Caffarelli–Silvestre Extension Method

• The Caffarelli–Silvestre extension is a method that recovers fractional Laplacians by translating nonlocal operators into local, degenerate-elliptic PDEs in one higher dimension.
• It employs weighted Sobolev spaces and variational techniques to equate the extension's energy with the Gagliardo semi-norm, providing a rigorous analytic framework.
• This technique underpins applications in inverse problems, regularity theory, and numerical schemes, and extends to variable-coefficient and higher-order operators.

The Caffarelli–Silvestre extension is a degenerate-elliptic technique that realizes nonlocal fractional powers of second-order differential operators, such as the fractional Laplacian $(-\Delta)^s$ for $0 < s < 1$, as Dirichlet-to-Neumann maps for local PDEs in one higher dimension. This extension method underpins the analytic, variational, probabilistic, and numerical frameworks for fractional elliptic operators and is foundational in the analysis of nonlocal PDEs, inverse problems, and stochastic processes.

1. Classical Formulation and Weighted Extension Problem

Given $u \in H^s(\mathbb{R}^n)$, the Caffarelli–Silvestre extension $U(x,y)$ is defined as the unique solution to the weighted degenerate elliptic equation

$$\begin{cases} \operatorname{div}(y^{1-2s}\nabla U) = 0, & (x,y) \in \mathbb{R}^n \times (0,\infty), \ U(x,0) = u(x), & x \in \mathbb{R}^n, \end{cases}$$

where the variable $y > 0$ is called the extension variable or "bulk" direction. The Dirichlet datum is imposed on $y = 0$, which acts as the "boundary" corresponding to the original domain.

The key property is the recovery of the fractional Laplacian via a weighted Neumann trace: $$-(\lim_{y \to 0^+} y^{1-2s} \partial_y U(x,y)) = C_s\, (-\Delta)^s u(x),$$ in the sense of $H^{-s}$–$H^s$ duality, with $C_s = 2^{1-2s}\Gamma(1-s)/\Gamma(s)$ (Covi et al., 2023). This provides a realization of the nonlocal operator $(-\Delta)^s$ as the Dirichlet-to-Neumann map of a local PDE in $\mathbb{R}^{n+1}_+$.

2. Analytic Structure and Weighted Sobolev Spaces

The extension problem is formulated in the weighted Sobolev space $\dot H^1(\mathbb{R}^{n+1}_+, y^{1-2s})$, which consists of functions $U$ with weak derivatives and finite energy

$$\|U\|_{\dot H^1(\mathbb{R}^{n+1}_+, y^{1-2s})}^2 = \int_{\mathbb{R}^{n+1}_+} y^{1-2s} |\nabla U(x,y)|^2\, dx\, dy < \infty,$$

and trace $U(\cdot,0) = u$ belonging to $H^s(\mathbb{R}^n)$. The mapping

$U \mapsto U|_{y=0}$

is continuous and surjective from $\dot H^1(\mathbb{R}^{n+1}_+, y^{1-2s})$ to $H^s(\mathbb{R}^n)$ (Chen et al., 2022).

The energy of the extension exactly captures the Gagliardo semi-norm of the fractional Sobolev space: $$E[U] = \int_{\mathbb{R}^{n+1}_+} y^{1-2s} |\nabla U|^2\, dx\, dy,$$ is equivalent, up to a constant, to

$$\iint_{\mathbb{R}^n \times \mathbb{R}^n} \frac{|u(x) - u(\xi)|^2}{|x - \xi|^{n + 2s}} \, dx\, d\xi.$$

This yields a variational characterization and enables minimization principles for nonlocal equations.

3. Dirichlet-to-Neumann Operators and Generalizations

For variable-coefficient elliptic operators, the extension builds on a degenerate-elliptic operator in higher dimensions. Given a symmetric, positive-definite matrix $a(x)$, the extension problem is

$$\operatorname{div}(y^{1-2s}\tilde a(x,y)\nabla \tilde U) = 0, \quad \tilde a(x,y) = \begin{pmatrix} a(x) & 0 \ 0 & 1 \end{pmatrix},$$

with appropriate boundary conditions. The resulting Dirichlet-to-Neumann map retrieves fractional elliptic operators of the form $(-\operatorname{div}(a \nabla))^s$ (Covi et al., 2023, Rüland, 2023).

The extension can equivalently be formulated for more general self-adjoint operators $L$, where the extension PDE becomes

$$\partial_{yy} U + \frac{1-2s}{y} \partial_y U = -L U, \qquad U|_{y=0} = f,$$

and the Dirichlet-to-Neumann map recovers $L^s f$ up to normalization. The abstract framework extends to generators of strongly continuous semigroups and Dirichlet forms, as in (Galé et al., 2012, Baudoin et al., 27 Mar 2024, Hauer et al., 2021), and to more general Banach-space settings via functional calculi based on complete Bernstein functions.

4. Applications: Inverse Problems, Regularity, and Numerical Methods

A notable application is in inverse problems for nonlocal elliptic operators. In the fractional Calderón problem, the Caffarelli–Silvestre extension provides a constructive pathway to reduce uniqueness proofs for nonlocal inverse problems to their local analogues (Covi et al., 2023, Rüland, 2023). This connection is explicit in the mapping of fractional Dirichlet-to-Neumann data to the local counterpart via vertical averaging of the extension solution: $$v(x') = \int_0^\infty y^{1-2s} \tilde U(x', y) \, dy,$$ which solves a local elliptic PDE and determines the local Dirichlet-to-Neumann map.

