Non-local Integral Functionals in Variational Analysis

• Non-local integral functionals are defined on function spaces by integrating pairwise interactions via nonlocal kernels, highlighting their role in variational calculus.
• They extend classical models, as seen in fractional Sobolev spaces, enabling rigorous treatment of convexity, relaxation, and weak sequential lower semi-continuity.
• Their applications span continuum mechanics, peridynamics, and homogenization, with results including compact embedding, Sobolev-type inequalities, and Γ-convergence.

Non-local integral functionals are functionals defined on function spaces—typically Lebesgue, Sobolev, or fractional Sobolev spaces—where the energy depends, via a nonlocal kernel or density, on pairs (or larger configurations) of points, often in the form of multiple integrals over the domain. These objects are central across modern analysis, the calculus of variations, continuum mechanics, materials science, and probability, as they encode long-range (nonlocal) interactions and allow for quantitative modeling beyond classical local or differential operators.

1. Fundamental Structures of Non-Local Integral Functionals

A prototypical non-local integral functional takes the form

$$F(u) = \iint_{\Omega \times \Omega} f(x, y, u(x), u(y))\,dx\,dy,$$

where $\Omega \subset \mathbb{R}^m$ is a bounded measurable set, $u$ is a function in $L^p(\Omega; \mathbb{R}^n)$ for some $1 \leq p \leq \infty$, and the density $f$ is a pairwise-symmetric, measurable function $$f: \Omega \times \Omega \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$$. The negative part $f^-$ must satisfy

$$\int_{\Omega} \int_{\Omega} f^-(x, y, u(x), u(y))\,dx\,dy < \infty$$

for all $u \in L^p(\Omega; \mathbb{R}^n)$ to ensure $F$ is well-defined (Elbau, 2011).

Variants include:

• Double-integral functionals with kernels $K(x-y)$,
• Functionals depending on finite differences or nonlinear “nonlocal gradients”,
• Nonlocal energies parameterized by interaction horizons or fractional exponents,
• Combinations with local (gradient) terms for hybrid models.

Special cases bridge to classical objects: for instance, if $f(x, y, w, z) = |w-z|^p |x-y|^{-d-sp}$, then $F$ is the Gagliardo seminorm squared for the fractional Sobolev space $W^{s,p}(\Omega)$.

2. Weak Sequential Lower Semi-Continuity, Convexity, and Relaxation

The core variational property of interest is sequential lower semi-continuity (s.l.s.c.) with respect to weak convergence. For double-integral non-local functionals as above, a central theorem asserts:

Let $f$ pairwise-symmetric, measurable, and continuous in $(w,z)$ for a.e. $(x, y)$, and satisfy suitable growth conditions. Then $F$ is weakly s.l.s.c. on $L^p(\Omega; \mathbb{R}^n)$ if and only if for every $\psi \in L^p(\Omega; \mathbb{R}^n)$ and a.e. $x$, the mapping

$$w \longmapsto \Phi_{x,\psi}(w) = \int_\Omega f(x, y, w, \psi(y))\,dy$$

is convex (Elbau, 2011). In the scalar case ($n=1$), this convexity is equivalent (modulo null Lagrangians) to separate convexity in $w$ and $z$ of the density $f$.

When convexity fails, minimizers may not exist. The process of relaxation then seeks the lower semicontinuous envelope $F^{\mathrm{rlx}}$, typically by integrating the separately convex envelope of $f$ (Kreisbeck et al., 2019), but in important cases this envelope may not itself be given by another double-integral functional. Counterexamples exhibit loss of double-integral form due to oscillatory microstructures beyond the reach of convexification.

3. Non-Local Function Spaces and Kernels

The appropriate function spaces for non-local theories are generalizations of Sobolev/BV spaces incorporating the interaction kernel.

