Unique Continuation and Backward Uniqueness

6B.11 Linear Parabolic Equations in Full proved that a parabolic equation forgets: its solutions become smooth instantly, and on a bounded domain they decay to equilibrium like $e^{-\lambda_1t}$. This chapter proves that they never forget completely. Two questions of uniqueness, both about how much a solution is determined by a small part of itself, run through PDE, and both turn out to matter for the Ricci flow.

Unique continuation. If a solution of an elliptic equation vanishes on a small open set, must it vanish everywhere? For harmonic functions the answer is yes, because they are analytic (6A.2 Harmonic Functions), and an analytic function that vanishes on an open set vanishes on its whole connected domain. For an equation $\operatorname{div}(A(x)\nabla u) = 0$ with merely smooth coefficients, solutions need not be analytic, and the question is genuinely hard. It was answered by Torsten Carleman in 1939, for two variables, with a method that is now the main tool of the subject: weighted estimates with exponential weights, Carleman estimates.

Backward uniqueness. If two solutions of the heat equation agree at time $T$, did they agree at all earlier times? The backward heat equation is ill posed (6A.1 What a PDE Is): the past cannot be computed from the present. But it can be determined by it: a solution that vanishes at time $T$ vanished all along. The proof is a convexity argument of Shmuel Agmon and Louis Nirenberg: the logarithm of the $L^2$ norm is a convex function of time. Convexity also measures exactly how badly the backward problem is posed, which is the starting point of the theory of inverse problems.

These are not refinements for specialists. Unique continuation is why eigenfunctions cannot vanish on open sets, why the conductivity inside a body is determined by measurements at its surface (the principle of electrical impedance tomography), and why one can steer the temperature of a whole room by heating a corner of it. Backward uniqueness is why a Ricci flow cannot acquire symmetries it did not have: Brett Kotschwar proved in 2010 that a Ricci flow is determined by its metric at any one time, so isometries of the final metric were isometries all along. And the mechanism of both, monotonicity of a frequency, is the one that 6A.13 Self-Similar Variables and Monotonicity found in the analysis of singularities.

By the end of this chapter you will be able to:

• distinguish weak and strong unique continuation and backward uniqueness, and give the counterexamples that show where hypotheses are needed;
• prove Hadamard's three-circle inequality for harmonic functions and deduce quantitative unique continuation;
• prove the Agmon–Nirenberg log-convexity theorem for $u' = Au$ with $A$ self-adjoint, and deduce backward uniqueness and Hölder stability under an a priori bound;
• explain the conjugation and commutator mechanism of a Carleman estimate, and prove one for the Laplacian with full details;
• use it to show that a Schrödinger operator with bounded potential has no compactly supported eigenfunctions;
• state the unique continuation theorems of Carleman, Aronszajn and Garofalo–Lin, and Pliś's counterexample;
• describe the role of unique continuation in nodal sets, inverse problems and control of the heat equation;
• state Kotschwar's backward uniqueness theorem for the Ricci flow and its consequences for symmetry and holonomy.

12.1Two kinds of uniqueness, and where they fail

Let $P$ be a linear differential operator on a connected open set $U$. $P$ has the weak unique continuation property if every solution of $Pu = 0$ that vanishes on a nonempty open subset of $U$ vanishes on $U$. It has the strong unique continuation property if every solution that vanishes to infinite order at a point (meaning $\int_{B_r(x_0)}u^2 = O(r^N)$ as $r \to 0$ for every $N$) vanishes on $U$. Strong implies weak. For the Laplacian both hold, by analyticity: a harmonic function vanishing to infinite order at a point has all its Taylor coefficients zero there.

For variable coefficients the answer depends on their regularity.

