Abstract
In this note we establish the weak unique continuation theorem for caloric functions on compact RCD(K, 2) spaces and show that there exists an RCD(K, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this note we study the weak and strong unique continuation properties for the heat equation on \({{\,\textrm{RCD}\,}}(K,N)\) spaces. Consider the equation
Let \(T > 0\). On an \({{\,\textrm{RCD}\,}}(K,N)\) space \((X, \textrm{d}, \mathfrak {m})\), a solution to (1.1) on [0, T] is defined as follows.
Definition 1
A function \(u: X\rightarrow \mathbb {R}\) is a solution (resp. sub-solution, super-solution) of (1.1) if \(u\in W^{1,2}_{loc} (X\times [0,T])\) and that, for all \(\phi \in {\text {Lip}}_0(X\times [0,T])\) which is compactly supported with \(\phi \ge 0\), we have
This paper is concerned with the validity of the strong and weak unique continuation property for the heat equation (1.1). To be precise, we say that
-
The heat equation satisfies the weak unique continuation property on an \({{\,\textrm{RCD}\,}}(K,N)\) space \((X, \textrm{d}, \mathfrak {m})\) if, for any solution \(u \in W^{1,2}(X \times [0, T])\) of (1.1), if u vanishes on any non-empty open subset of \(X \times [0,T]\), then \(u \equiv 0\).
-
A function \(u \in L^2(X \times [0,T])\) vanishes up to infinite order at some \((x_0, t_0) \in X \times [0,T]\) if there exists some \(R > 0\) so that for any integer \(N > 0\), there exists \(C(N) > 0\) so that
$$\begin{aligned} \int _{B_{r}(x_0) \times ((t_0 - r^2, t_0 + r^2) \cap [0, T])} |u|^2\, \, d\mathfrak {m}dt \le Cr^N, 0<r<1 \end{aligned}$$ -
The heat equation satisfies the strong unique continuation property on an \({{\,\textrm{RCD}\,}}(K,N)\) space \((X, \textrm{d}, \mathfrak {m})\) if, for any solution \(u \in W^{1,2}(X \times [0,T])\) of (1.1), if u vanishes up to infinite order at any \((x_0, t_0) \in X \times (0,T]\), then \(u \equiv 0\).
Note that we restrict \(t_0\) to be away from 0 in the definition of strong unique continuation as it is always possible to solve the heat equation starting from some initial data \(u(\cdot , 0)\), which vanishes up to infinite order (spatially) at some \(x_0 \in X\). This would easily imply that the heat flow also vanishes up to infinite order at \((x_0,0)\) in the smooth case . We mention that unique continuation type results are related to giving an upper bound for the measure of the nodal set of non-trivial solutions and that, recently, there has also been work to establish lower bounds in the nonsmooth setting, see [12, 24].
Our first result in Section 1 gives the validity of the weak unique continuation property for compact \({{\,\textrm{RCD}\,}}(K,2)\) spaces:
Theorem 1
Let \((X, \textrm{d}, \mathfrak {m})\) be a compact \({{\,\textrm{RCD}\,}}(K,2)\) space. The heat equation on X satisfies the weak unique continuation property.
As in [26], where the same theorem was shown for harmonic functions, the main idea to handle the spatial direction is to leverage the \(C^{0}\)-Riemannian structure of non-collapsed \({{\,\textrm{RCD}\,}}(K,2)\) spaces from [37, 44]. To handle the time direction, we also show the time analyticity of solutions of (1.1) following the recent [51].
In [26], an \({{\,\textrm{RCD}\,}}(K,4)\) space was constructed on which there exists non-trivial harmonic functions which vanish up to infinite order at some point. The heat flow of any such harmonic function would immediately give a counterexample to strong unique continuation for the heat equation as well. As such, we will be primarily interested in the strong unique continuation property for non-stationary solutions of the heat equation in this paper. We extend our result from [26] as follows in Sections 3 and 5 respectively:
Theorem 2
There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-trivial eigenfunction on it with eigenvalue \(\mu \ne 0\) which vanishes up to infinite order at one point.
Theorem 3
There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-stationary solution of the heat equation on it which vanishes up to infinite order at one point.
Besides these results, we also give frequency estimate on metric horns in Sections 2 and 4 and establish a strong unique continuation type result for eigenfunctions and caloric functions on metric horns, where the classical strong unique continuation property fails by the results of [26].
