Quantitative Unique Continuation

• Quantitative unique continuation is a framework that provides explicit, computable bounds on the rate at which nontrivial PDE solutions vanish near a point.
• It employs rigorous methods like Carleman estimates, frequency function techniques, and three-ball inequalities to derive precise dependencies on lower-order coefficients and geometric data.
• These quantifiable results enhance stability in inverse problems and observability in various settings including elliptic, parabolic, hyperbolic, and fractional discrete operators.

The quantitative unique continuation property (QUCP) provides explicit rates and exponents controlling the vanishing order, propagation of smallness, and observability for PDE solutions, notably beyond the qualitative unique continuation property (UCP) that only determines whether zero sets can propagate. In the quantitative setting, sharp dependencies on lower-order coefficients, geometric data, and the nature of the underlying operator (elliptic, parabolic, hyperbolic, higher-order, subelliptic, stochastic, or fractional discrete) are crucial. The development of these results has intertwined Carleman estimates, frequency function approaches, three-ball/doubling arguments, and parabolic or wave-specific techniques, generating a mature and interconnected array of results across diverse linear and some nonlinear PDE classes.

1. Fundamental Concepts and Definitions

The quantitative unique continuation property stipulates explicit, computable bounds on the order of vanishing or on how small nontrivial solutions to a given PDE can become near a point or in a domain, in terms of norms of the lower-order coefficients or right-hand side data. For a solution $u$ of a (typically linear) PDE $\mathcal L u = 0$ in a domain $\Omega$, the (local) order of vanishing $\mathcal{V}(x_0)$ at a point $x_0$ is defined via (e.g. in the elliptic case)

$$\mathcal{V}(x_0) = \inf\big\{ N>0 : \limsup_{r\to 0^+} r^{-N} \|u\|_{L^2(B_r(x_0))} > 0 \big\}.$$

Explicit upper bounds for $\mathcal{V}(x_0)$ are then derived in terms of quantities like $\|V\|_{L^t}$, $\|W\|_{L^s}$, or their Sobolev norms, for lower-order coefficients $V$, $W$ or their gradients.

The typical form of a quantitative lower bound is:

$\sup_{B_r(x_0)} |u| \geq c\, r^{\beta},$

with $\beta$ depending explicitly on norms of the coefficients, or an analogous propagation-through-domain estimate:

$$\|u\|_{L^2(\Omega')} \leq C \|u\|_{L^2(\omega)}^\theta \|u\|_{L^2(\Omega)}^{1-\theta}$$

for subdomains $\omega \Subset \Omega' \Subset \Omega$.

These estimates supersede classical SUCP by yielding not just uniqueness but control on how rapidly solutions can degenerate, underpinning stability in inverse problems and null controllability.

2. Elliptic Equations: Sharp Vanishing Order and Three-Ball Inequalities

For second-order elliptic equations, Bourgain–Kenig's sharp exponents for the vanishing order (and its extensions) are foundational. For $-\Delta u + W\cdot\nabla u + V u = 0$ on $\Omega\subset\mathbb{R}^n$,

• For $V\in L^t$ with $t>n/2$, $W\in L^s$ with $s>n$,
• The vanishing order at $x_0$ is bounded above by $$\mathcal{V}(x_0) \leq C\big(\|V\|_{L^t}^{\mu_0} + \|W\|_{L^s}^{\kappa_0}\big)$$ with explicit $\mu_0$, $\kappa_0$ (Davey, 2019, Caro et al., 2024).

When $V, W \in L^\infty$, the sharp exponents become $$\mathcal{V}(x_0) \leq C(\|V\|_{L^\infty}^{2/3} + \|W\|_{L^\infty}^2)$$ (Davey, 23 Jun 2025). This is reflected in both frequency function approaches and Carleman estimates. The "doubling index" (or three-ball lemma) states that

$$\|u\|_{L^2(B_{2r}(x_0))} \leq D \|u\|_{L^2(B_r(x_0))},$$

with $D$ exponential in these coefficient norms.

Modern approaches such as the frequency function method, generalizing Almgren's monotonicity, yield direct control on the doubling constant and, ultimately, the vanishing order. In variable-coefficient settings, the bound becomes

$N \leq C\left(1 + R\|W\|_{L^\infty} + R^2\|V\|_{L^\infty} + R(\text{Lipschitz norm of $A$}) \right)$

for generalized Schrödinger operators $$-\nabla\cdot(A(x)\nabla u) + W(x)\cdot\nabla u + V(x)u$$ on $B_R(x_0)$ (Davey, 23 Jun 2025).

For equations with rough or singular coefficients ($V\in L^t, t>n/2$), or including div-form lower order perturbations, explicit polynomial dependencies in the norm exponents are obtained by refined Carleman estimates and Wolff's measure concentration argument (Caro et al., 2024).