Non-local Sobolev–Gagliardo spaces: Given $K: \mathbb{R}^n \to [0, \infty)$, set

$$[u]_{W^{K,p}(\Omega)}^p = \iint_{\Omega \times \Omega} |u(x) - u(y)|^p K(x-y)\,dx\,dy,$$

$$W^{K,p}(\Omega) = \{u \in L^p(\Omega) : [u]_{W^{K,p}(\Omega)} < \infty\}$$

(Bessas et al., 8 Apr 2025).

Fractional and more exotic kernels (piecewise-fractional, oscillatory, log-fractional, etc.) are included, subject to integrability and decay assumptions ensuring nontrivial functionals and embedding properties.

Finite-horizon nonlocal/fractional gradients: For $u \in C_c^\infty(\mathbb{R}^n)$,

$$D^s_\delta u(x) = c_{n,s} \int_{|h| < \delta} \frac{u(x) - u(x+h)}{|h|} \frac{h}{|h|} \frac{w_\delta(h)}{|h|^{n-1+s}}\,dh,$$

with energy norm $$\|u\|_{X_{p,\delta}(\Omega)} = \|u\|_{L^p(\Omega_\delta)} + \|D^s_\delta u\|_{L^p(\Omega)}$$ (Bellido et al., 2022).

Nonlocal spaces with heterogeneous horizons: By varying the interaction scale near the boundary, one constructs spaces interpolating between classical and fractional Sobolev spaces, yielding trace and extension theorems adapted to “peridynamic” models (Scott et al., 2023).

4. Compactness, Inequalities, and Embedding Results

Non-local function spaces possess strong analytic properties akin to their local counterparts, under suitable kernel assumptions:

• Extension theorems: For $K$ satisfying decay, doubling, and other assumptions, $W^{K,p}(\Omega)$ admits bounded extension to $W^{K,p}(\mathbb{R}^n)$ (Bessas et al., 8 Apr 2025).
• Sobolev, Poincaré, Morrey, and Hardy inequalities: Non-local gradient or seminorm controls $L^p$-norms and increments; sharp embeddings and regularity statements (e.g., Hölder continuity if $sp>n$) follow (Bellido et al., 2022, Bessas et al., 8 Apr 2025).
• Compact embeddings: Under nontriviality of the kernel, $W^{K,p}(\Omega)$ embeds compactly into $L^q(\Omega)$ for appropriate $q$, mirroring the classical Rellich–Kondrachov theorem (Bessas et al., 8 Apr 2025, Scott et al., 2023).

5. Isoperimetry, Non-Local Perimeter, and Minimizability

Non-local functionals generate generalized notions of perimeter, crucial in geometric measure theory and phase transitions. Given $E \subset \mathbb{R}^n$ measurable,

$$\operatorname{Per}_K(E;\Omega) = \iint_{(E \cap \Omega) \times (E^c \cap \Omega)} K(x-y)\,dx\,dy + 2\iint_{(E \cap \Omega) \times (E^c \setminus \Omega)} K(x-y)\,dx\,dy,$$

with the global K-perimeter $$\operatorname{Per}_K(E) = \iint_{E \times E^c} K(x-y)\,dx\,dy$$ (Bessas et al., 8 Apr 2025).

Perimeter minimizers exist for integrable kernels, and isoperimetric inequalities, possibly up to constants, are inherited from the kernel. For radially decreasing kernels, balls minimize the nonlocal perimeter (Bessas et al., 8 Apr 2025). For non-symmetric or oscillatory kernels, optimal shapes can be notably nontrivial, presenting phenomena such as clustering or fragmentation.

Analogous nonlocal characterizations via asymptotic behavior have implications in stochastic geometry, e.g., for the variance of determinantal point processes, which connects the decay rate of the nonlocal energy to the Minkowski dimension of the boundary (Lin, 2023).

