Double Phase Model

• Double Phase Model is a framework that integrates two distinct growth regimes, capturing transitions between p- and q-growth in variational integrals and PDEs.
• It employs generalized Musielak–Orlicz spaces and phase-field formulations to analyze nonlinear phenomena in materials science and fracture mechanics.
• Advanced numerical and analytical methods, such as finite element schemes and De Giorgi iterations, ensure robust solutions across heterogeneous systems.

A double phase model refers to a mathematical formulation, physical theory, or computational framework characterized by the simultaneous presence, interaction, or coupling of two distinct growth regimes, phases, mechanisms, or field variables. Classical instances include double phase variational integrals in nonlinear PDEs, multi-phase flow models in fluid mechanics, double phase-field frameworks for fracture and pattern formation, and phase/energy double transitions in statistical physics or network theory. Substantial recent generalizations and applications have been developed in materials science, analysis, and mathematical physics.

1. Double Phase Variational Functionals and Musielak–Orlicz Spaces

The canonical double phase energy functional is

$$\mathcal{E}(u,\Omega)\;=\;\int_{\Omega}\left[|\nabla u(x)|^p + a(x)|\nabla u(x)|^q\right]\,dx,$$

for $u\colon \Omega\rightarrow\mathbb{R}$, $1<p<q<\infty$, and a modulating coefficient $a(x)\ge0$ that may be merely measurable. This structure captures a system with pure $p$-growth behaviour in regions where $a(x)=0$ and mixed $(p,q)$-growth where $a(x)>0$; thus, the energy density transitions between different polynomial regimes (Adamadze et al., 28 Jan 2026).

Such functionals, and their induced pseudo-norms, are described within generalized Musielak–Orlicz spaces $L^\varphi$, equipped with the Luxemburg norm

$$\|f\|_{L^\varphi} = \inf\left\{\lambda>0:\int_{\mathbb{R}^n} \varphi(x,|f(x)|/\lambda)\,dx\le1\right\},$$

where $\varphi(x,t)=t^p/p + a(x)t^q/q$. Critical growth conditions, such as the $\Delta_2$ property and $(p,q)$-growth

$$\varphi(x,st)\le \max\{s^p,s^q\}\varphi(x,t),\quad p\leq \frac{\varphi'(x,t)\,t}{\varphi(x,t)}\leq q,$$

ensure mathematical tractability.

Regularity and harmonic analysis in this setting are governed by a Muckenhoupt-type modular condition: $$[\varphi]_{\mathcal{A}} := \sup_{\text{cubes }Q} \frac{\|\mathbf{1}_Q\|_{L^\varphi}\, \|\mathbf{1}_Q\|_{L^{\varphi^*}}}{|Q|}<\infty,$$ which generalizes the classical $A_p$ theory for weighted $L^p$ spaces. Key analytic consequences include boundedness of the Hardy–Littlewood maximal operator and sharp Sobolev–Poincaré inequalities under minimal assumptions on $a(x)$ (Adamadze et al., 28 Jan 2026).

2. Double Phase Models in Nonlinear Partial Differential Equations

The double phase functional typically leads to degenerate or non-uniformly elliptic Euler–Lagrange equations of the type: $$-\text{div}\left(\varphi'(x,|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = 0,$$ with localized transitions between $p$- and $q$-growth regimes. Regularity analysis proceeds via Caccioppoli-type inequalities, oscillation decay, and De Giorgi iterations, where the main challenges are overcoming the lack of continuity in $a(x)$ and handling jump/gap conditions in the growth exponents.

Recent work has established that, under the modular Muckenhoupt condition $\varphi\in\mathcal{A}$, one can derive local boundedness, oscillation decay, and Hölder continuity up to an exponent $\beta$ determined by $n,p,q,[\varphi]_{\mathcal{A}}$. Crucially, no continuity or gap restriction is required on $a(x)$; the theory is as robust as the classical p-Laplacian weighted setting (Adamadze et al., 28 Jan 2026).

