A note about the strong maximum principle on $RCD$ spaces

TL;DR

This paper provides a straightforward proof of the strong maximum principle in finite dimensional RCD spaces, utilizing Laplacian comparison of the squared distance to establish the result.

Contribution

It offers a new, direct proof of the strong maximum principle specifically tailored for finite dimensional RCD spaces.

Findings

Proof based on Laplacian comparison of squared distance

Simplifies the understanding of maximum principles in RCD spaces

Enhances the theoretical framework of geometric analysis on metric measure spaces

Abstract

We give a quick and direct proof of the strong maximum principle on finite dimensional $RCD$ spaces based on the Laplacian comparison of the squared distance.

Equations

$$|\mathrm{D}f|=|\mathrm{D}g|\mathfrak{m}-a.e.\text{on }\{f=g\}\forall f,g\in W^{1,2}(\mathrm{X}).$$

$$\begin{array}{rl}{W}_{\mathsf{loc}}^{1,2}(\Omega ):=\{f\in L_{\mathsf{loc}}^2(\Omega ):& \text{for every}x\in \Omega \text{there exists}U\subset \Omega \text{neighbourhood of}x\\ & \text{and there exists}{f}_U\in W_{\mathsf{loc}}^{1,2}(\mathrm{X})\text{such that}f\text{lower}3.0pt{|}_U=f_U\}.\end{array}$$

$$|\mathrm{D}f|:=|\mathrm{D}{f}_U|\mathfrak{m}-\text{a.e. on }U,$$

$$W^{1,2}(\Omega ):=\{f\in W_{\mathsf{loc}}^{1,2}(\Omega ):f,|\mathrm{D}f|\in L^2(\Omega )\}.$$

$$\frac{1}{2}{\int }_{\Omega }|\mathrm{D}f{|}^2\mathrm{d}\mathfrak{m}\le \frac{1}{2}{\int }_{\Omega }|\mathrm{D}(f+g){|}^2\mathrm{d}\mathfrak{m}.$$

$$\begin{array}{rl}& \text{Let (X,d,m) be RCD(K,∞), K∈R, and f∈W1,2(X) be such that ∣Df∣∈L∞(X).}\\ & \text{Then there exists f~=f m-a.e. such that Lip(f~)≤∥Df∣∥L∞.}\end{array}$$

$$\underset{\Omega }{\sup }f\le \underset{\partial \Omega }{\sup }f,$$

$$\tilde{f}:=\min \{c,f\}$$

$$|\mathrm{D}\tilde{f}|={\text{raise}1.29167pt\chi }_{\{f<c\}}|\mathrm{D}f|$$

$${\int }_{\Omega }|\mathrm{D}f{|}^2\mathrm{d}\mathfrak{m}\le {\int }_{\Omega }|\mathrm{D}\tilde{f}{|}^2\mathrm{d}\mathfrak{m}\stackrel{\text{eqref}eq:tildef}{=}{\int }_{\{f<c\}\cap \Omega }|\mathrm{D}f{|}^2\mathrm{d}\mathfrak{m},$$

$$|\mathrm{D}f|=0$$

$$|\mathrm{D}g|={\text{raise}1.29167pt\chi }_{\Omega \cap \{f>c\}}|\mathrm{D}f|\stackrel{\text{eqref}eq:dz}{=}0.$$

$$\int g\mathrm{d}\Delta f\text{lower}3.0pt{|}_{\Omega }=-\int ⟨\nabla f,\nabla g⟩\mathrm{d}\mathfrak{m}.$$

$${\int }_{\Omega }|\mathrm{D}(f+\epsilon g){|}^2-|\mathrm{D}f{|}^2\mathrm{d}\mathfrak{m}\ge 0$$

$${\int }_{\Omega }⟨\nabla f,\nabla g⟩\mathrm{d}\mathfrak{m}\ge 0.$$

$$\text{$\big{\{}g\in{\rm LIP}_{c}(\Omega):g\leq 0\big{\}}$\quad is $W^{1,2}-$dense in\quad$\big{\{}g\in W^{1,2}(\Omega):\ g\leq 0\ \mathop{\rm supp}\nolimits(g)\subset\Omega\big{\}}$}.$$

$$|\mathrm{D}(f+g){|}^2-|\mathrm{D}f{|}^2\ge \underset{\epsilon \downarrow 0}{\lim }\frac{|\mathrm{D}(f+\epsilon g){|}^2-|\mathrm{D}f{|}^2}{\epsilon }=2⟨\nabla f,\nabla g⟩$$

$${\int }_{\Omega }|\mathrm{D}(f+g){|}^2-|\mathrm{D}f{|}^2\mathrm{d}\mathfrak{m}\ge 0$$

