y)}\frac{\textsf{d}^{\theta _4}_\textrm{Y}(m(x),m(y))}{\textsf{d}^{\theta _4}(x,y)}\right) . \end{aligned}$$

By $\theta _1+\theta _2+\theta _3+\theta _4-2\ge 1$ and by the choice of x, the right-hand side of (3.11) goes to 0 as $r\downarrow 0$. Thus, we confirm the validity of (3.9). Letting $r\downarrow 0$ in (3.8), we can derive the desired assertion. $\square$

We next prove the following (cf. [37, Lemma 1.14]):

Lemma 3.8

Assume that $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally Hilbertian. For $u\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o})),\eta \in \textsf{KS}^{1,2}(\Omega ,[0,1])$, we define $u_{\eta }:\Omega \rightarrow {\bar{\textrm{B}}}_{\rho }(\textrm{o})$ by $u_{\eta }:=\textsf{G}^{u,\textrm{o}}_{\eta }$ and $\textsf{d}:=\textsf{d}_\textrm{Y}(u,\textrm{o})$. Then we have $u_{\eta }\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o})), \textsf{d}\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$ and

$$\begin{aligned} \textrm{e}^2_{2}[u_{\eta }]\le \frac{\sin ^2(1-\eta )\textsf{d}}{\sin ^2 \textsf{d}} \left( \textrm{e}^2_{2}[u]-\textrm{e}^2_{2}[\textsf{d}] \right) +\textrm{e}^2_2[(1-\eta )\textsf{d}]\quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega . \end{aligned}$$

Proof

Let us verify $u_{\eta }\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o})), \textsf{d}\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$. Similarly to the proof of Lemma 3.7, the thesis for $\textsf{d}$ follows from the triangle inequality. By virtue of (3.5) for $l=\rho$, for all $x,y\in \Omega$,

$$\begin{aligned} \textsf{d}_\textrm{Y}(u_\eta (x),u_{\eta }(y))&\le c_{\rho }\,\textsf{d}_\textrm{Y}(u(x),u(y))+|\eta (x)-\eta (y)|\textsf{d}_\textrm{Y}(u(y),\textrm{o})\\&\le c_{\rho }\,\textsf{d}_\textrm{Y}(u(x),u(y))+\rho |\eta (x)-\eta (y)| \end{aligned}$$

for some constant $c_\rho >0$ depending only on $\rho$. Therefore,

$$\begin{aligned} \textsf{ks}^2_{2,r}[u_{\eta },\Omega ](x)\le 2c^2_{\rho }\,\textsf{ks}^2_{2,r}[u,\Omega ](x)+2\rho ^2\textsf{ks}^2_{2,r}[\eta ,\Omega ](x). \end{aligned}$$

Integrating this estimate over $\Omega$, and letting $r\downarrow 0$, we conclude $u_\eta \in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$.

Using Lemma 3.5, for all $x,y\in \Omega$ we have

$$\begin{aligned} \textsf{d}^2_\textrm{Y}(u_{\eta }(x),u_{\eta }(y))&\le \frac{\sin ^2(1-\eta (x))\textsf{d}(x)}{\sin ^2 \textsf{d}(x) } \left( \textsf{d}^2_{\textrm{Y}}(u(x),u(y))- (\textsf{d}(y)-\textsf{d}(x))^2 \right) \\&\quad +(1-\eta (x))^2(\textsf{d}(y)-\textsf{d}(x))^2+\textsf{d}^2(x)(\eta (y)-\eta (x))^2\\&\quad -2(1-\eta (x))\textsf{d}(x)(\eta (y)-\eta (x))(\textsf{d}(y)-\textsf{d}(x))\\&\quad +{\text {Cub}}(\eta (y)-\eta (x),\textsf{d}_\textrm{Y}(u(x),u(y)),\textsf{d}(y)-\textsf{d}(x),\textsf{d}_\textrm{Y}(u_{\eta }(x),u_{\eta }(y))). \end{aligned}$$