3. Parabolic and Stochastic Equations: Temporal Doubling, Frequency, and Observability

For parabolic equations such as $$\partial_t u - \Delta u = w_j(x,t)\partial_j u + v(x,t)u$$, with $v, w$ bounded, the rate of vanishing matches the elliptic situation up to exponents:

$$\mathcal{V}(x_0, t_0) \leq C\left(\|v\|_{L^\infty}^{2/3} + \|w\|_{L^\infty}^2\right)$$

(Camliyurt et al., 2017). The proof relies on Carleman inequalities with parabolic weights, three-cylinder (space-time) interpolation, and monotonicity of a heat-kernel-weighted frequency function.

On compact Riemannian manifolds or with variable coefficients of limited regularity, analogous results are derived where the vanishing order depends quadratically or polynomially on the $C^{1,1}$ or $L^\infty$ norms of the coefficients (Zhu, 2017).

For stochastic parabolic equations (e.g., stochastic heat equations), the propagation of observability and null controllability is quantitatively established through Carleman-type weighted energy estimates for the stochastic system, with explicit dependence on the bounds of the stochastic coefficients (Lu et al., 2013).

4. Hyperbolic and Wave Operators: Quantitative Propagation and Boundary Effects

For the wave equation $\square u + q u = f$ on a spacetime domain, the quantitative unique continuation can be expressed in terms of a geometric constant $\mathfrak{C}(\delta)$ controlling how smallness in a subdomain propagates to the maximal domain allowed by the finite speed of propagation (Filippas et al., 18 Feb 2025):

$$\|u\|_{L^2(D_\delta)} \leq \mathfrak{C}(\delta) \|u\|_{H^1(C)} \cdot [\log(\cdots)]^{-1},$$

with $\mathfrak{C}(\delta) \leq (1/\delta)^{N/\delta^4}$ as $\delta \to 0$, which is essentially optimal. The sharp rate of blowup in the stability constant is a signature of unique continuation up to the maximal geometric locus.

For anisotropic wave equations, similar propagation and strong unique continuation results hold, with quantitative interior estimates exhibiting logarithmic rates ("single-logarithm") (Vessella, 2014, Sincich et al., 2016). These results often involve reduction to auxiliary degenerate elliptic problems in higher dimension, Carleman inequalities with adapted weight functions, and three-cylinder inequalities across overlapping regions.

Boundary analogues (e.g., for wave equations with Robin condition) yield quantitative SUCP at the boundary, where the scale of logarithmic decay quantifies vanishing rate up to the boundary (Sincich et al., 2016).

5. Higher-Order, Subelliptic, and Discrete Operators

Higher-Order Equations

Quantitative unique continuation extends to polyharmonic or higher-order elliptic operators. For $$(-\Delta)^m u + \sum_{|\alpha| \leq 2m-1} V_\alpha(x) D^\alpha u + V_0(x) u=0$$, the vanishing order at $x_0$ is controlled linearly in

$$\beta = C\left(1 + \sum_{1\leq |\alpha|\leq 2m-1}\|V_\alpha\|_{L^\infty} + \|V_0\|_{L^s}\right),$$

with sharpness achieved in constant coefficient cases (Zhu, 2017). For bi-Laplacian equations with $V \in W^{1,\infty}$, the optimal vanishing order is $$C\big(\|V\|_{L^\infty}^{1/4} + \|\nabla V\|_{L^\infty} + 1\big)$$ (Liu et al., 2023).

Subelliptic Operators

For subelliptic (e.g., Grushin-type) operators of fourth order, the vanishing order bound is of doubly exponential character, a reflection of both degeneracy and operator order:

$$\|u\|_{L^\infty(B_r)} \geq c\, r^{C \overline{C}^{K_1} K_1^2}$$

for potentials $V$ satisfying $|V| \leq K_1, |ZV| \leq K_2 \psi$ (Qiu et al., 22 Nov 2025). This is less sharp than in elliptic cases but is dictated by the degeneracy structure.

Discrete and Fractional Discrete Laplacians

For the fractional discrete Laplacian, global unique continuation fails strictly, but a quantitative three-balls interpolation up to exponentially small (in mesh size) error holds:

$$\|\tilde{u}\|_{L^2(B^+_{r_0})} \leq C \|\tilde{u}\|_{L^2(B^+_1)}^{\alpha} \| \partial_t\tilde{u} \|_{L^2(B^+_1)}^{1-\alpha} + C e^{-c/h} \|u\|_{L^2(B_1)}$$

establishing nearly sharp control provided the mesh $h$ is small (Fernández-Bertolin et al., 2022).

6. Methodologies: Carleman Estimates, Frequency Functions, and Doubling

Carleman Estimates

Carleman inequalities, weighted $L^2$ (or $L^p-L^q$) inequalities for solutions of $\mathcal{L}u$, are the backbone for producing quantitative unique continuation. The weight functions are adapted to the operator and the geometry:

• For elliptic: logarithmic weights or $\phi(x) = \log r + \log(\log r)^2$
• For parabolic: backward heat kernel weights
• For hyperbolic: cone/adapted coordinates to the wave cone

These estimates are sharp enough to allow absorption of lower-order perturbations at the right exponent, yielding propagation of smallness and three-ball-type inequalities (and conditional observability in control theory).