• Lipschitz coefficients suffice. For $\operatorname{div}(A\nabla u) + b\cdot\nabla u + cu = 0$ with $A$ Lipschitz and uniformly elliptic and $b$, $c$ bounded, strong unique continuation holds: Carleman (1939) in two variables, Claus Müller (1954) and Heinz Cordes (1956) and Nachman Aronszajn (1957) in general, with Aronszajn, Krzywicki and Szarski (1962) for Lipschitz $A$; Nicola Garofalo and Fang-Hua Lin (1986) gave a proof by Almgren's frequency.
• Hölder coefficients do not. Andrzej Pliś (1963) constructed an elliptic equation in $\mathbb{R}^3$ with Hölder continuous coefficients (of every exponent less than one) and a nonzero smooth solution vanishing on an open set; Keith Miller (1974) gave an example in divergence form. So unique continuation is not a soft consequence of ellipticity, and it fails just below the threshold where the proofs work.
• Two dimensions are special. In the plane, solutions of divergence-form equations with merely bounded measurable coefficients have unique continuation, because they are related to quasiconformal maps, which compose with holomorphic functions (Bers and Nirenberg, 1954).

For the heat equation, the analogue of weak unique continuation in time is backward uniqueness: if $u_t = \Delta u$ on a bounded domain with Dirichlet conditions (or on a closed manifold) and $u(\cdot, T) = 0$, then $u \equiv 0$ for $0 \leq t \leq T$. It needs a boundary condition or a growth condition at infinity, for the same reason as the forward uniqueness of 6A.3 The Heat Equation on ℝⁿ: the proofs use integrals over the whole space, which must converge, and on unbounded domains the right growth conditions are a delicate matter (Escauriaza, Seregin and Šverák's theorem below is one answer).

12.2Log-convexity: three circles and backward uniqueness

The cleanest proofs of uniqueness are convexity statements. Start with harmonic functions.

The inequality is quantitative unique continuation. If $|u|$ is at most $1$ on the large circle and at most $\varepsilon$ on a small one, then on any intermediate circle it is at most $\varepsilon^{1-\theta}$: smallness on a small set propagates, with a loss measured by the exponent. Letting $\varepsilon \to 0$, a harmonic function vanishing near the centre vanishes on every circle. The same convexity holds in $\mathbb{R}^n$ with spherical harmonics in place of Fourier modes, and its general form is Almgren's frequency function (6A.13 Self-Similar Variables and Monotonicity): $\frac{d}{d\log r}\log M(r)$ is, up to normalisation, the frequency, and its monotonicity is the convexity of $\log M$. Garofalo and Lin's proof of unique continuation for Lipschitz coefficients shows that the frequency is almost monotone in that generality, which is enough.

Now the heat equation. Let $H$ be a real Hilbert space and $A$ a self-adjoint operator on it with $A \leq 0$, for example the Dirichlet Laplacian on a bounded domain with $H = L^2$ (6B.15 Spectral Theory and Heat Trace Asymptotics), or the Laplacian of a closed manifold.

The proof is the parabolic frequency computation of 6A.13 Self-Similar Variables and Monotonicity in another guise: $\frac{F'}{F} = -2\frac{\langle -Au, u\rangle}{\|u\|^2}$ is minus twice the frequency (the Rayleigh quotient of $-A$ at $u(t)$), and convexity of $\log F$ says the frequency is nonincreasing in time. A solution that is "mostly high frequency" decays fast, then becomes mostly low frequency and decays slowly; the frequency can only go down, and a solution that reached zero would need infinite frequency.

How ill-posed is the backward problem? The inequality is also a conditional stability estimate. Suppose we measure $u(T)$ with error $\varepsilon$ and know a priori that $\|u(0)\| \leq M$ (the initial temperature was not astronomically large). Two solutions consistent with these facts differ, at time $t$, by a solution $w$ with $\|w(T)\| \leq 2\varepsilon$ and $\|w(0)\| \leq 2M$, so

At any time $t > 0$ the error goes to zero as $\varepsilon \to 0$, with a Hölder rate $\varepsilon^{t/T}$; at $t = 0$ there is no control at all. This is Fritz John's (1960) notion of a problem that is ill posed but stable under an a priori bound, and the a priori bound is what regularisation methods (Tikhonov regularisation, truncated eigenfunction expansions) impose in practice. The heat equation itself shows the method: the bound $\|u(0)\| \leq M$ limits how much energy the high frequencies can have had, and only then can they be recovered.