Setting $\xi :=1-\eta$, we also have

$$\begin{aligned}&\textsf{d}^2_\textrm{Y}(u_{\eta }(x),u_{\eta }(y))\le \frac{\sin ^2\xi (x)\textsf{d}(x)}{\sin ^2 \textsf{d}(x) } \left( \textsf{d}^2_{\textrm{Y}}(u(x),u(y))-(\textsf{d}(y)-\textsf{d}(x))^2 \right) \nonumber \\&\qquad +\xi ^2(x)(\textsf{d}(y)-\textsf{d}(x))^2+\textsf{d}^2(x)(\xi (y)-\xi (x))^2\nonumber \\&\qquad +2\xi (x)\textsf{d}(x)(\xi (y)-\xi (x))(\textsf{d}(y)-\textsf{d}(x))\nonumber \\&\qquad +{\text {Cub}}(\xi (x)-\xi (y),\textsf{d}_\textrm{Y}(u(x),u(y)),\textsf{d}(y)-\textsf{d}(x),\textsf{d}_\textrm{Y}(u_{\eta }(x),u_{\eta }(y)))\nonumber \\&\quad = \frac{\sin ^2\xi (x)\textsf{d}(x)}{\sin ^2 \textsf{d}(x) } \left( \textsf{d}^2_{Y}(u(x),u(y))-(\textsf{d}(y)-\textsf{d}(x))^2 \right) \nonumber \\&\qquad +\xi ^2(x)(\textsf{d}(y)-\textsf{d}(x))^2+\textsf{d}^2(x)(\xi (y)-\xi (x))^2\nonumber \\&\qquad +\frac{\xi (x)\textsf{d}(x)}{2}\left[ \{(\xi +\textsf{d})(x)-(\xi +\textsf{d})(y)\}^2- \{(\xi -\textsf{d})(x)-(\xi -\textsf{d})(y)\}^2 \right] \nonumber \\&\qquad +{\text {Cub}}(\xi (x)-\xi (y),\textsf{d}_\textrm{Y}(u(x),u(y)),\textsf{d}(y)-\textsf{d}(x),\textsf{d}_\textrm{Y}(u_{\eta }(x),u_{\eta }(y))). \end{aligned}$$

(3.12)

Dividing the later expression by $r^2$ and integrating in y over $B_r(x)$ lead to

where ${\text {Cub}}(x,y)$ denotes the cubic terms in (3.12). In the same manner as in the proof of Lemma 3.7, we see

as $r\downarrow 0$. By virtue of Theorem 2.8, we obtain

$$\begin{aligned} \textrm{e}^2_{2}[u_{\eta }]&\le \frac{\sin ^2\xi \textsf{d}}{\sin ^2 \textsf{d}} \left( \textrm{e}^2_{2}[u]-\textrm{e}^2_{2}[\textsf{d}] \right) +\xi ^2 \textrm{e}^2_{2}[\textsf{d}]+\textsf{d}^2\textrm{e}^2_{2}[\xi ]+\frac{\xi \textsf{d}}{2}\left( \textrm{e}^2_{2}[\xi +\textsf{d}]-\textrm{e}^2_{2}[\xi -\textsf{d}] \right) \\&\quad {{\mathfrak {m}}}\hbox { -a.e. on } \Omega . \end{aligned}$$

Now, it suffices to prove that

$$\begin{aligned} \xi ^2 \textrm{e}^2_{2}[\textsf{d}]+\textsf{d}^2\textrm{e}^2_{2}[\xi ]+\frac{\xi \textsf{d}}{2}\left( \textrm{e}^2_{2}[\xi +\textsf{d}]-\textrm{e}^2_{2}[\xi -\textsf{d}] \right) =\textrm{e}^2_2[\xi \textsf{d}] \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega . \end{aligned}$$

Thanks to Proposition 2.9, for any $u\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$, we have $|D u|=c_d\, \textrm{e}_2[u]$ ${{\mathfrak {m}}}$-a.e. on $\Omega$, and hence it is enough to show that

$$\begin{aligned} \xi ^2 |D \textsf{d}|^2+\textsf{d}^2|D \xi |^2+\frac{\xi \textsf{d}}{2}\left( |D(\xi +\textsf{d})|^2-|D(\xi -\textsf{d})|^2 \right) =|D(\xi \textsf{d})|^2 \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega .\nonumber \\ \end{aligned}$$