$${\mathsf{d}}_{x_0}^2\in W_{\mathsf{loc}}^{1,2}(\mathrm{X})\text{ and }|\mathrm{D}({\mathsf{d}}_{x_0}^2){|}^2$$

$${\mathsf{d}}_{x_0}^2\in D(\mathbf{\Delta })\text{ and }\mathbf{\Delta }{\mathsf{d}}_{x_0}^2(x)$$

$$\mathsf{d}(x,y)=\underset{z\in C}{\min }\mathsf{d}(x,z).$$

$$\varphi(x):=\inf_{z\in C}\frac{{\sf d}^{2}(x,z)}{2}=\psi^{c}(x)\qquad\text{ where }\qquad\psi(y):=\left\{\begin{array}[]{ll}0,&\qquad\text{ if }y\in C,\\ -\infty,&\qquad\text{ if }y\in\mathrm{X}\setminus C.\end{array}\right.$$

$$\phi (x)+{\phi }^c(y)\ge \phi (x)+\psi (y)\stackrel{\text{eqref}eq:proj}{=}\frac{{\mathsf{d}}^2(x,y)}{2},$$

$$\Omega^{\prime}:=\big{\{}x\in\Omega\setminus C\ :\ {\sf d}(x,C)<{\sf d}(x,\partial\Omega)\big{\}}.$$

$$h(z):=e^{-A{\mathsf{d}}^2(z,x)}-e^{-A{r}^2},$$

$${\mathbf{\Delta}}h=A^{2}e^{-A{\sf d}^{2}_{x}}|\mathrm{D}{\sf d}_{x}^{2}|^{2}\,\mathfrak{m}-Ae^{-A{\sf d}^{2}_{x}}{\mathbf{\Delta}}{\sf d}_{x}^{2}\stackrel{{\scriptstyle\eqref{diffdist},\eqref{lapdist}}}{{\geq}}2e^{-A{\sf d}^{2}_{x}}\big{(}A^{2}{\sf d}_{x}^{2}-A\ell_{K,N}({\sf d}_{x})\big{)}\mathfrak{m}.$$

$$f_{\epsilon }(y)\le \underset{\partial B_{r^{\prime }}(y)}{\sup }{f}_{\epsilon },\forall \epsilon >0.$$

$$f_{\epsilon }(y)>f_{\epsilon }(z)\forall z\in \partial B_{r^{\prime }}(y)\setminus {\bar{B}}_r(x),\forall \epsilon >0.$$

$$f(y)>\underset{\partial B_{r^{\prime }}(y)\cap {\bar{B}}_r(x)}{\sup }f$$

$$f_{\epsilon }(y)>\underset{\partial B_{r^{\prime }}(y)\cap {\bar{B}}_r(x)}{\sup }{f}_{\epsilon }.$$

Full text

1. 1 Introduction
2. 2 Result

1 Introduction

In the context of analysis in metric measure spaces it is by now well understood that a doubling condition and a Poincaré inequality are sufficient to derive the basics of elliptic regularity theory. In particular, one can obtain the Harnack inequality for harmonic functions which in turns implies the strong maximum principle. We refer to [5] for an overview on the topic and detailed bibliography.

${\sf RCD}^{*}(K,N)$ spaces ([2], [11], see also [3], [8], [4]) are, for finite $N$, doubling ([16]) and supporting a Poincaré inequality ([15]) and thus in particular the above applies. Still, given that in fact such spaces are much more regular than general doubling&Poincaré ones, one might wonder whether there is a simpler proof of the strong maximum principle.

Aim of this short note it show that this is actually the case: out of the several arguments available in the Euclidean space, the one based on the estimates for the Laplacian of the squared distance carries over to such non-smooth context rather easily.

Beside such Laplacian comparison, the other ingredient that we shall use is a result about a.e. unique projection on closed subsets of ${\sf RCD}$ spaces which to the best of our knowledge has not been observed before and, we believe, is of its own interest: see Lemma 2.6 and Remark 2.7.

We remark that the present result simplifies the proofs of those properties of ${\sf RCD}(K,N)$ spaces which depend on the strong maximum principle, like for instance the splitting theorem ([9], [10]).

2 Result

All metric measure spaces $(\mathrm{X},\mathsf{d},\mathfrak{m})$ we will consider will be such that $(\mathrm{X},{\sf d})$ is complete and separable, $\mathfrak{m}$ is a Radon non-negative measure with $\mathop{\rm supp}\nolimits(\mathfrak{m})=\mathrm{X}$.