6. Γ-Convergence, Localization Limits, and Homogenization

Non-local functionals are central to approximation theories, relaxation, and homogenization:

• Γ-convergence: Parametrized (e.g., by horizon $\delta \to 0$ or exponent $s\to 1$) families of non-local functionals (convex or non-convex) are shown to $\Gamma$-converge in $L^p$ to local gradient or perimeter energies, possibly up to multiplicative constants (Brezis et al., 2016, Brezis et al., 2019, Cueto et al., 2023).
• Homogenization: Non-local energies with rapidly oscillating kernels (e.g., $f(x, y, \xi, \eta, (x-y)/\varepsilon)$) are homogenized via two-scale Young measures, yielding explicit formulas for the limit energy as a minimum over two-scale measures with prescribed barycenter (Bertazzoni et al., 18 Feb 2025).
• Interaction with local functionals: Nonlocal functionals can be combined with local energies; the interplay governs the effective limit, which may mix in a nontrivial way or reduce purely to local terms depending on the scaling/homogenization regime (Braides et al., 2022, Braides et al., 2023).

7. Analytical and Variational Features: Existence, Euler–Lagrange, and Pathologies

• Existence of minimizers: For convex densities in the nonlocal variable, direct methods (coercivity, lower semicontinuity, compactness) yield existence (Elbau, 2011, Bellido et al., 2022, Cueto et al., 2023). For scalar or particular one-dimensional cases, convexity is not even required for the direct method, due to the structure of the underlying fractional Sobolev spaces (Pedregal, 2020).
• Optimality conditions: The Euler–Lagrange equations associated with non-local functionals are typically nonlocal integral equations involving "nonlocal divergence" terms, fundamentally not reducible to PDEs except in localization limits (Pedregal, 2020).
• Relaxation and loss of double-integral structure: In general, the relaxation of a non-local double integral functional fails to admit a double-integral form, except for very special classes of kernels (e.g., product structures) (Kreisbeck et al., 2019). The fine structure of minimizing sequences can manifest hidden microscopic oscillations, making the set of relaxed energies strictly larger than those representable by a double integral.
• Failure of integral representation for non-quadratic forms: There exist convex sequences of non-local power $p$-functionals on $W^{1,p}_0$ (with $p \neq 2$) for which the $\Gamma$-limit cannot be represented by any double-integral of a measurable or continuous kernel, in sharp contrast to the quadratic/Dirichlet form case, where the Beurling–Deny decomposition applies (Braides et al., 2023).

• (Elbau, 2011) Peter Elbau, “Sequential Lower Semi-Continuity of Non-Local Functionals”
• (Kreisbeck et al., 2019) Loss of double-integral character during relaxation
• (Bessas et al., 8 Apr 2025) A note on non-local Sobolev spaces and non-local perimeters
• (Scott et al., 2023) Nonlocal problems with local boundary conditions I
• (Braides et al., 2022) Continuity of some non-local functionals with respect to a convergence of the underlying measures
• (Brezis et al., 2016) Non-local functionals related to the total variation and connections with Image Processing
• (Scilla et al., 2020) Non-local approximation of the Griffith functional
• (Lin, 2023) Nonlocal energy functionals and determinantal point processes on non-smooth domains
• (Braides et al., 2022) Compactness for a class of integral functionals with interacting local and non-local terms
• (Pedregal, 2020) On non-locality in the Calculus of Variations
• (Braides et al., 2023) Validity and failure of the integral representation of Γ-limits of convex non-local functionals
• (Brezis et al., 2019) Non-local, non-convex functionals converging to Sobolev norms
• (Cueto et al., 2023) A variational theory for integral functionals involving finite-horizon fractional gradients
• (Arroyo-Rabasa, 2022) Functional and variational aspects of nonlocal operators associated with linear PDEs
• (Bellido et al., 2022) Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings
• (Bertazzoni et al., 18 Feb 2025) Homogenization of non-local integral functionals via two-scale Young measures

These works jointly provide a rigorous theoretical foundation for non-local integral functionals, covering analytic, variational, and geometric aspects, and establish their central role in modern analysis, variational calculus, and applied mathematics.