(3.13)

Since $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally Hilbertian, Theorem 2.2 tells us that

$$\begin{aligned} |D(\xi \textsf{d})|^2=\langle \nabla (\xi \textsf{d}),\nabla (\xi \textsf{d}) \rangle =\xi ^2 |D \textsf{d}|^2+\textsf{d}^2|\nabla \xi |^2+2\xi \textsf{d}\langle \nabla \xi ,\nabla \textsf{d}\rangle \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega ,\nonumber \\ \end{aligned}$$

(3.14)

where we used the Leibniz rule. Furthermore, by the linearity of the gradient,

$$\begin{aligned}{} & {} |D(\xi +\textsf{d})|^2-|D(\xi -\textsf{d})|^2=\langle \nabla (\xi +\textsf{d}),\nabla (\xi +\textsf{d}) \rangle -\langle \nabla (\xi -\textsf{d}),\nabla (\xi -\textsf{d}) \rangle \nonumber \\{} & {} =4\langle \nabla \xi ,\nabla \textsf{d}\rangle \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on} \Omega . \end{aligned}$$

(3.15)

Combining (3.14) and (3.15) yields

$$\begin{aligned} |D(\xi \textsf{d})|^2=\xi ^2 |D \textsf{d}|^2+\textsf{d}^2|D \xi |^2+\frac{\xi \textsf{d}}{2}\left( |D(\xi +\textsf{d})|^2-|D(\xi -\textsf{d})|^2 \right) \quad \hbox {} {{\mathfrak {m}}}\hbox { -a.e. on } \Omega , \end{aligned}$$

and this is nothing but (3.13), thus concluding the proof. $\square$

We now prove that the energy $\textsf{E}^{\Omega }_{2}$ satisfies a certain kind of convexity (cf. [22, Proposition 1.15]):

Proposition 3.9

Assume that $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally Hilbertian. For $u,v\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$, we set $m:=\textsf{G}^{u,v}_{1/2}\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$ and $\textsf{d}:=\textsf{d}_\textrm{Y}(u,v),{{\mathfrak {d}}}:=\textsf{d}_{\textrm{Y}}(m,\textrm{o})\in \textsf{KS}^{1,2}(\Omega ,{\mathbb {R}})$. Let $\eta \in \textsf{KS}^{1,2}(\Omega ,[0,1])$ be a function determined by solving

$$\begin{aligned} \frac{\sin (1-\eta ){{\mathfrak {d}}}}{\sin {{\mathfrak {d}}}}=\cos \frac{\textsf{d}}{2}, \end{aligned}$$

(3.16)

and define $m_\eta :=\textsf{G}^{m,\textrm{o}}_{\eta }\in \textsf{KS}^{1,2}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$. Then we have

$$\begin{aligned} \textsf{E}^{\Omega }_{2}(m_{\eta })+\cos ^8 \rho \,\textsf{E}^{\Omega }_{2}\left( \frac{\tan \frac{\textsf{d}}{2}}{\cos {{\mathfrak {d}}}}\right) \le \frac{1}{2}\textsf{E}^{\Omega }_{2}(u)+\frac{1}{2}\textsf{E}^{\Omega }_{2}(v). \end{aligned}$$

Proof

Combining Lemmas 3.7 and 3.8, we have

$$\begin{aligned} \textrm{e}^2_2[m_{\eta }]&\le \frac{\sin ^2 (1-\eta ){{\mathfrak {d}}}}{\sin ^2 {{\mathfrak {d}}}}(\textrm{e}^2_2[m]-\textrm{e}^2_2[{{\mathfrak {d}}}])+\textrm{e}^2_2[(1-\eta ){{\mathfrak {d}}}]\nonumber \\&=\cos ^2 \frac{\textsf{d}}{2}(\textrm{e}^2_2[m]-\textrm{e}^2_2[{{\mathfrak {d}}}])+\textrm{e}^2_2[(1-\eta ){{\mathfrak {d}}}]\nonumber \\&\le \frac{1}{2}\left( \textrm{e}^2_2[u]+\textrm{e}^2_{2}[v] \right) -\frac{1}{4}\textrm{e}^2_2[\textsf{d}]-\cos ^2 \frac{\textsf{d}}{2}\,\textrm{e}^2_2[{{\mathfrak {d}}}]+\textrm{e}^2_2[(1-\eta ){{\mathfrak {d}}}] \quad {{\mathfrak {m}}}\text { -a.e. on } \Omega . \end{aligned}$$

