Quantitative Runge Approximation Theorem

• The quantitative Runge approximation theorem is a method to precisely approximate local PDE solutions with global ones while controlling error and norm growth.
• It employs duality, spectral truncation, and unique continuation methods to create explicit error bounds and manage the approximant’s norm inflation.
• Applications include inverse problems, control theory, and stability analysis across elliptic, parabolic, and nonlocal PDEs.

The Quantitative Runge Approximation Theorem is a central result in the theory of partial differential equations (PDEs) that extends the classical Runge density property to an explicit, stable approximation with quantitative control over the error and—equally critically—the norm of the global approximant. It plays a pivotal role in the analysis of inverse problems, unique continuation, control theory, and the propagation of smallness for elliptic, parabolic, hyperbolic, and nonlocal equations. Quantitative Runge theorems provide explicit estimates on how accurately solutions on a local domain can be approximated by globally defined solutions, systematically quantifying the trade-off between error and growth of the approximant.

1. Foundational Principles and General Formulation

A quantitative Runge approximation theorem asserts that, given a solution $v$ to a PDE locally (on a set $K$), one can approximate $v$ in a prescribed norm with a global solution $u$, with the approximation error and the norm of $u$ controlled explicitly in terms of the error parameter $\epsilon$. While the classical (qualitative) Runge theorem establishes density of global solutions in local solution spaces, the quantitative version establishes inequalities such as

$\|v-u\|_{L^2(K)} \leq \epsilon \|v\|_{H^1(L)}$

and

$\|u\|_{H^1(M)} \leq C \epsilon^{-s} \|v\|_{L^2(L)}$

for domains $K \subset L \subset M$ with $u$ global and all constants explicit and independent of $v$ (Rüland et al., 2017, Debrouwere et al., 2022).

The essential components are:

• Explicit approximation error bounds: The difference between the local target and the global approximant is controlled linearly in $\epsilon$.
• Norm growth estimates: The norm of the global approximant may grow polynomially or even exponentially in $1/\epsilon$, depending on geometry, operator, and boundary conditions.
• Loss of derivatives: Most results permit at most a loss of one derivative in the approximation norm due to technical limitations of unique continuation.

2. Core Methodologies and Proof Schemes

The construction of quantitative Runge approximants relies on:

• Duality and functional analytic reduction: The approximation problem is reframed via dual operators (often via the Poisson map) and singular value decomposition (SVD), reducing the approximation to truncating spectral expansions with explicit control (Rüland et al., 2017, García-Ferrero et al., 2021).
• Quantitative unique continuation: Sharp, explicit three-ball or Carleman-type inequalities are established for the adjoint problem, which provide the rate at which smallness can propagate from boundary data to the interior. The logarithmic character of the unique continuation estimate is reflected in the exponential norm growth in the approximant (García-Ferrero et al., 2021, Rüland et al., 2020).
• Spectral truncation and regularization: SVD techniques are used to truncate expansions at a singular value matched to $\epsilon$, yielding an explicit trade-off between approximation error and norm inflation.

These elements are synthesized in concrete settings to establish sharp rates: $\|v-u\|_{L^2(K)} \le \epsilon \|v\|_{H^1(K)}$

$$\|u\|_{H^1(M)} \le C \exp(C\epsilon^{-\mu})\|v\|_{L^2(K)}$$

with constants $C,\mu$ depending on geometry and the coefficients of the operator (Rüland et al., 2017, Rüland et al., 2020, García-Ferrero et al., 2021).

3. Representative Theorems and Model Operators

General Constant-Coefficient Operators

Debrouwere–Kalmes (Debrouwere et al., 2022) formalize the “quantitative Runge property” for constant-coefficient operators: For any $P(D)$, on convex sets,

$$\|f-h\|_{K,r_1} \leq \epsilon \|f\|_{L,r_1+1}, \quad \|h\|_{M,r_2} \leq C\epsilon^{-s}\|f\|_{L}$$

with explicit loss of one derivative.

Second Order Elliptic Operators

For elliptic divergence-form operators $L$ with measurable coefficients on bounded Lipschitz domains, one has (Rüland et al., 2017, Rüland et al., 2020): $$\|h-u\|_{L^2(D_1)} \leq \epsilon \|h\|_{H^1(D_1)},\qquad \|u\|_{H^{1/2}(\partial D_2)} \leq C\exp(C\epsilon^{-\mu}) \|h\|_{L^2(D_1)}.$$ If $h$ extends beyond $D_1$, a polynomial dependence in $1/\epsilon$ is possible (Rüland et al., 2017).

Helmholtz and Maxwell Systems

For the Helmholtz operator $L_k u = \Delta u + k^2 q(x) u + V(x) u$, the quantitative Runge bound in non-convex domains is

$$\|u-v\|_{L^2(\Omega_1)} \leq \epsilon \|v\|_{H^1(\Omega_1)},\quad \|u\|_{H^{1/2}(\partial\Omega_2)} \leq C \exp(C k^s \epsilon^{-\mu})\|v\|_{L^2(\Omega_1)}$$

While for Maxwell’s equations, an explicit $j^{-\theta}$ algebraic rate in the number $j$ of singular modes is obtained, with double-exponential norm growth in $j$ (Pohjola, 2021).

