Carleman Estimate: Theory & Applications

• Carleman estimate is a weighted a priori inequality for PDEs that uses exponential weights and large parameters to rigorously control solution behavior.
• It underpins unique continuation, control theory, and inverse problems, with applications across elliptic, parabolic, hyperbolic, fractional, and degenerate systems.
• Technical methods such as conjugation, integration by parts, and microlocal analysis yield precise stability estimates and quantitative vanishing order results.

A Carleman estimate is a weighted a priori inequality for solutions of partial differential equations (PDEs), typically using an exponential weight with a large parameter. These estimates are quantitatively coercive and are instrumental in unique continuation, control theory, inverse problems, and quantitative analysis of PDEs. Carleman inequalities are sensitive to the structure of the operator—order, type, coefficients, geometry—and thus have developed into a flexible technology embracing elliptic, parabolic, hyperbolic, fractional, degenerate, and coupled systems.

1. Canonical Formulation and Operator Classes

Carleman estimates combine a strong weight (often exponential in a properly chosen pseudoconvex function) with the principal part of an operator and a large parameter. The standard prototype is for a second-order elliptic or parabolic operator $L$ on $\Omega \subset \mathbb{R}^n$: $$\int_\Omega e^{2s\varphi} \left( s^2 |u|^2 + s|\nabla u|^2 \right) dx \leq C \int_\Omega e^{2s\varphi} |L u|^2 dx$$ where $\varphi$ is the Carleman weight and $s \gg 1$.

Notable operator types for which Carleman estimates have been rigorously established with explicit weights, large parameter behavior, and precise coefficient assumptions include:

• Second Order Elliptic: With $C^2$ or $W^{1,\infty}$ coefficients and lower order terms, extending to variable or “degenerate” ellipticities (Li et al., 2023, Nakić et al., 2015, Banerjee et al., 2020).
• Parabolic: Including classical and degenerate cases, often with rough coefficients or interior singularities (Fragnelli et al., 2015).
• Hyperbolic/Wave Operators: In bounded domains and unbounded geometry, for exact controllability and unique continuation (Baudouin et al., 2011, Sun, 2017).
• First-Order Transport and Networks: Utilizing new partition-of-domain strategies and combinatorial graph-theoretic properties (Cannarsa et al., 23 Jul 2025, Ding et al., 2023).
• Higher-Order and Coupled Systems: For $a\partial_t+\partial_x^n$, Stokes/Navier-Stokes, viscoelasticity, thermoacoustics, and population models (Fu et al., 22 Apr 2024, Imanuvilov et al., 2021, Imanuvilov et al., 2017, Shang et al., 2020, Uesaka et al., 2014).
• Fractional/Nonlocal in Time: For Caputo or Riemann-Liouville derivatives, including anomalous diffusion (Lin et al., 2013, Li et al., 2017, Fu et al., 2021).
• Degenerate and Subelliptic Operators: Examples include Baouendi-Grushin and higher-rank stratified structures (Banerjee et al., 2020).
• Discrete and Stochastic-Difference Equations: For full-space/time-discrete approximations and semidiscrete stochastic parabolic equations (Zhu et al., 26 Mar 2025, Wu et al., 28 Mar 2024).
• Free Boundary/Obstacle Problems: For thin obstacle problems with minimal coefficient regularity and applications to blow-up and growth analysis (Koch et al., 2015).

2. Structure of the Weight and Large Parameter

The Carleman weight function is tailored to the PDE, domain, and unique continuation geometry. Typical forms include:

• Quadratic/Gaussian: $\varphi(x) = |x-x_0|^2 - \beta t^2$ for control/observation near a point or region (Baudouin et al., 2011, Shang et al., 2020, Fu et al., 22 Apr 2024).
• Radial/Conic: $\varphi(x) = \ln p(x)$, $p$ a gauge for degenerate or subelliptic geometry (Banerjee et al., 2020).
• Piecewise: For transport on domains admitting a partition into cells satisfying graph-theoretical constraints (Cannarsa et al., 23 Jul 2025).
• Singular in Time or Space: To absorb boundary and lower-order effects, notably in time-fractional or high-order operators (Fu et al., 2021, Lin et al., 2013).
• Limiting Weights: Satisfying degenerate pseudoconvexity for critical unique continuation and CGO constructions (Li et al., 2023).