(3.17)

We now observe that the following hold:

$$\begin{aligned}{} & {} \textrm{e}^2_2[(1-\eta ){{\mathfrak {d}}}]+\frac{\cos ^4 \frac{\textsf{d}}{2}\,\cos ^4 {{\mathfrak {d}}}}{1-\cos ^2 \frac{\textsf{d}}{2}\,\sin ^2 {{\mathfrak {d}}}}\,\textrm{e}^2_2 \left[ \frac{\tan \frac{\textsf{d}}{2}}{ \cos {{\mathfrak {d}}}} \right] \nonumber \\{} & {} \quad =\cos ^2 \frac{\textsf{d}}{2}\,\textrm{e}^2_2[{{\mathfrak {d}}}]+\frac{1}{4}\textrm{e}^2_2[\textsf{d}] \quad {{\mathfrak {m}}}\text { -a.e. on } \Omega . \end{aligned}$$

(3.18)

In view of Proposition 2.9, this is equivalent to the following:

$$\begin{aligned} |D((1-\eta ){{\mathfrak {d}}})|^2+\frac{\cos ^4 \frac{\textsf{d}}{2}\,\cos ^4 {{\mathfrak {d}}}}{1-\cos ^2 \frac{\textsf{d}}{2}\,\sin ^2 {{\mathfrak {d}}}}\,\left| D\left( \frac{\tan \frac{\textsf{d}}{2}}{ \cos {{\mathfrak {d}}}}\right) \right| ^2=\cos ^2 \frac{\textsf{d}}{2}\,|D{{\mathfrak {d}}}|^2+\frac{1}{4}|D \textsf{d}|^2 \quad {{\mathfrak {m}}}\text { -a.e. on } \Omega . \end{aligned}$$

Since $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ is infinitesimally Hilbertian, it can be derived from Theorems 2.1, 2.2 (chain rule, Leibniz rule), the definition of $\eta$, and a straightforward calculation. Combining (3.17) and (3.18), we obtain

$$\begin{aligned} \textrm{e}^2_2[m_{\eta }]+\cos ^8 \rho \,\textrm{e}^2_2 \left[ \frac{\tan \frac{\textsf{d}}{2}}{ \cos {{\mathfrak {d}}}} \right]&\le \textrm{e}^2_2[m_{\eta }]+\cos ^4 \frac{\textsf{d}}{2}\,\cos ^4 {{\mathfrak {d}}}\,\textrm{e}^2_2 \left[ \frac{\tan \frac{\textsf{d}}{2}}{ \cos {{\mathfrak {d}}}} \right] \\&\le \textrm{e}^2_2[m_{\eta }]+\frac{\cos ^4 \frac{\textsf{d}}{2}\,\cos ^4 {{\mathfrak {d}}}}{1-\cos ^2 \frac{\textsf{d}}{2}\,\sin ^2 {{\mathfrak {d}}}}\,\textrm{e}^2_2 \left[ \frac{\tan \frac{\textsf{d}}{2}}{ \cos {{\mathfrak {d}}}} \right] \\&\le \frac{1}{2}\left( \textrm{e}^2_2[u]+\textrm{e}^2_{2}[v] \right) \quad {{\mathfrak {m}}}\text { -a.e. on } \Omega . \end{aligned}$$

By integrating this inequality over $\Omega$, we obtain the desired one. $\square$

3.4 Proof of theorem 1.2

We are now in a position to prove Theorem 1.2.