For time-dependent operators $A(t)$, Agmon and Nirenberg (1963) proved the same backward uniqueness under conditions on $\frac{d}{dt}A(t)$, and Lees and Protter (1961) obtained parabolic backward uniqueness by an energy method; these are the versions needed for parabolic equations with time-dependent coefficients, such as the heat equation of a metric evolving by the Ricci flow.

12.3Carleman estimates

Log-convexity uses a structure (self-adjointness, a spectral decomposition, a radial variable) that general equations lack. Carleman's idea is to prove uniqueness from an inequality. Choose a weight function $\varphi$ and a large parameter $\tau$, and prove, for all smooth $u$ with compact support in a region,

with $C$ independent of $\tau$. For large $\tau$, the weight $e^{2\tau\varphi}$ magnifies enormously the region where $\varphi$ is largest. If a solution of $Pu = 0$ is cut off so that $P(\chi u)$ is supported only where $\varphi$ is smaller than it is in some region $V$, the estimate forces $u$ to be exponentially small in $V$ compared with an exponentially larger weight, and letting $\tau \to \infty$ forces $u = 0$ in $V$. The art is in the weight.

Conjugation. Write $v = e^{\tau\varphi}u$, so $e^{\tau\varphi}Pu = P_\tau v$ with the conjugated operator $P_\tau = e^{\tau\varphi}Pe^{-\tau\varphi}$. For $P = \Delta$ a direct computation gives

Split $P_\tau = S + B$ into its symmetric and antisymmetric parts in $L^2$:

Then $\|P_\tau v\|^2 = \|Sv\|^2 + \|Bv\|^2 + \langle[S, B]v, v\rangle$, because the cross terms combine into a commutator ($\langle Sv, Bv\rangle + \langle Bv, Sv\rangle = \langle(SB - BS)v, v\rangle$, using $S^* = S$ and $B^* = -B$). Everything depends on the sign of the commutator, and the commutator is positive when the weight is convex in the right sense. For the simplest convex weight it is explicitly positive.

Here is a first consequence, with physical content. In quantum mechanics a bound state of a particle in a potential $V$ is an eigenfunction, $-\Delta u + Vu = Eu$. A particle can be confined to a region by an infinite potential wall, but can it be confined, with zero probability outside a bounded set, by a bounded potential? No.

The corollary is a unique continuation result "from infinity": a solution that vanishes outside a compact set vanishes. The weight $|x|^2$, largest far out, is what points the argument outwards. To prove unique continuation from a small ball outwards, or across a hypersurface, one needs a weight that is largest inside the region of interest and still makes the commutator positive. A decreasing radial weight such as $\varphi = |x|^{-\lambda}$ works for large $\lambda$: it is not convex, but its Hessian is positive on the directions that matter (Hörmander's condition of pseudoconvexity), and the commutator computation goes through with more terms. The classical weights for strong unique continuation are of the form $\varphi = -\log|x| + \varepsilon(\log|x|)^2$ or $|x|^{-\lambda}$; with them, Carleman's method proves all the positive results listed above, for operators with Lipschitz leading coefficients and bounded lower-order terms.

Parabolic Carleman estimates. The same method, with weights depending on $(x, t)$, proves unique continuation and backward uniqueness for parabolic equations with variable coefficients. Luis Escauriaza, Gregory Seregin and Vladimír Šverák (2003) proved backward uniqueness for the heat operator with bounded lower-order terms on a half-space, with no boundary condition at all, by two parabolic Carleman estimates, and used it to show that a solution of the three-dimensional Navier–Stokes equations that stays bounded in $L^3$ cannot blow up: if it did, a blow-up limit would be a solution of a heat-type inequality that vanishes at the final time in a half-space, hence everywhere, a contradiction. The structure of that argument, blow up, find a limit with a rigidity property, and contradict, is the structure of 6A.13 Self-Similar Variables and Monotonicity.