Frequency Function Techniques

Frequency function approaches generalize Almgren's monotonicity formula to various operators, yielding explicit monotonicity and control over the "growth rate" or doubling exponent of solutions. In variable coefficient or lower-order settings, they provide transparent dependence on boundedness/Lipschitz/other norms (Davey, 23 Jun 2025Zhu, 2013).

Wolff's Measure Concentration and Parametrix

To treat equations with minimal integrability (nonregular) lower order terms, Wolff’s lemma is used in combination with scale-sharp Carleman estimates and parametrices, producing three-ball inequalities with explicit exponents in the Lebesgue norms of the coefficients (Caro et al., 2024).

7. Applications, Generalizations, and Open Problems

Applications of quantitative unique continuation include:

• Stability and logarithmic observability in inverse problems (e.g., Calderón problem, recovery of coefficients from solution observations)
• Null controllability and observability for parabolic, wave, and stochastic equations (1305.38882001.01882)
• Sharp bounds on nodal set sizes for eigenfunctions and random waves
• Lower bounds for decay in Landis’ conjecture or limitations thereof (minimum decay rates at infinity for Schrödinger, drift equations and their complex/degenerate analogues) (Davey et al., 2020, Davey, 23 Jun 2025).

Open questions involve:

• Critical integrability thresholds for lower-order terms where unique continuation fails or where only qualitative versions hold
• Optimal exponents for vanishing order in variable coefficient or non-uniformly elliptic/higher-order settings
• Extension to fully nonlinear PDE, systems, operators on manifolds or rough domains
• Relationships to spectral theory, random media, and the approximation theory in the discrete-to-continuum limit

Table: Summary of Key Quantitative UCP Results Across Operator Classes

Operator/Class	Vanishing Order/Decay Bound	Main Dependencies
$-\Delta u + W\cdot\nabla u + Vu=0$, $L^\infty$	$C(\|V\|^{2/3} + \|W\|^2)$	$L^\infty$ norms (Davey, 23 Jun 2025)
$-\Delta u + Vu=0$, $V\in L^t$,$t>n/2$	$C \|V\|_{L^t}^\mu$, explicit $\mu$	$L^t$ norm (Davey, 2019)
Parabolic, bounded coefficients	$C(\|v\|^{2/3}+\|w\|^2)$	$L^\infty$ bounds (Camliyurt et al., 2017Zhu, 2017)
Polyharmonic, lower order $L^\infty/L^s$	$C(1+\sum\|V_\alpha\| + \|V_0\|_{L^s})$	$L^\infty$, $L^s$ (Zhu, 2017)
Fourth-order Grushin (subelliptic)	$r^{C \overline{C}^{K_1}K_1^2}$	$\|V\|_{L^\infty},\, \|ZV\|$ (Qiu et al., 22 Nov 2025)
Stochastic Heat	$$\|y(T)\|_{G}\leq C\|y(0)\|_{G}^{1-\delta}\|y(T)\|_{G_0}^{\delta}$$	$L^\infty$ bounds (Lu et al., 2013)
Wave, maximal UCP domain	$C(\delta)\leq (1/\delta)^{N/\delta^4}$	Distance to propagation boundary (Filippas et al., 18 Feb 2025)
Parabolic/forward inequality	$\|u(\cdot,0)\|_{L^2(\Omega)} \leq \cdots$	$M, T,$ geometric parameters (Zheng et al., 2020)
Fractional discrete Laplacian	Lower bound up to $e^{-c/h}$	Lattice mesh $h$ (Fernández-Bertolin et al., 2022)

• (Davey, 23 Jun 2025) A frequency function approach to quantitative unique continuation for elliptic equations
• (Caro et al., 2024) Quantitative unique continuation for non-regular perturbations of the Laplacian
• (Davey, 2019) Quantitative unique continuation for Schrödinger operators
• (Camliyurt et al., 2017) Quantitative unique continuation for a parabolic equation
• (Zhu, 2017) Quantitative uniqueness of solutions to parabolic equations
• (Qiu et al., 22 Nov 2025) Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential
• (Liu et al., 2023) Quantitative unique continuation property for solutions to a bi-Laplacian equation with a potential
• (Zhu, 2017) Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients
• (Zhu, 2013) Quantitative uniqueness of elliptic equations
• (Lu et al., 2013) Unique Continuation for Stochastic Heat Equations
• (Filippas et al., 18 Feb 2025) On the blowup of quantitative unique continuation estimates for waves and applications to stability estimates
• (Zheng et al., 2020) A unique continuation property for a class of parabolic differential inequalities in a bounded domain
• (Fernández-Bertolin et al., 2022) On (Global) Unique Continuation Properties of the Fractional Discrete Laplacian

These works collectively form the backbone of modern quantitative unique continuation theory, establishing rigorous dependencies of smallness propagation on lower-order coefficients and providing a platform for further studies in control, inverse problems, and model analysis in both deterministic and stochastic frameworks.