Parabolic, Fractional, and Stokes Problems

For the (fractional) heat equation and nonstationary Stokes system, analogous results hold with explicit cost bounds, derived via dual variational formulations and semigroup/contour integral representations (Rüland et al., 2017, Higaki, 20 Nov 2025, Higaki et al., 30 Aug 2024). The Runge approximant for the 3D unsteady Stokes system globally exhibits at least exponential spatial growth, due to the presence of parasitic (Serrin-type) solutions (Higaki et al., 30 Aug 2024).

4. Geometric and Operator-Dependent Phenomena

Quantitative Runge theorems display significant sensitivity to geometry and the operator class:

• Interior–exterior configuration: Exponential (or worse) growth of the norm of the approximant is generic when the “gap” between local and global region is nonempty and non-convex (García-Ferrero et al., 2021).
• Convexity and monotonicity: Convex geometries or monotonic coefficients can lead to polynomial (or even linear) dependence upon $\epsilon$ or frequency parameters, as shown for Helmholtz in radially-increasing media (García-Ferrero et al., 2021).
• Nonlocal and parabolic cases: Fractional or nonlocal operators involving exterior data require Runge theorems in lower regularity spaces, and their density result may be sharp only in non-smooth spaces like $L^2(0,T;\widetilde H^s(\Omega))$ (Zimmermann, 3 Dec 2024, Rüland et al., 2017).

Operator/class	Approximation error	Norm growth of approximant	Geometry/regularity
Elliptic, general	$\epsilon$	$\exp(C \epsilon^{-\mu})$	Non-convex, Lipschitz
Elliptic, convex	$\epsilon$	Polynomial in $1/\epsilon$	Convex, monotone coefficients
Parabolic/fractional	$\epsilon$	$C\exp(C\epsilon^{-\sigma})$	Minimal $\widetilde H^s$ interior space
Maxwell	$j^{-\theta}$	$C\exp(Cj^{2/m})$	$L^p$ localized modes
Stokes (unsteady)	$\epsilon$	At least exponential	Exponential growth forced by nonuniqueness

5. Applications in Inverse Problems and Control

Quantitative Runge theorems underpin the stability theory in several inverse boundary value problems:

• Local Calderón problem: They enable the reduction of stability from boundary to interior via recursive application and CGO solutions, with the modulus of continuity dictated by the Runge cost (Rüland et al., 2017, Rüland et al., 2020).
• Schrödinger and heat inverse problems: The exponential scaling is propagated in chain-estimates for Lipschitz or logarithmic stability.
• Nonlocal inverse problems: Precise Runge bounds enable conversion from qualitative uniqueness to explicit stability estimates involving the boundary operator difference (Rüland et al., 2017, Zimmermann, 3 Dec 2024).
• Controllability and observability: Quantitative Runge estimates are equivalent to controllability cost in parabolic and nonlocal PDEs (Rüland et al., 2017).

6. Optimality, Limitations, and Extensions

Optimality

The generally exponential or super-exponential nature of the norm growth is sharp, as demonstrated by explicit families of spherical harmonics or high-frequency modes: approximation of highly oscillatory local solutions necessitates exponentially large global data (Rüland et al., 2017, García-Ferrero et al., 2021, Higaki et al., 30 Aug 2024).

Limitations

• Minimum reachable interior regularity: For nonlocal wave equations, the optimal Runge approximation set is exactly $L^2(0,T;\widetilde H^s(\Omega))$, and cannot be extended to higher regularity or $L^2$ norms in the whole domain (Zimmermann, 3 Dec 2024).
• Temporal regularity: In parabolic and Stokes cases, the absence of control over time-smoothness is an inherent restriction due to the possible presence of parasitic or residual modes (Higaki et al., 30 Aug 2024).
• Boundary data inflation: In exterior control problems, the norm of the required boundary/exterior data grows uncontrollably as the target error shrinks.

Extensions

• Variable coefficient and non-elliptic operators: Quantitative Runge theory extends, sometimes with analytic, topological, or PDE-theoretic adjustments, to variable coefficient settings, semi-elliptic, and parabolic/hyperbolic equations (Debrouwere et al., 2022).
• Nonlinear problems and quasi-linearization: Linear Runge theory underpins nonlinear inverse problems through iterative linearization, although explicit quantitative estimates in this setting are only partially developed (Debrouwere et al., 2022, Zimmermann, 3 Dec 2024).

7. Fundamental Research Directions and Open Problems

Current research directions include:

• Optimal cost determination: Improving the exponential dependence to polynomial or linear rates, particularly in convex or monotonic geometric settings (García-Ferrero et al., 2021, Debrouwere et al., 2022).
• Nonlocal and multi-scale operators: Extending robust, operator-independent methodologies to fractional, pseudo-differential, and nonlocal equations with rough coefficients (Rüland et al., 2017, Zimmermann, 3 Dec 2024).
• Density in higher regularity spaces: Characterization and limitations for Runge approximation in Sobolev spaces $H^s$ with $s>1$.
• Interaction with control theory: Tighter linkages between Runge cost, control cost for PDEs, and propagation-of-smallness phenomena.

The theory of quantitative Runge approximation has become a unifying framework for stability, control, and unique continuation in PDE analysis, with significant ramifications for both theoretical and applied problems in mathematical physics, engineering, and data science (Rüland et al., 2017, Debrouwere et al., 2022, García-Ferrero et al., 2021, Rüland et al., 2017, Higaki, 20 Nov 2025, Zimmermann, 3 Dec 2024, Higaki et al., 30 Aug 2024, Rüland et al., 2020).