To keep the presentation short we assume the reader familiar with the definition of ${\sf RCD}^{*}$ spaces and with calculus on them. Here we only recall those definitions and facts that will be used in the course of the proofs. In particular, we shall take for granted the notion of $W^{1,2}(\mathrm{X})$ space on the metric measure space $(\mathrm{X},\mathsf{d},\mathfrak{m})$ and, for $f\in W^{1,2}(\mathrm{X})$, of the minimal weak upper gradient $|\mathrm{D}f|$. Recall that the minimal weak upper gradient is a local object, i.e.:

[TABLE]

Then the notion of Sobolev space over an open set can be easily given:

Definition 2.1** **(Sobolev space on an open subset of ${\rm X}$).

Let $(\mathrm{X},\mathsf{d},\mathfrak{m})$ be a metric measure space and let $\Omega\subset{\rm X}$ open. Then we define

[TABLE]

For $f\in W^{1,2}_{\mathsf{loc}}(\Omega)$ the function $|\mathrm{D}f|\in L^{2}_{\mathsf{loc}}(\Omega)$ is defined as

[TABLE]

where $|\mathrm{D}f_{U}|$ is the minimal weak upper gradient of $f_{U}$ and the locality of this object ensures that $|\mathrm{D}f|$ is well defined.

Then we set

[TABLE]

The definition of (sub/super)-harmonic functions can be given in terms of minimizers of the Dirichlet integral (see [5] for a throughout discussion on the topic):

Definition 2.2** **(Subharmonic/Superharmonic/Harmonic functions).

Let $(\mathrm{X},\mathsf{d},\mathfrak{m})$ be a metric measure space and $\Omega$ be an open subset in ${\rm X}$. We say that $f$ is subharmonic (resp. superharmonic) in $\Omega$ if $f\in\mathit{W}^{1,2}(\Omega)$ and for any $g\in\mathit{W}^{1,2}(\Omega)$, $g\leq 0$ (resp. $g\geq 0$) with $\mathop{\rm supp}\nolimits g\subset\Omega$, it holds

[TABLE]

$f$* is harmonic in $\Omega$ if it is both subharmonic and superharmonic.*

On ${\sf RCD}(K,\infty)$ spaces, the weak maximum principle can be deduced directly from the definition of subharmonic function and the following property, proved in [2]:

[TABLE]

We can now easily prove the following:

Theorem 2.3** **(Weak Maximum Principle).

Let $(\mathrm{X},\mathsf{d},\mathfrak{m})$ be an ${\sf RCD}(K,\infty)$ space, $\Omega\subset\mathrm{X}$ open and let $f\in\mathit{W}^{1,2}(\Omega)\cap\mathit{C}(\bar{\Omega})$ be subharmonic. Then

[TABLE]

to be intended as ‘$f$ is constant’ in the case $\Omega=\mathrm{X}$.

proof We argue by contradiction. If (2.0.4) does not hold, regardless of weather $\Omega$ coincides with $\mathrm{X}$ or not, we can find $c<\sup_{\Omega}f$ such that the function

[TABLE]

agrees with $f$ on $\partial\Omega$. The locality of the differential grants that

[TABLE]

and from the assumption that $f$ is subharmonic and the fact that $\tilde{f}\leq f$ we deduce that

[TABLE]

which forces

[TABLE]

Now consider the function g:=\max\{c,{\raise 1.29167pt\hbox{\chi}}_{\Omega}f\}, notice that our assumptions grant that $g\in C(\mathrm{X})$ and that the locality of the differential yields

[TABLE]

Hence property (2.0.3) gives that $g$ is constant, i.e. $f\leq c$ on $\Omega$. This contradicts our choice of $c$ and gives the conclusion. $\square$

We remark that in the finite-dimensional case one could conclude from (2.0.7) by using the Poincaré inequality in place of property (2.0.3).

To prove the strong maximum principle we need to recall few facts. The first is the concept of measure-valued Laplacian (see [11]), for which we restrict the attention to proper (=closed bounded sets are compact) and infinitesimally Hilbertian (=$W^{1,2}(\mathrm{X})$ is an Hilbert space, see [11]) spaces:

Definition 2.4** **(Measure valued Laplacian).

Let $({\rm X},{\sf d},\mathfrak{m})$ be proper and infinitesimally Hilbertian, $\Omega\subset{\rm X}$ open and $f\in\mathit{W}^{1,2}(\Omega)$. We say that $f$ has a measure valued Laplacian in $\Omega$, and write $f\in D({\mathbf{\Delta}},\Omega)$, provided there exists a Radon measure, that we denote by \mathbb{\Delta}f\lower 3.0pt\hbox{|_{\Omega}}, such that for every $g\colon\mathrm{X}\rightarrow$ Lipschitz with support compact and contained in $\Omega$ it holds

[TABLE]

If $\Omega=\mathrm{X}$ we write $f\in D({\mathbf{\Delta}})$ and ${\mathbf{\Delta}}f$.