Regularity results, Almgren-type monotonicity formulas, and optimal Hölder estimates for obstacle and free boundary problems are naturally formulated in the extension framework, facilitating the treatment of degenerate elliptic equations and providing access to weighted harmonic measure theory (Petrosyan et al., 2014).

In computational mathematics, the extension transforms nonlocal PDEs into local, albeit higher-dimensional, variational problems, enabling the design of efficient numerical methods, including deep Ritz approaches in high dimensions (Gu et al., 2021). The degeneracy and singularity near $y=0$ are addressed by specialized approximation spaces and integration schemes based on the structure of the extension.

5. Extensions: Higher-Order, Discrete, and Non-Euclidean Generalizations

Higher-order fractional Laplacians, $(-\Delta)^s$ with $s > 1$, admit an analogous extension, but the related PDE becomes higher-order in $y$ and requires additional boundary conditions, involving both Dirichlet and conormal data (Yang, 2013, Felli et al., 2018). The Dirichlet-to-Neumann characterization persists, though with increased complexity in the formulation.

On discrete structures, such as $\mathbb{Z}$ or graphs, a "discrete Caffarelli–Silvestre extension" realizes fractional discrete Laplacians as boundary traces of local operators defined on higher-dimensional graphs (e.g., $\mathbb{Z} \times \mathbb{N}$), enabling the analysis of discrete nonlocal models via local Gaussian free fields on layered structures (Garban, 2023).

For subelliptic and CR geometries (e.g., the Heisenberg group), the extension operates in the context of non-Euclidean metrics and sub-Laplacians, often linked to scattering theory and the geometry of the Siegel upper half-space. The Dirichlet-to-Neumann map for the extension again recovers conformally covariant fractional powers of subelliptic operators, with identical normalization to the Euclidean case (Frank et al., 2013).

6. Limitations, Obstructions, and Inverse Correspondence

While the extension provides a surjective reduction from fractional to local operators at the level of Dirichlet-to-Neumann data, the inverse is generally obstructed. In the Calderón-type framework, key obstructions arise from two independent one-dimensional averaging effects: (i) vertical averaging (losing $y$-dependence of the extension on the boundary), and (ii) tangential averaging related to density arguments for traces from "partial" to "full" boundary data. Consequently, the mapping is surjective only up to closure and cannot be continuously inverted in general (Covi et al., 2023).

7. Further Generalizations and Connections

Recent developments generalize the extension method to higher-order and logarithmic operators (via Hadamard regularization), discrete settings, general Dirichlet forms, and nonlocal problems involving drift terms or singular potentials (Lee, 15 Feb 2025, Petrosyan et al., 2014, Baudoin et al., 27 Mar 2024). The functional calculus based on extension operators characterizes a broad class of spectral multipliers for which the extension paradigm applies, including Phillips–Bochner subordination and complete Bernstein functions (Hauer et al., 2021).

The extension also underpins a rich theory of trace inequalities, embeddings, and capacity theory in function spaces (e.g., Sobolev, Besov, and Lebesgue spaces) and interfaces with stochastic analysis, via its probabilistic realization as the boundary value of degenerate diffusions with suitable Bessel-type processes in the extension variable (Cavina, 2023, Li et al., 2022, Li et al., 2020).

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References:

• "A Reduction of the Fractional Calderón Problem to the Local Calderón Problem by Means of the Caffarelli-Silvestre Extension" (Covi et al., 2023)
• "Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift" (Petrosyan et al., 2014)
• "An extension problem of higher order operators and operators of logarithmic type via renormalization" (Lee, 15 Feb 2025)
• "Revisiting the Anisotropic Fractional Calderón Problem Using the Caffarelli-Silvestre Extension" (Rüland, 2023)
• "On a family of integral operators on the ball" (Tian, 2021)
• "Extension problem and fractional operators: semigroups and wave equations" (Galé et al., 2012)
• "Functional Calculus via the extension technique: a first hitting time approach" (Hauer et al., 2021)
• "A note on a stochastic approach to Caffarelli-Silvestre Theorem" (Cavina, 2023)
• "Quantization of nonlocal fractional field theories via the extension problem" (Frassino et al., 2019)
• "Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian" (Garban, 2023)
• "Singularities of fractional Emden's equations via Caffarelli-Silvestre extension" (Chen et al., 2022)
• "Deep Ritz method for the spectral fractional Laplacian equation using the Caffarelli-Silvestre extension" (Gu et al., 2021)
• "Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials" (Li et al., 2020)
• "Unique continuation principles for a higher order fractional Laplace equation" (Felli et al., 2018)
• "Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals" (Baudoin et al., 27 Mar 2024)
• "Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension" (Li et al., 2022)
• "On higher order extensions for the fractional Laplacian" (Yang, 2013)
• "An extension problem for the CR fractional Laplacian" (Frank et al., 2013)
• "An extension problem related to inverse fractional operators" (Teso, 2016)