We have to use the geometry setting of this equation to deal with the difficulty of coefficients which are non-Lipschitz. For discussions on smooth manifolds with Ricci curvarure lower bound and some related questions, see for example [9,10,11, 13,14,15, 19,20,21]. For references on the theory of \({{\,\textrm{RCD}\,}}\) spaces, see [1,2,3, 7, 8, 16,17,18, 23, 25, 27, 28, 34, 38, 43, 47, 48].
2 Weak unique continuation of caloric functions on \({{\,\textrm{RCD}\,}}(K,2)\) spaces
In this section we will consider the case of \({{\,\textrm{RCD}\,}}(K,2)\) spaces. In the smooth setting, the Ricci curvature in 2 dimensions completely determines sectional curvature and hence a Ricci lower bound translates to a sectional curvature lower bound. This is also true for any non-collapsed \({{\,\textrm{RCD}\,}}(K,2)\) space \((X,\textrm{d}, \mathfrak {m})\).
Theorem 2.1
([37]) If \(\mathfrak {m}= c\mathcal {H}^2\) with \(c \ge 0\) (i.e. \((X, \textrm{d}, \mathfrak {m})\) is non-collapsed), then \((X,\textrm{d})\) is an Alexandrov space with curvature bounded below by K.
Therefore, it suffices to consider 2-dimensional Alexandrov spaces of curvature at least K and collapsed \({{\,\textrm{RCD}\,}}(K,2)\) spaces. Alexandrov spaces are known to have a generalized Riemannian structure by [44]. We refer to [44] for relevant definitions.
Theorem 2.2
([44]) Let \((X,\textrm{d})\) be an n-dimensional Alexandrov space and denote \(S_X\) as the set of singular points. Then there exists a \(C^0\)-Riemannian structure on \(X\setminus S_X\subset X\) satisfying the following:
-
(1)
There exists an \(X_0\subset X\setminus S_X\) such that \(X\setminus X_0\) is of n-dimensional Hausdorff measure zero and that the Riemannian structure is \(C^{\frac{1}{2}}\)-continuous on \(X_0\subset X\);
-
(2)
The metric structure on \(X\setminus S_X\) induced from the Riemannian structure coincides with the original metric of X.
In a coordinate neighborhood U given by the \(C^0\)-Riemannian structure of X, with corresponding metric \(g_{ij}\), a solution u of (1.1) satisfies
for all \(\phi \in Lip_0(X\times [0,T])\) supported in \(U\times [0,T]\) (Cf. [35]). Note that since \(g^{ij}\) may not be Lipschitz, the results of [36] do not apply. We will instead use techniques which are special in the 2-dimensional case. For more details, see for example [22].
We proceed by taking local isothermal coordinates. To be precise, consider functions \(\sigma ,\rho :U \rightarrow \mathbb {R}^2\) satisfying
where derivatives are taken with respect to the coordinate chart for U. The existence of such functions is given by [41], see also [6, 22, 32]. This is equivalent to solving the complex equation
where \(\mu =\frac{g^{yy}-g^{xx}-2ig^{xy}}{g^{xx}+g^{yy}+2\sqrt{g^{xx}g^{yy}-(g^{xy})^2}}\). Note that
We have
and so
After rearranging the terms,
and so in the new coordinates, we have
Thus in the coordinate \((\rho ,\sigma )\) the equation becomes
where a is measurable and Hölder on a full measure subset. Note that as now we have a possibly discontinuous coefficient in front of \(\partial _t\), the result in [36] cannot be used. We will instead use a geometrical argument to deal with this difficulty.
Now we are ready to state the main result of this section.
Theorem 1
Let \(u\in W^{1,2} (X\times [0,T])\) be a solution of (1.1) on an \({{\,\textrm{RCD}\,}}(K,2)\) space. If u vanishes on a non-empty open set \(\Gamma \subset X\times (0,T]\), then \(u\equiv 0\) on \(\text {proj}_X(\Gamma )\times [0,T]\). Moreover if X is compact, then \(u\equiv 0\) on \(X\times [0,T]\).
Proof
By a similar discussion as in [26], since all the collapsed cases are trivial we can only consider the non-collapsed case. We assume that u vanishes on a non-empty open set \(V\times (t_1,t_2)\).
Since u satisfies the heat equation, we have the following a priori estimates (cf.[29, Remark 5.2.11]):
for any \(t \in (0,T]\).
