Quantitative Unique Continuation for PDEs

• Quantitative unique continuation for PDEs is a framework that defines explicit bounds on the vanishing orders of solutions by linking them to coefficient properties and domain geometry.
• It employs methods such as Carleman estimates, frequency function monotonicity, and three-ball inequalities to quantify the propagation of smallness.
• This approach enhances stability in inverse problems, control theory, and spectral geometry by offering actionable quantitative estimates for PDE solutions.

Quantitative unique continuation (QUC) for partial differential equations (PDEs) is a framework characterizing the extent to which nontrivial solutions can vanish at a point or on a set, with explicit estimates relating the order of vanishing and propagation of smallness to quantitative properties of coefficients and domains. QUC augments the classical (qualitative) unique continuation property (UCP) with explicit bounds—typically involving Carleman inequalities, frequency function monotonicity, or interpolation theorems. The subject connects elliptic and parabolic regularity theory, inverse problems, control theory, spectral geometry, stochastic analysis, and mathematical physics.

1. Foundational Concepts and Historical Milestones

The classical unique continuation property, formalized through results of Aronszajn, Carleman, and others, asserts that nontrivial solutions to certain analytic or elliptic PDEs cannot vanish to infinite order at any interior point unless identically zero locally. The modern development of quantitative unique continuation, starting with work of Jerison–Kenig, Bourgain–Kenig, Vessella, and Bakri–Lebeau, focuses on deriving explicit bounds on the vanishing order and on the propagation of smallness for solutions, making the dependence on the geometry and potential coefficients explicit (Logunov et al., 2019, Zhu, 2013, Malinnikova et al., 2010).

This quantitative perspective also yields powerful geometric applications, such as Donnelly–Fefferman’s nodal volume estimates for eigenfunctions and stability estimates in inverse problems.

2. Core Methodologies: Carleman Inequalities and Frequency Functions

The quantitative analysis of unique continuation builds on several deep analytic tools:

• Carleman Estimates: Weighted integral inequalities with exponential or singular weights serve as the backbone for proving QUC. In prototypical form, for $Lu = \operatorname{div}(A(x)\nabla u) + c(x) u$,

$$\tau\| |x|^{-\tau-1}u \|_{L^2(B_r)} + \| |x|^{-\tau}\nabla u \|_{L^2(B_r)} \leq C \| |x|^{-(\tau-1)}Lu \|_{L^2(B_R)}$$

with explicit dependence of constants on ellipticity, regularity of $A$, and $c$ (Logunov et al., 2019, Malinnikova et al., 2010). Generalizations accommodate singular lower-order terms ($b \in L^n$, $c \in L^{n/2}$) and degenerate or anisotropic settings (Malinnikova et al., 2010, Banerjee et al., 2016).

• Frequency Function Methods: Almgren-type monotonicity schemes track the ratio of Dirichlet energy to boundary mass, producing doubling inequalities and controlling vanishing order (Logunov et al., 2019, Zhu, 2013, Yu, 2016). For $u$ solving $Lu = 0$, define

$$N(r) = \frac{r \int_{|x|=r} (A\nabla u, \nabla u)}{\int_{|x|=r} u^2}$$

and show near-monotonicity or exponential bounds for $N(r)$, feeding into three-ball and doubling inequalities.

• Three-Ball/Three-Sphere and Remez-Type Inequalities: Log-convexity of $H(r)$, the $L^2$ mass on spheres, yields

$$\|u\|_{L^2(S_r)} \leq C \|u\|_{L^2(S_{r_1})}^\alpha \|u\|_{L^2(S_{r_2})}^{1-\alpha}$$

with exponents and constants derived from the analytic structure (Logunov et al., 2019, Malinnikova et al., 2010).

• Propagation of Smallness and Chains of Balls: QUC translates local smallness on a set of positive measure or interior data into explicit smallness elsewhere, by iterative use of three-ball inequalities, covering arguments, and measure-theoretic chains (Malinnikova et al., 2010, Logunov et al., 2019).

3. Representative Quantitative Results

The scope of QUC has been characterized with explicit inequalities in diverse contexts:

• Elliptic Operators (Second Order): For weak solutions $Lu = 0$, with $A$ Lipschitz and $c$ bounded, one finds (Logunov et al., 2019)

$\|u\|_{L^\infty(K)} \leq C \varepsilon^\gamma$

where $\varepsilon$ is the smallness on a set $E\subset\Omega$, and $\gamma$ depends on measure of $E$, distance to boundary, and ellipticity. Similar Remez-type bounds show exponential decay of measure of small values in terms of vanishing order.

• Higher-Order Elliptic Equations: For $(-\Delta)^m u + V u = 0$ with $V$ bounded (Zhu, 2013, Huang et al., 2015), the vanishing order satisfies

$$\text{ord}_{x_0}(u) \leq C(1 + \|V\|_{L^\infty}^{1/2})$$

and lower bounds at infinity of the type

$$\inf_{|y| = R} \sup_{B_1(y)} |u| \geq C \exp(-A R^\alpha \ln R)$$

with exponents dependent on $m, n$.

• Equations with Singular or Unbounded Potentials: For $Lu = \Delta u + q(x)u$ with $q \in L^{n/2}$ and small norm, quantitative SUCP holds with constants tracing the $L^{n/2}$ norm of $q$ (Choulli, 2023). The local estimate

$$C \|u\|_{L^2(B_{\lambda/2})} \leq 2^{-(X+1)} \|(\Delta+q)(\phi_\lambda u)\|_{L^p(B)} + 3^m \lambda^{m-(X+3)} O_m(u)$$

is sharp in the Lebesgue scale, and generalizes to global statements and more general elliptic operators.