Much like in the smooth case, it turns out that being subharmonic is equivalent to having non-negative Laplacian. This topic has been investigated in [11] and [12], here we report the proof of this fact because in [12] it has been assumed the presence of a Poincaré inequality, while working on proper infinitesimally Hilbertian spaces allows to easily remove such assumption.

Theorem 2.5**.**

Let $({\rm X},{\sf d},\mathfrak{m})$ be a proper infinitesimally Hilbertian space, $\Omega\subset{\rm X}$ open and $f\in\mathit{W}^{1,2}(\Omega)$.

Then $f$ is subharmonic (resp. superharmonic, resp. harmonic) if and only if $f\in D({\mathbf{\Delta}},\Omega)$ with {\mathbf{\Delta}}f\lower 3.0pt\hbox{|{\Omega}}\geq 0 (resp. {\mathbf{\Delta}}f\lower 3.0pt\hbox{|{\Omega}}\leq 0, resp. {\mathbf{\Delta}}f\lower 3.0pt\hbox{|_{\Omega}}=0).

proof

Only if Let ${\rm LIP}_{c}(\Omega)\subset W^{1,2}(\Omega)$ be the space of Lipschitz functions with support compact and contained in $\Omega$. For $g\in{\rm LIP}_{c}(\Omega)$ non-positive and $\varepsilon>0$ apply (2.0.2) with $\varepsilon g$ in place of $g$ to deduce

[TABLE]

and dividing by $\varepsilon$ and letting $\varepsilon\downarrow 0$ we conclude

[TABLE]

In other words, the linear functional ${\rm LIP}_{c}(\Omega)\ni g\mapsto-\int_{\Omega}\left<\nabla f,\nabla g\right>\,{\mathrm{d}}\mathfrak{m}$ is positive. It is then well known, see e.g. [6, Theorem 7.11.3], that the monotone extension of such functional to the space of continuous and compactly supported functions on $\Omega$ is uniquely represented by integration w.r.t. a non-negative measure, which is the claim.

If Recall from [1] that on general metric measure spaces Lipschitz functions are dense in energy in $W^{1,2}$; since infinitesimally Hilbertianity implies uniform convexity of $W^{1,2}$, we see that in our case they are dense in the $W^{1,2}-$norm. Then by truncation and cut-off argument we easily see that

[TABLE]

Now notice that the convexity of $g\mapsto\frac{1}{2}\int_{\Omega}|\mathrm{D}g|^{2}\,{\mathrm{d}}\mathfrak{m}$ grants that for any $g\in W^{1,2}(\Omega)$ it holds

[TABLE]

and thus from the assumption {\mathbf{\Delta}}f\lower 3.0pt\hbox{|_{\Omega}}\geq 0 we deduce that

[TABLE]

for every $g\in{\rm LIP}_{c}(\Omega)$ non-positive. Taking (2.0.9) into account we see that (2.0.10) also holds for any $g\in W^{1,2}(\Omega)$ non-negative with $\mathop{\rm supp}\nolimits(g)\subset\Omega$, which is the thesis. $\square$

For $x\in\mathrm{X}$ we write ${\sf d}_{x}$ for the function $y\mapsto{\sf d}(x,y)$. We shall need the following two properties of the squared distance function valid on ${\sf RCD}(K,N)$ spaces, $N<\infty$:

[TABLE]

where $\ell_{K,N}:[0,+\infty)\to[0,+\infty)$ is some continuous function depending only on $K,N$. Property (2.0.11) can be seen as a consequence of Cheeger’s work [7]: recall that ${\sf CD}(K,N)$ spaces are doubling ([16]) and support a 1-2 weak Poincaré inequality ([15]) and notice that, being geodesic, the local Lipschitz constant of ${\sf d}_{x}$ is identically 1. An alternative proof, more tailored to the ${\sf RCD}$ setting, passes through the fact that ${\sf d}_{x_{0}}^{2}/2$ is $c$-concave and uses the regularity of $W_{2}$-geodesics, see for instance [14] for the details of the proof.

The Laplacian comparison estimate (2.0.12) is one of the main results in [11]. Notice that in [11] such inequality has been obtained in its sharp form, but for our purposes the above formulation is sufficient.