We first prove that u vanishes on \(V\times [0,T]\) by showing that \(u(x,\cdot )\) is analytic with respect to t. For the corresponding arguments on Riemannian manifolds see for example [51]. For the reader’s convenience we sketch the argument here.
As in [39], for any \((x_0,t_0)\) with \(0< t_0 < T\), \(k>0\) and \(j\le k\), we consider
So that u is defined on these sets, we extend u to be a solution of (1.1) on \(X \times [0, \infty )\).
By [43], see also [21], we may choose a cutoff function \(\phi _1(x,t)\) supported in \(H^2_j\) such that \(\phi _1=1\) on \(H^1_j\), satisfying
Indeed, \(\phi _1\) may be chosen to be a product of two cutoff functions on the spatial and time coordinates respectively, so that the spatial and time derivatives are well-defined without further measure theoretic arguments. It follows that
where the last inequality follows from integration by parts, Young’s inequality and (2.17). This gives that
Similarly, we may choose a cutoff function \(\phi _2(x,t)\) supported in \(H^1_{j+1}\) such that \(\phi _2=1\) on \(H^2_j\), satisfying
Arguing as before, we have
and so
By using (2.18),(2.21) inductively on j, we obtain
where C at the end also depends on the \(L^2\)-norm of \(u_0\).
From [5, 31, 46, 49] we know that the (p, p)-Poincaré inequality holds on \({{\,\textrm{RCD}\,}}(K,N)\) space for any \(p \ge 1\) (see, for example, [33, Section 4] for a careful discussion of Poincaré inequalties in the \({{\,\textrm{RCD}\,}}(K,N)\) setting). Thus Moser iteration ([31, Lemma 3.10]) shows that for \(R<1\) there exist a mean value inequality
Thus by taking \(R=\frac{\sqrt{t_0}}{\sqrt{2k}}\) and using (2.22), we obtain
Using volume comparison, this implies that \(u(x_0,\cdot )\) is analytic with respect to t. In particular, since u vanishes on \(V \times (t_1,t_2)\) by assumption, u also vanishes on \(V\times [0,T]\).
We now prove the second assertion of the theorem by contradiction. Assume X is compact and \(u\not \equiv 0\).
Let \(\phi _k\) be eigenfunctions of \(-\Delta \) corresponding to eigenvalues \(\lambda _k\) with \(\Vert \phi _k\Vert _{L^2}=1\) and \(0 = \lambda _0\lambda _1 \le \lambda _2 \le ... \rightarrow \infty \), see [30] for a discussion on this. It follows from the estimates obtained in the appendix of [4] that u admits the representation
Let j be the first index for which \(a_j \ne 0\). As it does not affect the argument, we assume for simplicity that the dimension of the eigenspace corresponding to \(\lambda _j\) is 1. This gives that for any \(x_0\in V\),
where we have used [4, Proposition 7.1] to bound the values of \(\phi _k(x_0)\) for \(k > j\) to obtain the second equality. As we assumed that \(a_j\ne 0\), we conclude that \(\phi _j(x_0)=0\). This shows that \(\phi _j(x_0)=0\) for all k and \(x_0\in V\). From [26], this implies that \(\phi _j\equiv 0\), which is a contradiction. It follows that \(u \equiv 0\). Note that similar arguments also work for Dirichlet problem on non-compact space.
Remark 1
The previous argument actually shows the time analyticity of caloric functions with respect to time on any \({{\,\textrm{RCD}\,}}(K,N)\) space, since that part of the argument does not require any assumptions on dimension.
Remark 2
In [36], Fourier transform was used to reduce the problem to a solution of an elliptic equation on \(X \times \mathbb {R}\). In this case, one does not have weak unique continuation for elliptic equations on \({{\,\textrm{RCD}\,}}(K,3)\) spaces, so a different argument had to be used.
Finally, we recall a well-known counterexample given by Miller [40] which indicates that in general we cannot expect weak unique continuation for parabolic operators with time-dependent coefficients even if the time-slices of the corresponding metric have a uniform Ricci curvature bound.
Proposition 1
([40]) There exists a smooth function \(u:\mathbb {R}^2\times [0,\infty )\rightarrow \mathbb {R}\) such that:
where \(A=C=a=b=c=u=0\) on \(t\ge T\), a, b, c, u are smooth on \(\mathbb {R}^2\times [0,\infty )\), A(t), C(t) are smooth on (0, T) and Hölder on \([0,\infty )\). Moreover, u, a, b, c are period in x and y with period \(2\pi \).