• Fractional and Subelliptic Equations: QUC for nonlocal/fractional Laplacians exploits extension methods and frequency function monotonicity (Yu, 2016), yielding

$$N(r) = \frac{r D(r)}{H(r)} \quad \text{is almost nondecreasing}$$

and bounds vanishing order accordingly.

• Parabolic and Stochastic PDEs: QUC for heat-type equations and stochastic heat equations is proved via Carleman weights, frequency function arguments, and stochastic Itô calculus. For solutions to

$dy - \Delta y dt = a(x,t)y dt + b(x,t)y dW(t)$

with $a \in L^\infty$, $b \in W^{1,\infty}$, one obtains interpolation inequalities

$$\mathbb{E}\int_{\mathbb{R}^n} |y(x,T)|^2 dx \leq C \left(\mathbb{E}\int_\omega |y|^2\right)^{1-\theta} \left(\mathbb{E}\int_{\mathbb{R}^n} |y_0|^2 dx\right)^\theta$$

with explicit dependence on norms of $a$ and $b$, and exponent $\theta$ (Liu et al., 20 Feb 2024, Lu et al., 2013, Duan et al., 2022).

• Complex Drift and Beltrami-Type Equations: For $\Delta u + W \cdot \nabla u = 0$ with complex-valued drifts, the global decay and local vanishing order can be sharply bounded via a reduction to Beltrami systems and weighted Carleman estimates, with the rate at infinity tuned by the decay properties of the imaginary part of the drift (Davey et al., 2020).

4. Applications, Limitations, and Extensions

The explicit bounds of QUC have broad consequences:

• Inverse Problems: Stability and identifiability in inverse boundary-value and scattering problems rely on propagation-of-smallness (quantitative UCP), translating into quantitative stability of reconstructions (Logunov et al., 2019, Choulli, 2023).
• Spectral Geometry: Bounds on nodal sets of Laplace–Beltrami eigenfunctions and vanishing order drive results on eigenvalue distribution and quantum chaos—see Donnelly–Fefferman and Logunov–Malinnikova’s combinatorial propagation techniques (Logunov et al., 2019).
• Control Theory: Observability and null-controllability for parabolic and stochastic PDEs in both bounded and unbounded domains are derived from QUC interpolation and telescoping arguments, with precise dependence on measurement subsets (Liu et al., 20 Feb 2024, Lu et al., 2013, Duan et al., 2022).
• Higher Order and Singular Equations: Carleman estimates and QUC extend efficiently to polyharmonic and subelliptic operators, equations with rough, degenerate, or singular coefficients, and evolutionary PDEs (Banerjee et al., 2016, Zhu, 2017, Huang et al., 2015).

Limitations arise from criticality and regularity thresholds:

• For Schrödinger operators, $L^{n/2}$ is the sharp scale for potentials—failure for lower integrability (Meshkov-type counterexamples).
• In dimension $n=2$, several QUC results become false or are optimal only up to logarithmic corrections (Choulli, 2023, Davey et al., 2020).
• The constants in QUC can degenerate rapidly as coefficients become large or near-borderline.

Possible extensions encompass:

• Nonlinear and system-level PDEs, using a priori $L^\infty$ bounds to reduce semi-linear systems to linear problems (Wang et al., 2021).
• Fractional and nonlocal equations, via extension and frequency methods (Yu, 2016).
• Boundary-point estimates, such as a QUC form of Hopf’s lemma (Choulli et al., 2021).

5. Representative Quantitative Inequality Table

Context	Typical QUC Estimate	Notable Constants/Exponent Dependence
Schrödinger ($q\in L^{n/2}$)	$\|u\|_{L^2(B_r)}\ge C r^{N}$	$N\sim 1 + \|q\|_{L^{n/2}}$ (Choulli, 2023, Zhu, 2013)
Elliptic, bounded lower order terms	$\|u\|_{L^\infty(K)}\le C \varepsilon^\gamma$	$\gamma$ depends on measure, doubling index (Logunov et al., 2019, Malinnikova et al., 2010)
Parabolic/stochastic equations	$E\|y(T)\|_{L^2}\le C\cdots$ interpolation	$C$ and exponent depend on $a,b$ norms, time parameter (Liu et al., 20 Feb 2024, Lu et al., 2013, Duan et al., 2022)

These inequalities are central in practical applications requiring quantitative propagation of information (stability, control, inverse reconstruction).

6. Advanced Directions and Current Research Frontiers

Recent work focuses on fully quantitative SUCP with minimal regularity, making all constants explicit and sharp in terms of Lebesgue/sobolev norms of lower-order coefficients and geometric distances [(Choulli et al., 10 Apr 2025) (abstract), (Choulli, 2023)]. The interplay with control theory, stochastic analysis, and nonlocal/fractional PDEs is rapidly evolving, and QUC is a critical tool for understanding fundamental limits in data-driven inverse problems, controllability of random media, propagation of chaos, and measurement-based system identification.

Extensions to systems, equations on Riemannian manifolds, degenerate or subelliptic structures, and fully nonlinear PDEs require further development of both analytic and combinatorial propagation methods, as well as innovations in Carleman weights, multi-scale frequency monotonicity, and sharp interpolation schemes (Banerjee et al., 2016, Zhu, 2017, Zhu, 2017).

7. Connections to Related Areas

QUC for PDEs is foundational for:

• Spectral theory and nodal set geometry (eigenfunction localization, nodal volumes),
• Inverse boundary value problems (Calderón-type questions),
• Quantitative control and observability results,
• Stochastic PDE and random media analysis,
• Carleman estimate theory and analytic microlocal analysis.

It remains a unifying analytic tool for understanding rigidity, uncertainty, and propagation phenomena across analysis, geometry, probability, and applied mathematics.