Quantitative Unique Continuation Principles

• Quantitative UCP is a framework that offers explicit bounds on the vanishing order of PDE solutions by linking coefficient norms to decay rates.
• It combines Carleman estimates, frequency function methods, and three-ball inequalities to derive scale-free and sharp quantitative estimates.
• Its applications extend to spectral theory, control, and inverse problems, enabling robust analysis in multiscale and singular coefficient scenarios.

Quantitative Unique Continuation Principles (UCP) are a family of results that provide explicit, quantitative estimates for the strong unique continuation property of solutions to partial differential equations (PDEs). While qualitative UCP asserts that a solution which vanishes on a set of positive measure or to infinite order at a point must be identically zero, quantitative UCP characterizes the maximal vanishing order or rate at which a solution can decay given specific bounds on the equation's coefficients. This quantification is crucial in spectral theory, control theory, inverse problems, and the paper of random operators. Modern approaches combine advanced Carleman estimates, frequency function monotonicity, and spectral-theoretic methods to produce sharp global and local bounds, including scale-free results for multiscale structures and systems with singular or rough coefficients.

1. Quantitative UCP for Elliptic and Parabolic Equations

Quantitative UCP for elliptic operators is typically formulated as three-ball inequalities, doubling estimates, or explicit power-law vanishing bounds based on coefficient norms. For a second-order elliptic operator $-\nabla\cdot(A\nabla u)+W\cdot\nabla u+Vu=0$ with uniformly elliptic, Lipschitz-continuous matrix $A$, $W\in L^\infty$, $V\in L^\infty$, quantitative SUCP takes the form (Davey, 23 Jun 2025, Davey, 2019, Caro et al., 28 Nov 2024, Choulli, 2022): $$\|u\|_{L^2(B_r(x_0))} \geq r^{C'(K^2+M^{2/3}+\eta+1) e^{c\eta}},$$ where $M = \|V\|_\infty$, $K = \|W\|_\infty$, $\eta$ is the Lipschitz constant of $A$, and $C',c$ are explicit in the dimension and ellipticity. For equations with non-regular lower order terms (e.g., $V\in L^{q_0}$, $W_1\in L^{q_1}$, $W_2\in L^{q_2}$) the vanishing order $N_{\max}$ satisfies (Caro et al., 28 Nov 2024): $$N_{\max} = O\Big( \|V\|_{L^{q_0}}^{\frac{2q_0}{2q_0-d}} + \|W_1\|_{L^{q_1}}^{\frac{q_1}{q_1-d}} + \|W_2\|_{L^{q_2}}^{\frac{q_2}{q_2-d}} \Big),$$ with $q_0 > d/2, q_1 > d, q_2 > d$.

For parabolic equations, using the frequency function approach with Gaussian weight, one establishes an interpolation inequality at a fixed time $T_0$ such as (Zheng et al., 2020, Lu et al., 2013): $$\|u(\cdot,0)\|_{L^2(\Omega)}^2 \leq C\,\exp\{C(M T + M^2 T^2)\}\,\|u(\cdot,T)\|_{L^2(\omega)}^{\gamma}\,\|u(\cdot,T)\|_{L^2(\Omega)}^{1-\gamma},$$ where $\omega\subset\Omega$ is an arbitrary interior observation set, $M$ bounds the lower order coefficients.

2. Carleman Estimates and Frequency Function Methods

Carleman estimates are weighted $L^2$ inequalities crucial for quantitative UCP, especially with rough or singular coefficients (Caro et al., 28 Nov 2024, Zhu, 2017, Davey, 23 Jun 2025, Klein et al., 2014). In higher order or singular coefficient cases, the structure of the Carleman weight and its convexity or pseudoconvexity is fundamental. For $$(-\Delta)^m u + \sum_{|a|\leq 2m-1} V_a(x) D^a u + V_0(x)u = 0$$, the maximal vanishing order satisfies a sharp power-law bound in terms of Lebesgue and uniform norms of $V_a,V_0$, explicitly dictated by the Carleman embedding gain and available Sobolev regularity (Zhu, 2017): $$\|u\|_{L^\infty(B_r)} \geq r^{C \mathcal{M}^\alpha},$$ where $\mathcal{M}$ collects all coefficient norms and $\alpha$ depends on dimension and integrability exponents. For singular (e.g., $L^\infty+L^p$) potentials, vanishing order and three-ball inequalities track subexponential dependence on $K$ and geometric scale $Q$ (Klein et al., 2014).

Frequency function approaches, pioneered in (Davey, 23 Jun 2025, Zhu, 2013), establish monotonicity properties of a solution's local energy-to-boundary flux ratio, bypassing heavy Fourier-analytic Carleman machinery and directly producing three-ball inequalities and doubling bounds.