12.4Where unique continuation is used

Nodal sets. An eigenfunction $\Delta\phi + \lambda\phi = 0$ cannot vanish on an open set, so its zero set, the nodal set, is a closed set with empty interior, in fact a hypersurface away from a set of codimension two. Ernst Chladni made nodal lines visible in 1787 by sprinkling sand on vibrating plates; the sand collects where the plate does not move. Courant's nodal domain theorem, that the $k$-th Dirichlet eigenfunction has at most $k$ nodal domains, uses unique continuation in its proof. Shing-Tung Yau conjectured that the $(n-1)$-dimensional measure of the nodal set of $\phi$ on a closed $n$-manifold is comparable to $\sqrt\lambda$; Harold Donnelly and Charles Fefferman (1988) proved this for analytic metrics, using quantitative unique continuation (doubling inequalities, the frequency of this chapter in quantitative form), and Alexander Logunov (2018) proved the lower bound for smooth metrics.

Inverse problems. Alberto Calderón asked in 1980 whether the electrical conductivity $\gamma(x)$ inside a body is determined by boundary measurements: the map sending a boundary voltage $f$ to the current $\gamma\partial_\nu u$ that flows out, where $\operatorname{div}(\gamma\nabla u) = 0$ in the body and $u = f$ on the boundary. This is the mathematical model of electrical impedance tomography, used in medical imaging of the lungs and in geophysics. John Sylvester and Gunther Uhlmann (1987) proved uniqueness for smooth conductivities in dimension $n \geq 3$; Adrian Nachman (1996) in dimension two, and Kari Astala and Lassi Päivärinta (2006) for bounded measurable conductivities in the plane. Unique continuation enters at the start (boundary data determine solutions near the boundary and, by continuation, far inside) and at the end (stability). The stability is only logarithmic (Giovanni Alessandrini, 1988), and Niculae Mandache (2001) showed that no better rate is possible in general: the analogue of the Hölder exponent $t/T$ of the backward heat equation degenerates to a logarithm. That is why EIT images are blurry, and why it is used where its low cost, speed and safety matter more than resolution.

Control of the heat equation. Can one bring the temperature in a whole room to zero at time $T$ by heating and cooling only a small region $\omega$? For the heat equation on a bounded domain the answer is yes, for any $\omega$ and any $T > 0$: Gilles Lebeau and Luc Robbiano (1995) and Andrei Fursikov and Oleg Imanuvilov (1996) proved null controllability by Carleman estimates. By duality, controllability is equivalent to an observability inequality for the backward (adjoint) equation, $\|w(0)\|^2 \leq C\int_0^T\int_\omega w^2$, a quantitative form of unique continuation from $\omega$; the Carleman estimate is how one proves it.

12.5Backward uniqueness for the Ricci flow

The Ricci flow is a heat equation for the metric, and it is natural to ask whether it, too, never completely forgets. The difficulty is the one met in 6A.4 Fourier Methods and Scale Space and 6B.10 Harmonic Coordinates and the Regularity of Metrics: the flow is not strictly parabolic, because of its diffeomorphism invariance, and DeTurck's trick, which fixes this going forward, requires solving a harmonic map heat flow, which cannot be solved backward. Brett Kotschwar found a way around it.

Kotschwar's proof avoids the gauge problem by working not with the metrics but with the differences of their curvature tensors, their connections and their metrics, which satisfy a coupled system: parabolic inequalities for the curvature differences and ordinary differential inequalities in time for the connection and metric differences. He proved Carleman-type estimates for such mixed parabolic–ordinary systems, in the spirit of those of Escauriaza, Seregin and Šverák, and concluded that all the differences vanish. In 2015 he gave a much shorter proof, for a class of geometric flows that includes the Ricci flow, in which the Carleman estimates are replaced by the logarithmic convexity of an energy quantity, following Agmon and Nirenberg: the method of Theorem 12.2, adapted to the mixed system.

The consequences are geometric.