3 Elliptic frequency estimate on Metric horn
In this section we will give a frequency estimate on the metric horn, which allows us to prove a form of unique continuation. Recall from [26] that the standard formulation of strong unique continuation does not hold at the horn tip. The form of unique continuation we will prove in this section will therefore assume a higher order of decay at the horn tip, see Remark 1 at the end of the section for more details.
On a weighted warped product \((X,dr^2+f^2(r)g_{S^{n-1}},e^{-\psi (r)}dvol)\), given function \(\varphi \), we have that
away from \(r=0\). In the case of the standard metric horn,
and so,
From the equation of Laplacian on metric horn, in particular, for \(\varphi = r^\alpha \), we have that
and
If we take \(\alpha =2\), then
Given an eigenfunction u with \(\Delta u=\lambda u\), define scale-invariant quantities I(r), E(r) and the frequency function U(r) with respect to the level sets of the distance function \(d_p\), where p is the horn tip. For \(r>0\), we denote
and define
where \(\mathfrak {m}= e^{-\psi }(r)dvol\) is the weighted volume measure of the metric horn and \(\mathfrak {m}_r = e^{-\psi }(r)dvol_r\) is the corresponding weighted area measure on \(\partial B_r\).
We first compute the derivative of I. We remark that all computations are done away from the cone tip p, so no regularity issues will arise from p itself. Let \(\phi \) be any smooth function compactly supported on \((0,\infty )\), we have
This shows that
and so by using (3.5),
For E, from the definition (3.11) and the coarea formula, we have that
Now consider the vector field \(X=\frac{\langle \nabla (d_p^2),\nabla u\rangle \nabla u-\frac{1}{2}|\nabla u|^2\nabla (d_p^2)}{|\nabla d_p|^3}\), we have that
By Integrating both sides in \(B_r\) we have that
Combining (3.15) and (3.17) we have that
Now consider the frequency function U(r). We have that
From Cauchy-Schwarz inequality we know that
Combining (3.19) and (3.20) we have that
As we have that
it follows that
This shows that
and so
which is equivalent to
Combining with (3.14) we obtain
This gives
and so in all
Remark 1
From [26], we know that strong unique continuation fails on metric horn. In particular, any harmonic function with \(u(0)=0\) satisfies \(u(x)=O(e^{-C(\log r)^2})\). However the discussion above shows that if an eigenfunction, in particular a harmonic function, vanishes at the tip up to order \(u(x)=O(e^{-Cr^{-2\epsilon }})\), then \(u\equiv 0\).
4 Failure of strong unique continuation of eigenfuctions on metric horn
In this section we will prove the following theorem:
Theorem 4.1
There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-trivial eigenfunction on it with eigenvalue \(\mu \ne 0\), which vanishes up to infinite order at one point.
Proof
Consider any modified metric horn X constructed in [26, Section 6], which uses techniques in [50]. Let us denote the eigenfunctions on \(S^{n-1}\) as \(\{\varphi _i\}_{i=1}^\infty \) such that
Assume \(\varphi \) is an \(L^2\) function which is smooth away from the tip, then \(\varphi \) may be decomposed as
Therefore, for any eigenfunction on metric horn with eigenvalue \(-\mu \) we have, for the decomposition (4.3) and each i,
for r sufficiently close to 0. We note that we do not have this formula for arbitrary r since the modified metric horn was constructed using a gluing procedure.
