Fractional Elliptic Systems Overview

• Fractional elliptic systems are PDEs where each component is governed by a nonlocal operator like the fractional Laplacian, capturing long-range interactions.
• Extension techniques, such as the Caffarelli–Silvestre method, reformulate these nonlocal problems as local degenerate elliptic equations for effective geometric and variational analysis.
• Symmetry results, optimal energy estimates, and Hamiltonian identities play key roles in classifying solutions and understanding phase transitions and stability in complex systems.

A fractional elliptic system is a system of partial differential equations in which each component is governed by a nonlocal (fractional) elliptic operator—typically a variant of the fractional Laplacian—and the components interact through a nonlinear coupling. Such systems model a broad range of phenomena exhibiting both long-range (nonlocal) spatial effects and vectorial or multicomponent structure, and are prominent in mathematical physics, phase transitions, and analysis of coupled diffusive processes. The system is characterized by the operator $$(-\Delta)^{\mathbf{s}}\mathbf{u} = \nabla H(\mathbf{u})$$ in $\mathbb{R}^n$, where $\mathbf{u}:\mathbb{R}^n\to\mathbb{R}^m$ and the fractional Laplacian $(−\Delta)^{\mathbf{s}}$ acts componentwise with possibly different fractional orders $s_i\in(0,1)$.

1. Fractional Laplacian and Extension Techniques

Each component $u_i$ of the unknown vector field $\mathbf{u}$ is governed by the fractional Laplacian of order $s_i$: $$(-\Delta)^{s_i} u_i(x) = \text{P.V.} \int_{\mathbb{R}^n} \frac{u_i(x)-u_i(y)}{|x-y|^{n+2s_i}}\,dy.$$ This inherently nonlocal operator encodes long-range interactions and is realized as a Dirichlet-to-Neumann map via the Caffarelli–Silvestre extension: for each $u_i$, one introduces an extension $v_i(x,y)$ on the upper half-space $\mathbb{R}_+^{n+1}$ satisfying

$$\operatorname{div}(y^{a_i}\nabla v_i)=0 \quad\text{in}\quad \mathbb{R}^{n+1}_+, \qquad a_i=1-2s_i,$$

and

$$-\lim_{y\to0} y^{a_i}\partial_y v_i = d_{s_i} \partial_{v_i} H(v)\big|_{y=0},$$

where $H\in C^{2,\gamma}(\mathbb{R}^m)$ is the system potential and $d_{s_i}$ is an explicit normalization constant. The extension reformulates the nonlocal problem as a local degenerate elliptic problem in higher dimensions, facilitating geometric and variational methods.

2. Symmetry Results and De Giorgi-Type Theorems

A central result is the classification of bounded entire solutions under symmetry and monotonicity hypotheses. A De Giorgi-type theorem holds: if $n\leq 3$, $\min\{s_i\}\geq 1/2$, and the solution $\mathbf{u}$ is $H$-monotone in a fixed direction (i.e., each component is strictly monotone and the monotonicity aligns appropriately with the nonlinearity of $H$), then $\mathbf{u}$ is one-dimensional—there exists $\Gamma\in\mathbb{S}^{n-1}$ such that $v_i(x,y) = v_i^*(\Gamma\cdot x, y)$. This extends analogues of the De Giorgi conjecture for the local Allen–Cahn and multi-component elliptic systems to the fractional and vector-valued case.

The proof combines variational and geometric tools adapted to the nonlocal context, notably including a Liouville theorem for stable solutions and a geometric Poincaré inequality derived for the degenerate extension.

3. Optimal Energy Estimates and Monotonicity Formulae

The authors establish sharp energy growth estimates for extended solutions: $$E_R(v) = \sum_{i=1}^m \left[ \frac{1}{2d_{s_i}}\int_{\mathcal{C}_R} y^{a_i}|\nabla v_i|^2\,dX - \int_{B_R} H(v)\,dx \right].$$ There are three regimes, depending on $s_* = \min\{s_i\}$:

• If $1/2 < s_* < 1$: $E_R(v) \leq C R^{n-1}$,
• If $s_* = 1/2$: $E_R(v) \leq C R^{n-1}\log R$,
• If $0 < s_* < 1/2$: $E_R(v) \leq C R^{n-2s_*}$.