3. Double Phase Relaxation Models and Phase Transitions

In continuum thermodynamics, double phase relaxation models parameterize phase coexistence, metastability, and finite-rate mass exchange. For example, in isothermal two-phase fluid models governed by van der Waals equation of state, the double phase structure is encoded at the level of conservative variables $(\rho, \rho_1, \rho_2, m)$ coupled to a dynamical relaxation system: $$\dot\rho_1= -(\rho-\rho_1)(\rho-\rho_2)\,f(\rho_2|\rho_1), \quad \dot\rho_2=+(\rho-\rho_1)(\rho-\rho_2)\,f(\rho_1|\rho_2),$$ where $f(\rho_2|\rho_1)$ is the relative Helmholtz free energy. This ensures Lyapunov stability and allows chemically metastable states, pure phases, and mixed equilibrium, with transitions between regimes determined by the Maxwell construction (James et al., 2015, James et al., 2014).

The resulting PDE system combines hyperbolic convection (Eulerian transport) and stiff ODE-type relaxation: $\partial_t W + \partial_x F(W) = S(W),$ where $S(W)$ involves the above double phase mass-exchange rates. Fractional-step and IMEX finite-volume methods are employed to manage stiffness and accurately capture phase transitions, nucleation, and metastable persistence.

4. Double Phase-Field Formulations and Mixed-Mode Fracture

Double phase-field models utilize multiple phase parameters to resolve competing mechanisms. In mixed-mode fracture mechanics, two phase-fields $(d_1,d_2)$ encode tensile (mode I) and shear (mode II) decohesion: $$\Pi[u, d_1, d_2] = \int \left( g_1(d_1) W_1^+(\epsilon) + g_2(d_2) W_2^+(\epsilon) + W^-(\epsilon) + G_{c,I}\Gamma(d_1,\nabla d_1) + G_{c,II}\Gamma(d_2,\nabla d_2)\right) \, dV.$$ Energy splits and degradation functions allow direct use of experimental material strengths without sensitivity to regularization length scales. Variational inequalities enforce irreversibility, and a local energy-based criterion selects the dominant fracture mode dynamically. Finite-element schemes are used for numerical solution and validated against patterns of crack propagation and coalescence in rocks (Fei et al., 2020).

5. Double Phase Transitions in Statistical Physics and Networks

A double phase transition arises when a system exhibits two distinct critical phenomena, often due to hierarchical or modular structure. Example: the Ising model on core–periphery networks reveals an intermediate, inhomogeneous phase where core nodes order at low temperature $T_{c1}$, while the entire network undergoes a true order-disorder transition at higher $T_{c2}$. The mean-field analysis and Monte Carlo simulations show that susceptibility peaks diverge differently depending on how the core–periphery connections scale with system size; in certain scaling regimes, both transitions are singular, constituting a bona fide double phase transition (Chen et al., 2018).

6. Double Phase Models in Materials and Image Processing

Double phase structures are increasingly used in materials science for nucleation and growth in two-phase media (modified Becker–Döring kinetics with $\phi$-dependent rates), and in computational imaging. Adaptive double phase denoising, formulated as a convex variational integral combining TV and quadratic gradient penalties via a spatially varying adaptive weight, yields staircasing-free reconstructions preserving edges while suppressing noise in flat regions. The solver employs accelerated primal-dual methods and achieves superior SSIM and PSNR over single-phase models (Blesgen et al., 2019, Górny et al., 5 Oct 2025).

7. Extensions: Two-Scale, Multi-Regime, and Field-Theoretic Double Phase Models

Recent advances expand the double phase paradigm to multi-regime models (disperse–separated flows), physically realistic multiphase field models (e.g., for phase separation, ferromagnetism, and electrohydrodynamics), and generalized kinetic and field-theoretic formulations. For example, in Double Field Theory, the phase space is doubled $(X^M,P_N)$, enabling $O(D,D)$-covariant kinetic equations and Boltzmann formulations for matter tensors (Lescano et al., 2020).

Developments include two-scale Hamiltonian formulations for mixtures involving separated and disperse phases (with capillary area density, curvature moments, and polydispersity closure via GeoMOM), and rigorous homogenization of double porosity media capturing the multiscale interaction of matrix blocks and fractures (Loison et al., 2023, Amaziane et al., 2016).

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Double phase models unify distinct growth or physical regimes in a single analytic, computational, or physical framework. Their technical and conceptual underpinnings span advanced harmonic analysis, PDE regularity, optimization, statistical mechanics, and computational physics, enabling the rigorous study of systems with heterogeneous or competing behaviours.