3. Scale-Free and Multiscale Quantitative UCP

For Schrödinger and elliptic operators on cubes $\Lambda_L$ and multiscale geometries, “scale-free” UCP results have the form (Borisov et al., 2014, Täufer et al., 2016, Peyerimhoff et al., 2017): $$\int_{S_L} |\psi|^2 \geq C \int_{\Lambda_L} |\psi|^2,$$ where $S_L$ is a union of $\delta$-balls with equidistributed centers, $C$ is uniform in $L$ and quantitative in $\delta,\|V\|,\text{energy}$. For spectral subspaces or band-limited states, the constant depends polynomially or subexponentially on the energy or spectral gap, but not on the cube's size. A critical threshold is established: scale-free bounds require exponential decay of spectral tails; polynomial decay is insufficient (Täufer et al., 2016).

These results underpin Wegner-type estimates for random Schrödinger operators, optimal observability for control, and rigorous uncertainty principles.

4. UCP for Hyperbolic Systems and Wave/Elasticity Equations

Quantitative UCP has been extended to wave-type and elasticity systems on manifolds, including cases with jumps in coefficients or transmission interfaces (Filippas, 2022, Hoop et al., 2022). For a wave operator with a piecewise smooth metric or speed $c(x)$ across an interface $S$: $$P = \partial_t^2 - \operatorname{div}_g(c(x) \nabla_g),$$ under suitable geometric and regularity hypotheses, one obtains for all $\mu\geq \mu_0$: $$\|(u_0,u_1)\|_{L^2 \times H^{-1}} \leq C e^{\kappa \mu} \|u\|_{L^2((0,T)\times\omega)} + \frac{C}{\mu} \|(u_0,u_1)\|_{H^1\times L^2},$$ where $\omega$ is an interior observation set and $T > 2\mathcal{L}(\mathcal{M},\omega)$ with $\mathcal{L}$ the largest observation distance in the metric $g_c$ (Filippas, 2022).

In elasticity, UCP yields double-logarithmic stability bounds for recovery of traces and traction at rupture surfaces; explicit constants depend on the minimal speeds, domain geometry, and system coupling bounds (Hoop et al., 2022).

5. Three-Ball Inequalities, Doubling Properties, and Vanishing Order

Three-ball (or three-sphere) inequalities interpolate the size of a solution between nested balls, giving rise to the doubling property and explicit vanishing order bounds. For general elliptic operators with rough lower-order coefficients $A,B,W,V$, one obtains for any $x$ in the domain and $0$$\|u\|_{L^2(B(x,2r))} \leq C \|u\|_{L^2(B(x,4r))}^a \|u\|_{L^2(B(x,r))}^{1-a},$$ with constants explicit in coefficient norms, ellipticity, and geometric parameters (Choulli, 2022). Covering and patching arguments propagate smallness and establish global quantitative UCP bounds. For eigenfunctions of Laplace–Beltrami operators or Schrödinger operators, this leads to uniform lower bounds in terms of eigenvalues or energy bands.

6. Applications: Spectral Theory, Control, Inverse Problems

Quantitative UCP forms the backbone of numerous advanced results:

• Spectral Theory: Scale-free UCP yields optimal Wegner estimates, continuity/Hölder bounds for the integrated density of states, and localization for random Schrödinger operators (Borisov et al., 2014, Peyerimhoff et al., 2017, Täufer et al., 2016).
• Control Theory: Observability and approximate/null controllability results for heat, stochastic heat, and wave equations depend on explicit UCP bounds, with constants quantifying the cost of control and duality arguments (Lu et al., 2013, Zheng et al., 2020, Filippas, 2022).
• Inverse Problems: Stability in inverse spectral problems, e.g., determination of metrics from partial boundary data, relies on UCP-powered lower bounds for eigenfunctions/solutions, ensuring gauge equivalence when spectral data agree in a boundary neighborhood (Choulli, 2022).
• Nodal Set Geometry: Vanishing order controls the doubling index and hence the Hausdorff measure of nodal sets and eigenfunction growth.

7. Optimality, Extensions, and Open Problems

Sharpness of exponents is validated by Meshkov-type counterexamples and Donnelly–Fefferman bounds. The constants in quantitative UCP tend to $2/3$ in bounded potential regimes and blow up near critical Lebesgue exponents. Open directions remain in further reducing regularity assumptions (e.g., interface transmission for matrix-valued coefficients, systems with divergence-form lower order terms), refining sharpness of constants/time bounds in hyperbolic settings, and exploring propagation phenomena in nonconvex geometries and systems.

Quantitative UCP continues to be a central tool in rigorously analyzing PDEs in high-dimensional, random, and multiscale environments, with its implications permeating through spectral theory, control, and inverse problems.