Beside these facts, we shall need the following geometric property of ${\sf RCD}$ spaces, which we believe is interesting on its own:

Lemma 2.6** (a.e. unique projection).**

Let $K\in\mathbb{R}$, $N\in[1,\infty)$, $(\mathrm{X},\mathsf{d},\mathfrak{m})$ an ${\sf RCD}(K,N)$ space and $C\subset\mathrm{X}$ a closed set. Then for $\mathfrak{m}$-a.e. $x\in\mathrm{X}$ there exists a unique $y\in C$ such that

[TABLE]

proof Existence follows trivially from the fact that $\mathrm{X}$ is proper. For uniqueness define

[TABLE]

Since $\varphi^{c}=\psi^{cc}\geq\psi$, if $x\in\mathrm{X}$ and $y\in C$ are such that (2.0.13) holds, we have

[TABLE]

i.e. $y\in\partial^{c}\varphi(x)$. Conclude recalling that since $\varphi$ is $c$-concave and real valued, Theorem 3.4 in [13] grants that for $\mathfrak{m}$-a.e. $x$ there exists a unique $y\in\partial^{c}\varphi(x)$. $\square$

Remark 2.7**.**

The simple proof of this lemma relies on quite delicate properties of ${\sf RCD}$ spaces, notice indeed that the conclusion can fail on the more general ${\sf CD}(K,N)$ spaces. Consider for instance $\mathbb{R}^{2}$ equipped with the distance coming from the $L^{\infty}$ norm and the Lebesgue measure $\mathcal{L}^{2}$. This is a ${\sf CD}(0,2)$ space, as shown in the last theorem in [17]. Then pick $C:=\{(z_{1},z_{2}):z_{1}\geq 0\}$ and notice that for every $(x_{1},x_{2})\in\mathbb{R}^{2}$ with $x_{1}<0$ there are uncountably many minimizers in (2.0.13).

• $\blacksquare$*

We can now prove the main result of this note:

Theorem 2.8** **(Strong Maximum Principle).

Let $K\in\mathbb{R}$, $N\in[1,\infty)$ and $(\mathrm{X},\mathsf{d},\mathfrak{m})$ an ${\sf RCD}(K,N)$ space. Let $\Omega\subset\mathrm{X}$ be open and connected and let $f\in\mathit{W}^{1,2}(\Omega)\cap\mathit{C}(\bar{\Omega})$ be subharmonic and such that for some $\bar{x}\in\Omega$ it holds $f(\bar{x})=\max_{\bar{\Omega}}f$. Then $f$ is constant.

proof Put $m:=\sup_{\Omega}f$, $C:=\{x\in\bar{\Omega}:f(x)=m\}$ and define

[TABLE]

By assumption we know that $C\cap\Omega\neq\emptyset$ and that $\Omega$ is connected, thus since $C$ is closed, either $C\supset\Omega$, in which case we are done, or $\partial C\cap\Omega\neq\emptyset$, in which case $\Omega^{\prime}\neq\emptyset$. We now show that such second case cannot occur, thus concluding the proof.

Assume by contradiction that $\Omega^{\prime}\neq\emptyset$, notice that $\Omega^{\prime}$ is open and thus $\mathfrak{m}(\Omega^{\prime})>0$. Hence by Lemma 2.6 we can find $x\in\Omega^{\prime}$ and $y\in C$ such that (2.0.13) holds. Notice that the definition of $\Omega^{\prime}$ grants that $y\in\Omega$, put $r:={\sf d}(x,y)$ and define

[TABLE]

where $A\gg 1$ will be fixed later. By the chain rule for the measure-valued Laplacian (see [11]) we have that $h\in D({\mathbf{\Delta}})$ with

[TABLE]

Hence we can, and will, choose $A$ so big that {\mathbf{\Delta}}h\lower 3.0pt\hbox{|{B{r/2}(y)}}\geq 0. Now let $r^{\prime}<r/2$ be such that $B_{r^{\prime}}(y)\subset\Omega$ and notice that for every $\varepsilon>0$ the function $f_{\varepsilon}:=f+\varepsilon h$ is subharmonic in $B_{r^{\prime}}(y)$ and thus according to Theorem 2.3 we have

[TABLE]

Since $\{h<0\}={\rm X}\setminus\bar{B}_{r}(x)$ and $h(y)=0$ we have

[TABLE]

On the other hand, $\partial B_{r^{\prime}}(y)\cap\bar{B}_{r}(x)$ is a compact set contained in $\Omega\setminus C$, hence by continuity and the definition of $C$ we have

[TABLE]

and thus for $\varepsilon>0$ sufficiently small we also have

[TABLE]

This inequality, (2.0.15) and the continuity of $f_{\varepsilon}$ contradict (2.0.14); the thesis follows. $\square$