The large parameter $s$ or $\tau$ features in the exponent of the weight and powers the main terms:

• Absorption Principle: Allows domination of lower-order terms by increasing the weight parameter, thus closing the estimate for rough coefficients or critical potentials (Fragnelli et al., 2015, Li et al., 2023, Baudouin et al., 2011).
• Quantitative Control: Growth estimates, doubling bounds, or vanishing order can be explicitly computed as a function of $s$ (Nakić et al., 2015, Banerjee et al., 2020, Koch et al., 2015).

3. Main Inequalities and Technical Principles

Carleman estimates are often proved by:

• Conjugation/Multiplier Identity: Transforming $u$ to $e^{s\varphi}u$ and analyzing the conjugated operator.
• Integration by Parts: Extraction of positive-definite “bulk” terms with powers of the large parameter, management of boundary/trace contributions.
• Microlocal or Pointwise Structure: For low regularity and nonlocal/pseudodifferential operators, using partition of unity in frequency space or explicit pointwise calculations (Li et al., 2023, Banerjee et al., 2020, Koch et al., 2015).
• Weighted Hardy/Rellich-Poincaré Estimates: Controlling degeneracy/singularity and critical potential terms near points or subdomains (Fragnelli et al., 2015, Banerjee et al., 2020).
• Cut-off/Localization and Gluing: Adapting the argument to domains with coefficients or geometry changing across interfaces (Cannarsa et al., 23 Jul 2025, Ding et al., 2023).

Typical forms (simplified) include: $$\int e^{2s\varphi} (s^k|D^m u|^2 + \dots)\,dxdt \leq C \int e^{2s\varphi}|L u|^2\,dxdt + \text{(boundary/obs)}$$ The precise exponents $k$, derivative order $m$, and absorbing structure depend on operator class and weight.

4. Applications: Unique Continuation, Control, and Inverse Problems

Carleman estimates have generated or underpinned a broad range of sharp results:

• Strong/Quantitative Unique Continuation: For elliptic, parabolic, degenerate, and fractional order systems: vanishing in a region with suitable Carleman geometry implies global vanishing, with explicit vanishing order (Nakić et al., 2015, Li et al., 2023, Banerjee et al., 2020, Lin et al., 2013, Sun, 2017, Li et al., 2017).
• Exact and Null Controllability: Construction of controls for waves, parabolic, and higher-order equations, with energy cost estimates scaling with Carleman parameters (Fragnelli et al., 2015, Baudouin et al., 2011, Imanuvilov et al., 2021, Fu et al., 22 Apr 2024, Fu et al., 2021, Zhu et al., 26 Mar 2025).
• Observation and Observability Inequalities: Forward or adjoint problems with measurement on partial boundary, directly linked to the Carleman weight region (Baudouin et al., 2011, Fragnelli et al., 2015, Ding et al., 2023, Cannarsa et al., 23 Jul 2025).
• Inverse Source and Coefficient Problems: Global Lipschitz or Hölder stability for coefficient or source reconstruction under partial observation, using Carleman-based conditional stability mechanisms (Ding et al., 2023, Imanuvilov et al., 2017, Imanuvilov et al., 2021, Wu et al., 28 Mar 2024, Li et al., 2017, Koch et al., 2015, Shang et al., 2020).
• Blow-ups, Quantitative Vanishing, and Regularity for Free-Boundary Problems: Establishment of semi-continuity of vanishing order, uniform two-sided growth, and almost-optimal regularity for problems with minimal regularity (Koch et al., 2015).
• Discrete and Stochastic PDEs: Robust Carleman technology for numerical schemes (full-discretizations, semi-discretizations), transferring unique continuation and control to the finite-dimensional setting (Zhu et al., 26 Mar 2025, Wu et al., 28 Mar 2024).