When all $s_i$ coincide and $H\leq 0$, an associated rescaled energy is non-decreasing: $$I(R) = \frac{1}{R^{n-2s}}\left[ \frac{1}{2}\int_{B_R^+} \sum_i y^a|\nabla v_i|^2\,dX - \int_{B_R\times\{y=0\}} H(v)\,dx \right].$$ Such monotonicity formulas are crucial for blow-up analysis, energy classification, and as an integral part of the symmetry and rigidity arguments.

4. Hamiltonian Identities and Liouville-Type Theorems

In the layered one-dimensional regime, a Hamiltonian identity is obtained: $$\sum_{i=1}^m \int_0^\infty y^{1-2s}\left[ (\partial_x v_i)^2 - (\partial_y v_i)^2 \right]dy = 2d_s[H(v(x,0)) - H(\alpha)],$$ where $\alpha = \lim_{x\to\infty} v(x,0)$. For radial solutions, the (non-increasing) radial energy function is constructed.

Using a combination of these identities, energy bounds, and spectral analysis, various Liouville-type results are derived: if $\nabla H\geq 0$, any bounded pointwise-stable solution is trivial ($u$ constant) in dimensions $n \leq 2(1 + s_*)$; for $s_*\geq 1/2$ this covers $n\leq 3$. This yields strong classification of solutions, which is a fundamental input to symmetry results and nonexistence theorems.

5. Orientable Systems and $H$-Monotone Solutions

Two structural notions are pivotal:

• Orientable Systems: The system is orientable if there exist fixed-sign functions $\theta_i$ such that for all $i\neq j$:

$$\partial_{u_i u_j}H(u)\,\theta_i(x)\,\theta_j(x) \geq 0\quad\forall x\in\mathbb{R}^n.$$

This ensures a favorable sign in mixed derivatives, imposing a form of cooperativity and is essential for the derivation of a geometric Poincaré inequality.

• $H$-Monotone Solutions: These are solutions such that each $\partial_\nu u_i(x)\neq 0$ in a fixed direction and

$$\partial_{u_i u_j} H(u)\,\partial_\nu u_i(x)\,\partial_\nu u_j(x)\geq 0,$$

again encoding compatibility between monotonicity and the structure of $H$.

These hypotheses allow one to utilize sliding methods and geometric inequalities, and—via quotient arguments (expressing ratios between directional derivatives and positive solutions of the linearized problem)—imply that all components vary along the same direction, leading to one-dimensional symmetry.

6. Applications, Specific Models, and Open Problems

The theoretical framework applies in particular to multi-component models including nonlinear Schrödinger systems (e.g., the Gross–Pitaevskii system for Bose–Einstein condensates) and phase separation phenomena where fractional diffusive effects are relevant. Under the symmetry, $H$-monotonicity, and regularity assumptions, Liouville theorems extend directly to these models, implying that bounded entire pointwise-stable solutions are trivial or one-dimensional.

Open problems highlighted include:

• The De Giorgi-type symmetry result for $n=3$ when $s_*\in(0,1/2)$ remains unsettled.
• Extension to higher dimensions ($n\geq4$) and to systems beyond orientable and symmetric cases.
• Identification of sharp dimensions and parameter ranges where Liouville-type results hold for more general nonlinearities and coupling structures.

7. Tables: Principal Regimes and Symmetry Results

Spatial Dimension $n$	Minimal Fractional Order $s_*$	Symmetry/Classification Result
$n=2$	$0 < s_i < 1$	1D symmetry (stable or $H$-monotone)
$n=3$	$1/2 \leq s_*$	1D symmetry ($H$-monotone)
$n\geq4$	Any	Open problem

The De Giorgi-type symmetry and classification of solutions in the table refer to entire bounded stable or $H$-monotone solutions to fractional elliptic systems with orientable (or symmetric) nonlinearities.

---

A fractional elliptic system, as illustrated by the analytic and geometric results above, provides a rigorous mathematical structure for studying nonlocal vectorial PDEs with spatially long-range interactions, rich symmetry phenomena, variational structure, and challenging classification problems. These features are central to both theoretical research and modeling in nonlocal physical systems, phase transitions, and nonlinear analysis (Fazly et al., 2014, Fazly, 2015).