For the radial part, that is for \(\mu _0=0\),
Denoting \(c=(n-1)(1+\epsilon )+(N-n)(1-\eta )\), we have
where \(J_\nu ,\,Y_\nu \) are Bessel functions. For a discussion of the properties of Bessel functions, see [42, Appendix B]. We will use the fact that \(J_{\frac{c-1}{2}}(r) \approx r^{\frac{c-1}{2}}\) and \(Y_{\frac{c-1}{2}}(r) \approx r^{\frac{1-c}{2}}\) as \(r \rightarrow 0\). Since \(\varphi \) is assumed to be in \(L^2\), we have that \(f_0\) is in a weighted \(L^2\) space where the weight for small r is of the form \(r^{c}\). Taking this into consideration with the asymptotics of J and Y, we conclude that
For \(\mu _i=i(n+i-2)\) and denoting \(c=(n-1)(1+\epsilon )+(N-n)(1-\eta )\), we obtain
Define
Then
Thus
If we further define
then
and so
If we consider the solution with data \(k_i^{(1)}(r_\mu )=1=(k_i^{(1)})'(r_\mu )\), where \(r_\mu \) is chosen such that \(1\ge \frac{c-1-\epsilon }{2\epsilon }(\frac{c-1-\epsilon }{2\epsilon }+1)r^{-2}-\frac{\mu }{\epsilon ^2} r^{-\frac{2}{\epsilon }-2}\ge 0\) for \(r\ge r_\mu \), that is
So for \(r>r_\mu \) from \((\frac{4\mu _i}{\epsilon ^2}+1)k^{(1)}_i(r)\ge (k^{(1)}_i)''(r)\ge \frac{4\mu _i}{\epsilon ^2}k^{(1)}_i(r)\), we have
Then we have that
is also a solution of (4.12). So we get
And also
From ODE theory we have that \(k_i^{(1)},\,k_i^{(2)}\) form a basis of the solutions of (4.12). This tells us that
As before, we have that \(f_i\) is in a weighted \(L^2\) space from assumption, this tells us that
From (4.15) we have the decay rate \(f_i(r)=O(e^{-Cr^{-\epsilon }})\). This tells us that there exists eigenfunctions on \({{\,\textrm{RCD}\,}}(K,4)\) spaces which are not zero but vanish up to infinite order at one point.
Remark 1
In fact this argument tells us that any eigenfunction on this metric horn with zero integral on each link \(\{r=r_0\}\) vanishes up to infinite order.
Remark 2
Since
the normalized coefficient \(c_{\mu ,i}\) for the eigenfunction \(f_i(r)\varphi _i(\theta )\) with \(L^2\)-norm 1 corresponding to eigenvalue \(\mu \) satisfies
Remark 3
Let us use a more careful argument on \(\varphi \). That is, as we know that
Thus if we denote the pasting radius as \(\tilde{r}\), by the discussion we have
As we know that when transform back
As outside \(\tilde{r}\) the space is a cone, the function is of the form \(f(r) = k_1 r^{(1 - b)/2} J_{1/2 \sqrt{(b - 1)^2 + 4 a}}(\sqrt{c} r) + k_2 r^{(1 - b)/2} Y_{1/2 \sqrt{(b - 1)^2 + 4 a}}(\sqrt{c} r)\). From the asymptotics of Bessel functions we can see that if we consider the eigenfunction on \(B_R\), the normalized coefficient satisfies
5 Parabolic frequency estimate on metric horn
In this section we give a parabolic frequency estimate and the corresponding unique continuation type result on the metric horn. For simplicity, we consider the metric horn which is not modified. For previous discussions on parabolic frequency see for example [20, 45].
Consider the metric horn \((\mathbb {R}^n,\,\,g_{\epsilon ,n},\,\,e^{(N-n)(1-\eta )log(r)}dvol)\) with \( g_{\epsilon ,n}=dr^2+(\frac{1}{2}r^{1+\epsilon })^2g_{S^{n-1}}\). Note that we will use X to denote the metric horn in the following. Let p(x, y, t) be the heat kernel. Following [45], for \(x_0 \in X\) and \(t_0 \ge 0\), we define the backward heat kernel \(G_{x_0, t_0}: X \times (-\infty , t_0) \rightarrow \mathbb {R}\) as \(G_{x_0,t_0}(x,t):= p(x,x_0,t_0-t)\). For simplicity of notation, we denote \(G(x,t) = G_{o,0}(x,t)\). Let u be a solution of (1.1) on \(X \times [-R_0^2, 0]\) for some \(R_0 > 0\).
For \(R_0 \ge R > 0\), define
From the definition of G, we have that
and so
Therefore,
Consider the vector field \(X=\langle \nabla G,\,\nabla u\rangle \nabla u\), then
On the metric horn,
and so
So combining with (5.6) we have that
Moreover,
so from integration by parts we have that
From Cauchy–Schwarz inequality we obtain
So we have that
and so
which is equivalent to that
Since
we obtain
So finally
This gives a strong unique continuation type result for solution of heat equation on the metric horn.
6 Failure of strong unique continuation property of caloric on metric horn
In this section we will prove the following theorem:
Theorem 6.1
There exists an \({{\,\textrm{RCD}\,}}(K,4)\) space and a non-stationary solution of heat equation on it which vanishes up to infinite order at one point.