5. Technical Innovations and Extensions

Key methodological advances in recent works include:

• Elementary Pointwise/Combinatorial Proofs: Replacing microlocal analysis with pointwise identities and combinatorial “back-propagation” arguments, thus increasing generality and accessibility (Li et al., 2023, Fu et al., 22 Apr 2024).
• Piecewise Weight Design: Tailoring the weight function to cell decompositions respecting transport/flow structure or interface-geometric conditions (Cannarsa et al., 23 Jul 2025, Ding et al., 2023).
• Handling Degeneracy and Non-Smoothness: Carleman estimates for operators with low-regularity coefficients, degenerate or singular weights, and interior degeneracy not previously covered (Fragnelli et al., 2015, Banerjee et al., 2020).
• Stochastic and Discrete Calculus: Carleman inequalities adapted to Itô calculus and spatial/temporal difference operators, with uniform control as mesh parameters vanish (Wu et al., 28 Mar 2024, Zhu et al., 26 Mar 2025).
• Regularity-Wise Minimality: Extension to coefficients in $W^{1,p}$, $p>n+1$, Sobolev or even weaker metrics, while maintaining robust inequality structure (Koch et al., 2015).

6. Notable Case Studies and Theoretical Table

Paper (arXiv)	Operator Class	Weight Structure	Carleman Property	Application/Nuance
(Fragnelli et al., 2015)	Degenerate/singular parabolic	Two-variable, singular at t→0,T	Weighted Hardy-Poincaré absorption	Null-controllability with interior singularity
(Li et al., 2023)	Elliptic, arbitrary dim.	Limiting weight, degenerate pseudoconvexity	Elementary pointwise, no microlocal	CGO construction, unique continuation, inverse problems
(Cannarsa et al., 23 Jul 2025)	1st-order transport	Piecewise–quadratic, cell-based	No global half-space, directed graph	Lipschitz stability for inverse/observable problems
(Imanuvilov et al., 2021)	Linearized Navier–Stokes	Parabolic/elliptic coupled weights	Double Carleman, curl–based splitting	Lateral Cauchy, inverse divergence-free source
(Koch et al., 2015)	Thin obstacle (Signorini)	Log-radial	$W^{1,p}$ coefficients, minimal regularity	Vanishing order, growth, compactness, regularity
(Imanuvilov et al., 2017)	Linear viscoelasticity	Pseudoconvex, spatial exponential	Pseudodifferential cut-off, microlocal factorization	Inverse source stability with Neumann data
(Wu et al., 28 Mar 2024)	Stochastic parabolic (semi-discrete)	Quadratic in $x$, exponential-in-$t$	Discrete Itô, uniform $h$ control	Discrete random source, Cauchy stability
(Banerjee et al., 2020)	Degenerate elliptic, Baouendi–Grushin	Log-gauge $p(z,t)$	Subelliptic, critical Hardy potentials	Quantitative vanishing, strong unique continuation

7. Outlook and Contemporary Developments

Carleman inequalities remain at the core of contemporary analysis for several reasons:

• Inverse Problems: Quantitative stability in the recovery of coefficients or sources under partial and noisy data is almost always grounded in some variant of a Carleman inequality for the underlying PDE, often translated to the numerical or stochastic context as well (Ding et al., 2023, Wu et al., 28 Mar 2024).
• Design of Control and Observation: Both theoretical and constructive algorithms for state transfer or data assimilation use Carleman-weighted functionals for optimization and energy localization (Baudouin et al., 2011, Zhu et al., 26 Mar 2025).
• Low Regularity and Geometry: The extension of Carleman theory to rougher domains, minimal coefficients, and degenerate or geometric operators continues to broaden the class of phenomena amenable to rigorous unique continuation or stability analysis (Banerjee et al., 2020, Koch et al., 2015).
• Emerging Directions: Interactions with stochastic PDEs, fully discrete models, geometric flows, and hybrid nonlocal-local systems are active ongoing areas, with adaptation of Carleman structure for new settings (Wu et al., 28 Mar 2024, Fu et al., 22 Apr 2024, Sun, 2017).

Recent works have also pushed Carleman estimates into the field of quantitative vanishing order bounds, three-ball inequalities, and scale-free unique continuation, exploiting explicit parameter dependence and effective constants (Nakić et al., 2015, Banerjee et al., 2020, Koch et al., 2015).