• Symmetries cannot be created. If $\phi$ is an isometry of $g(T)$, then $\phi^*g(t)$ and $g(t)$ are two Ricci flows that agree at time $T$, so they agree for all $t$: $\phi$ was an isometry of $g(t)$ for every earlier time. The forward statement, that a flow preserves the isometries of its initial metric, follows from forward uniqueness; Kotschwar's theorem says the isometry group cannot grow either. A flow that becomes round at the end, such as Hamilton's flow of positive Ricci curvature in dimension three (11A.11 Hamilton’s 1982 Theorem), does not become round at any finite time unless it started round.
• Holonomy and Kähler structures. Kotschwar (2014) used backward uniqueness to show that the holonomy group of the metric is constant along a complete Ricci flow with bounded curvature, so, for instance, a flow that is Kähler at a positive time was Kähler from the start.
• Solitons are rigid. Uniqueness arguments for solitons and for ancient solutions, which compare a given flow with a model at the end of time, use backward uniqueness to propagate agreement from one time slice to the whole flow.

Perelman's proof uses the conjugate heat equation, solved backward in time (9B.7 The Heat Equation on a Manifold, 12A.3 The 𝓦-Entropy); that is a well-posed problem, the adjoint of the forward heat equation, and not a backward heat equation at all. Backward uniqueness is a different and more delicate statement, about the forward equation run in the wrong direction, and the Ricci flow version came eight years after Perelman's preprints.

12.6History

Hadamard's three-circle theorem for analytic functions dates from 1896, and Holmgren's uniqueness theorem from 1901. Carleman proved unique continuation in two variables with his weighted estimates in 1939; Müller (1954), Cordes (1956) and Aronszajn (1957) extended it to higher dimensions, and Aronszajn, Krzywicki and Szarski (1962) to Lipschitz coefficients. Bers and Nirenberg treated the plane in 1954. Hörmander developed the general theory of Carleman estimates from 1958. Pliś's counterexample appeared in 1963 and Miller's in 1974. Lees and Protter proved parabolic backward uniqueness in 1961, Agmon and Nirenberg the log-convexity method in 1963, and John formulated stability under a priori bounds in 1960. Chladni published his figures in 1787. Calderón posed his inverse problem in 1980; Sylvester and Uhlmann solved it for smooth conductivities in 1987, Nachman in the plane in 1996 and Astala and Päivärinta for bounded conductivities in 2006; Alessandrini's logarithmic stability dates from 1988 and Mandache's optimality from 2001. Garofalo and Lin's frequency proof appeared in 1986, Donnelly and Fefferman's nodal estimates in 1988, Lebeau and Robbiano's controllability in 1995, Escauriaza, Seregin and Šverák's Navier–Stokes theorem in 2003, Kotschwar's backward uniqueness for the Ricci flow in 2010 and Logunov's lower bound in 2018.

12.7Exercises

Solution

(a) $\Delta(x_j\partial_ju) = 2\partial_j\partial_ju + x_j\partial_j\Delta u$ (the first term from $\partial_k\partial_k$ hitting $x_j$ twice, once from each factor), so $\Delta Eu - E\Delta u = 2\Delta u$. (b) $E(|x|^2u) = 2|x|^2u + |x|^2Eu$, so $|x|^2Eu - E(|x|^2u) = -2|x|^2u$. (c) $\int(x_j\partial_ju)w = -\int u\,\partial_j(x_jw) = -\int u(nw + x_j\partial_jw)$. (d) For $u = x_1^2$: $Eu = 2x_1^2$, $\Delta Eu = 4$, $E\Delta u = E(2) = 0$, difference $4 = 2\Delta u$. For $u = e^{x_1}$: $Eu = x_1e^{x_1}$, $\Delta Eu = (x_1 + 2)e^{x_1}$, $E\Delta u = x_1e^{x_1}$, difference $2e^{x_1} = 2\Delta u$.