Similar as in the elliptic case, the heat equation becomes like
Denote the eigenfunctions on \(S^{n-1}\) as \(\{\varphi _i\}_{i=1}^\infty \) such that
We can decompose it as \(\varphi (r,\theta ,t)=f_0(r,t)+\sum _{i=1}^\infty f_i(r,t)\varphi _i(\theta )\). So we have
For \(\mu _i=i(n+i-2)\) and denote \(c=(n-1)(1+\epsilon )+(N-n)(1-\eta )\)
We first consider the solution of heat equation on the compact modified metric horn constructed in [26, Section 6], where p is the tip of the metric horn. We denote the eigenfunctions with eigenvalue \(\nu _j\) with spherical components \(\varphi _i\) as \(q_{\nu _j}(r,\theta )=g_j(r)\varphi _i(\theta )\).
Then we have that
Denote
By [52] we have \(C_1j^{\frac{2}{N}}\le \nu _j\le C_2j^{2}\), thus \(C_1j^{\frac{\epsilon }{N}}\le r_{\nu _j}\le C_2j^{\epsilon }.\)
From (4.15) we have that given any \(100r_{\nu _{k}}\le r^{-\epsilon } <100r_{\nu _{k+1}}\), which indicates that \(\frac{C_1}{k+1}<r<C_2k^{-\frac{1}{N}}\), we have
As the estimate is independent of k, we can see that \(f_i\) vanishes up to infinite order at tip.
Remark 5.9
The argument above actually shows that all caloric functions on the modified metric horn which does not contain the radial part \(f_0(r,t)\) vanishes up to infinite order at the tip.
References
Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric measure spaces with \(\sigma \)-finite measure. Trans. Amer. Math. Soc. 367, 4661–4701 (2015)
Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405–1490 (2014)
Ambrosio, L., Gigli, N., Savaré, G.: Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43, 339–404 (2015)
Ambrosio, L., Honda, S., Portegies, W., Tewodrose, D.: Embedding of \(RCD^{*}(K, N)\)-spaces in \(L^2\) via eigenfunctions. J. Funct. Anal. 280(10), 108968 (2018)
Björn, A., Björn, J.: Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, (2011). xii+403 pp. ISBN: 978-3-03719-099-9
Bers, L., Nirenberg, L.: On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954, pp. 111-140. Edizioni Cremonese, Roma, (1955)
Bruè, E., Naber, A., Semola, D.: Boundary regularity and stability for spaces with Ricci bounded below. Invent. Math. 228(2), 777–891 (2022)
Bruè, E., Semola, D.: Constancy of the dimension for RCD(K, N) spaces via regularity of Lagrangian flows. Comm. Pure Appl. Math. 73(6), 1141–1204 (2020)
Cheeger, J., Colding, T.-H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. 46(3), 406–480 (1997)
Cheeger, J., Colding, T.-H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom. 54(1), 13–35 (2000)
Cheeger, J., Colding, T.-H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differential Geom. 54(1), 37–74 (2000)
Cavalletti, F., Farinelli, S.: Indeterminacy estimates and the size of nodal sets in singular spaces. Adv. Math. 389, 107919 (2021)
Colding, T.-H., Minicozzi, W.P., II.: Harmonic functions with polynomial growth. J. Diff. Geom. 46(1), 1–77 (1997)
Colding, T.-H., Minicozzi, W.P., II.: Harmonic functions on manifolds. Ann. Math. (2) 146(3), 725–747 (1997)
Colding, T.-H., Minicozzi, II, W.P.: A course in minimal surfaces, Graduate studies in mathematics, 121. American mathematical society, Providence, RI, (2011). xii+313 pp. ISBN: 978-0-8218-5323-8
Cavalletti, F., Mondino, A.: Almost Euclidean isoperimetric inequalities in spaces satisfying local Ricci curvature lower bounds. Int. Math. Res. Not. IMRN 5, 1481–1510 (2020)
Cavalletti, F., Mondino, A.: New formulas for the Laplacian of distance functions and applications. Anal. PDE 13(7), 2091–2147 (2020)
Cavalletti, F., Milman, E.: The globalization theorem for the curvature-dimension condition. Invent. Math. 226(1), 1–137 (2021)
Colding, T.-H., Minicozzi II, W.P.: Singularities of Ricci flow and diffeomorphisms, preprint (2021)
Colding, T.-H., Minicozzi, W.P., II.: Parabolic frequency on manifolds. Int. Math. Res. Not. IMRN 15, 11878–11890 (2022)
Colding, T.-H., Naber, A.: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math. (2) 176(2), 1173–1229 (2012)
Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. II: Partial differential equations, (Vol. II by R. Courant.) Interscience Publishers (a division of Wiley), New York-London, (1962) xxii+830 pp
Deng, Q.: Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching, preprint, (2020)
De Ponti, N., Farinelli, S.: Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances. Calc. Var. Partial Diff. Eq. 61(4), 1–17 (2022)
De Philippis, G., Gigli, N.: Non-collapsed spaces with Ricci curvature bounded from below. J. de l’École polytechnique Mathématiques 5, 613–650 (2018)
Deng, Q., Zhao, X.: Failure of strong unique continuation for harmonic functions on RCD Spaces, Preprint, (2021)
Erbar, M., Kuwada, K., Sturm, K.-T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)
Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc. 236, vi–91 (2015)
Gigli, N.: Lectures on nonsmoooth differential geometry. Springer, Berlin (2020)
Honda, S.: Bakry-Émery conditions on almost smooth metric measure spaces. Anal. Geom. Metr. Spaces 6(1), 129–145 (2018)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré, (English summary) Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp
Kenig, C.E.: Some recent applications of unique continuation, Recent developments in nonlinear partial differential equations, 25-56, Contemp. Math., 439, Amer. Math. Soc., Providence, RI, (2007)
Ketterer, C.: Cones over metric measure spaces and the maximal diameter theorem. J. de Mathématiques Pures et Appliquées 103(5), 1228–1275 (2015)
Kapovitch, V., Mondino, A.: On the topology and the boundary of N-dimensional RCD(K, N) spaces. Geometry Topol 25(1), 445–495 (2021)
Kuwae, K., Machigashira, Y., Shioya, T.: Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces. Math. Z. 238, 269–316 (2001)
Lin, F.H.: A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. 43(1), 127–136 (1990)
Lytchak, A., Stadler, S.: Ricci curvature in dimension 2, preprint, (2018)
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169, 903–991 (2009)
Lin, F.H., Zhang, Q.S.: On ancient solutions of the heat equation. Comm. Pure Appl. Math. 72(9), 2006–2028 (2019)
Miller, K.: Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients. Arch. Rational Mech. Anal. 54, 105–117 (1974)
Morrey, C.B., Jr.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43(1), 126–166 (1938)
Maindardi, F.: Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, Imperial College Press (2010)
Mondino, A., Naber, A.: Structure theory of metric-measure spaces with lower Ricci curvature bounds. J. Eur. Math. Soc. (JEMS) 21(6), 1809–1854 (2019)
Otsu, Y., Shioya, T.: The Riemannian structure of Alexandrov spaces. J. Differential Geom. 39(3), 629–658 (1994)
Poon, C.: Unique continuation for parabolic equations. Comm. Partial Diff. Eq. 21(3–4), 521–539 (1996)
Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differential Eq. 44(3–4), 477–494 (2012)
Sturm, K.-T.: On the geometry of metric measure spaces I. Acta Math. 196, 65–131 (2006)
Sturm, K.-T.: On the geometry of metric measure spaces II. Acta Math. 196, 133–177 (2006)
von Renesse, M.: On local Poincar�� via transportation. Math. Z. 258(1), 21–31 (2008)
Wang, B., Zhao, X.: Canonical diffeomorphisms of manifolds near spheres, Preprint (2021)
Zeng, C.: Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations. Comm. Pure Appl. Anal. 21(3), 749–783 (2022)
Zhang, H., Zhu, X.: Weyl’s law on RCD(K, N) metric measure spaces. Comm. Anal. Geom. 27(8), 1869–1914 (2019)
Acknowledgements
We are very grateful to Prof. Tobias Colding for his interest on this unique continuation problem and his constant encouragements. Xinrui Zhao is supported by NSF Grant DMS 1812142 and NSF Grant DMS 1811267.
Funding
Open Access funding provided by the MIT Libraries.
Author information
Authors and Affiliations
Contributions
QD and XZ wrote the main manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Deng, Q., Zhao, X. Unique continuation problem on RCD Spaces. I. Geom Dedicata 218, 42 (2024). https://doi.org/10.1007/s10711-024-00890-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10711-024-00890-7