Proof of Theorem 1.2

Let $(\textrm{X},\textsf{d},{{\mathfrak {m}}})$ be an infinitesimally Hilbertian, locally uniformly doubling, strongly rectifiable space satisfying a Poincaré inequality, and let $\Omega$ be a bounded open subset of $\textrm{X}$ with ${{\mathfrak {m}}}(\textrm{X}{\setminus } \Omega )>0$. Let $\textrm{Y}_{\textrm{o}}=(\textrm{Y},\textsf{d}_\textrm{Y},\textrm{o})$ be a pointed ${\text {CAT}}(1)$ space, and let ${\bar{\textrm{B}}}_{\rho }(\textrm{o})$ be a regular ball. Let $(u_n)_n\subset \textsf{KS}^{1,2}_{{{\bar{u}}}}(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$ stand for a minimizing sequence of $\textsf{E}^{\Omega }_{2,{\bar{u}}}$. By Proposition 2.10, the functional $\textsf{E}_{2,{{\bar{u}}}}^\Omega$ is lower semicontinuous, and hence it is sufficient to show that $(u_n)_n$ is an $L^2(\Omega ,{\bar{\textrm{B}}}_{\rho }(\textrm{o}))$-Cauchy sequence.

Set $I:=\lim _n \textsf{E}_{2,{{\bar{u}}}}^\Omega (u_n)=\inf \textsf{E}_{2,{{\bar{u}}}}^\Omega$. We also set

$$\begin{aligned} m_{n,m}:=\textsf{G}^{u_n,u_m}_{1/2},\quad \textsf{d}_{n,m}:=\textsf{d}_\textrm{Y}(u_n,u_m),\quad {{\mathfrak {d}}}_{n,m}:=\textsf{d}_\textrm{Y}(m_{n,m},\textrm{o}),\quad m_{n,m,\eta }:=\textsf{G}^{m_{n,m},\textrm{o}}_{\eta _{n,m}}, \end{aligned}$$

where $\eta _{n,m}\in \textsf{KS}^{1,2}(\Omega ,[0,1])$ is defined as (3.16). Proposition 3.9 yields

$$\begin{aligned} \cos ^8 \rho \,\textsf{E}^{\Omega }_{2,{\bar{u}}}\left( \frac{\tan \frac{\textsf{d}_{n,m}}{2}}{\cos {{\mathfrak {d}}}_{n,m}}\right)&\le \frac{1}{2}\textsf{E}^{\Omega }_{2,{\bar{u}}}(u_{n})+\frac{1}{2}\textsf{E}^{\Omega }_{2,{\bar{u}}}(u_m)-\textsf{E}^{\Omega }_{2,{\bar{u}}}(m_{n,m,\eta })\\&\le \frac{1}{2}\textsf{E}^{\Omega }_{2,{\bar{u}}}(u_{n})+\frac{1}{2}\textsf{E}^{\Omega }_{2,{\bar{u}}}(u_m)-I; \end{aligned}$$

in particular,

$$\begin{aligned} \varlimsup _{n,m\rightarrow \infty }\int _\Omega \left| D\left( \frac{\tan \frac{\textsf{d}_{n,m}}{2}}{\cos {{\mathfrak {d}}}_{n,m}}\right) \right| ^2\,\textrm{d}{{\mathfrak {m}}}=0. \end{aligned}$$

Furthermore, a Poincaré inequality under Dirichlet boundary conditions (see e.g., [22, Lemma 6.3], [2, Subsection 5.5]) together with Proposition 2.10 leads us to

$$\begin{aligned} \lim _{n,m\rightarrow \infty }\int _\Omega \left| \frac{\tan \frac{\textsf{d}_{n,m}}{2}}{\cos {{\mathfrak {d}}}_{n,m}} \right| ^2\,\textrm{d}{{\mathfrak {m}}}=0. \end{aligned}$$

Thus, $\lim _{n,m\rightarrow \infty }\int _\Omega |\textsf{d}_{n,m}|^2\,\textrm{d}{{\mathfrak {m}}}=0$ since $|\textsf{d}_{n,m}|^2 \le 4|\tan (\textsf{d}_{n,m}/2)|^2$ and $|\cos {{\mathfrak {d}}}_{n,m}|^2\le 1$ over $\Omega$. This concludes the proof. $\square$