Solution

$F(t) = \sum a_ke^{-\lambda_kt}$ with $a_k = |\hat g(k)|^2 \geq 0$ and $\lambda_k = 8\pi^2|k|^2$. Cauchy–Schwarz gives $(\sum a_k\lambda_ke^{-\lambda_kt})^2 \leq (\sum a_ke^{-\lambda_kt})(\sum a_k\lambda_k^2e^{-\lambda_kt})$, which is $F'^2 \leq FF''$. If $u(T) = 0$, then $F(T) = 0$, so every $a_k = 0$ (each term is nonnegative and $e^{-\lambda_kT} > 0$), and $g = 0$. Equality in Cauchy–Schwarz holds when $\lambda_k$ is constant on the support of $(a_k)$: $\log F$ is linear exactly when $\hat g$ is supported on one sphere $|k| = \rho$, that is when $g$ is an eigenfunction of the Laplacian. Then $F = F(0)e^{-8\pi^2\rho^2t}$.

Solution

By the conditional stability estimate, the difference $w$ of two consistent solutions has $\|w(t)\| \leq (2M)^{1-t/T}(2\varepsilon)^{t/T}$. At $t = 0.05$, $t/T = \frac12$: $\|w\| \leq \sqrt{200\cdot0.002} = \sqrt{0.4} \approx 0.63$. At $t = 0.01$, $t/T = 0.1$: $\|w\| \leq 200^{0.9}\cdot0.002^{0.1} \approx 118\cdot0.54 \approx 64$, hardly better than the a priori bound $2M = 200$. At $t = 0$ the bound is $2M$: no information. (The estimate is sharp in order of magnitude, as the eigenfunction expansion shows: the mode $\sin k\pi x$ is damped by $e^{-k^2\pi^2T}$, so data error $\varepsilon$ allows amplitude up to $\varepsilon e^{k^2\pi^2T}$ at time $0$, limited only by $M$.)

Solution

Take $r_1 = 0.01$ and $r_2 \to 1$. For $r = 0.1$: $\theta = \frac{\log(10)}{\log(100)} = \frac12$, so $M(0.1) \leq 10^{-3}$. For $r = 0.5$: $\theta = \frac{\log 50}{\log100} \approx 0.849$, so $M(0.5) \leq (10^{-6})^{0.151} \approx 0.12$. Smallness near the centre forces smallness further out, but with an exponent that tends to zero as $r$ approaches the outer radius: harmonic continuation outward is possible in principle and ill-conditioned in practice, which is Hadamard's example of 6B.5 Power Series and the Cauchy–Kovalevskaya Theorem in quantitative form.

Solution

(a) Each term has $-\Delta = \pi^2(1 + 4) = 5\pi^2$. (b) $\sin2\theta = 2\sin\theta\cos\theta$ gives $\phi = 2\sin\pi x\sin\pi y\cos\pi y + 2\sin\pi x\cos\pi x\sin\pi y$. Inside the square $\sin\pi x\sin\pi y > 0$, so $\phi = 0$ iff $\cos\pi x = -\cos\pi y = \cos(\pi - \pi y)$, iff $x = 1 - y$. (c) Eigenfunctions are analytic in the interior (the Laplacian has constant coefficients), or by unique continuation for $\Delta + \lambda$. The function $\phi + \varepsilon\sin\pi x\sin\pi y$ is a combination of eigenfunctions with different eigenvalues ($5\pi^2$ and $2\pi^2$), so it is not an eigenfunction: an eigenfunction belongs to a single eigenvalue, and adding a mode of another eigenvalue destroys that, however small $\varepsilon$ is.

Solution

(a) $\phi^*g(t)$ is a Ricci flow (the Ricci tensor is natural under diffeomorphisms) with bounded curvature, and $\phi^*g(T) = g(T)$; by backward uniqueness $\phi^*g(t) = g(t)$ for all $t \leq T$. (b) By (a), $\operatorname{Isom}(g(T)) \subseteq \operatorname{Isom}(g(t))$ for $t \leq T$. Conversely, if $\phi$ is an isometry of $g(t_0)$, then $\phi^*g(t)$ and $g(t)$ agree at $t_0$, so by forward uniqueness they agree for $t \geq t_0$. So the group is constant. (c) The round metric is reached only after normalisation and as $t \to T_{\max}$, where the flow is not defined; at every finite time the metric is not round unless it was round initially, and the theorem is consistent with symmetry appearing only in the limit.