Variational Perron Method

• Variational Perron Method is defined by reformulating eigenvalue and solution concepts as extrema of energy functionals, extending classical Perron techniques.
• It applies minimax and variational frameworks in operator theory, degenerate and nonlocal PDEs, and nonlinear settings, ensuring robust uniqueness and stability.
• The method supports advanced computational techniques, including neural-network-based solvers and fine potential theory, to tackle complex boundary and eigenvalue problems.

The Variational Perron Method encompasses a collection of variational, energy-minimization, and minimax frameworks for solving problems that were traditionally within the scope of the classical Perron method. It has been developed in several directions: operator theory (particularly for the Perron–Frobenius theorem), degenerate and nonlocal PDEs, nonlinear and metric settings, and more recently, computational and neural-network-based applications. Its unifying feature is the use of variational (minimization or minimax) principles replacing or extending the order-theoretic or maximum-principle methodologies underlying the classical Perron construction.

1. Foundational Framework and Variational Characterizations

The core insight of the variational Perron method is the reformulation of the principal eigenvalue or solution concept as an extremum of an energy or quotient over appropriate classes.

For the Perron–Frobenius context, given $A\in M_n(\mathbb{R})$ and a cone $C\in K(\mathbb{R}^n)$ (closed, convex, solid, self-dual), one defines the two-vector "extended Rayleigh quotient" $\lambda(u,v) = (A u, v)/(u, v)$. The set of quasi-eigenvalues is specified variationally:

$$\lambda_C(A) = \sup_{u \in C} \inf_{v \in C^\circ} \lambda(u,v) = \inf_{v \in C^\circ} \sup_{u \in C} \lambda(u,v),$$

where $C^\circ$ is the dual cone, coinciding with $C$ in the self-dual case. In the matrix case, if $C$ is the nonnegative orthant $S_+$, this recovers the classical Birkhoff–Varga formula and, under irreducibility, the equivalence with the Perron root $\rho(A)$ (Il'yasov et al., 9 Feb 2024).

In the setting of elliptic PDEs, the variational Perron solution leverages function-space minimization. For second-order elliptic operators $A$ in divergence form, the Perron operator $T$ maps continuous boundary data $\varphi\in C(\partial \Omega)$ to the unique minimizer $u_P$ of the energy in the affine space with approximate trace $\varphi$, i.e.,

$u_P = \arg\min_{w \in V_\varphi} \Re a(w, w)$

with $$V_\varphi := \{ w \in H^1(\Omega): \text{tr } w = \varphi \}$$ (Arendt et al., 2023).

For nonlocal and nonlinear (fractional Laplacian, etc.) equations, the weak solution/energy minimizer framework is imposed in fractional Sobolev spaces, often supplemented by sub/supersolution envelope constructions and obstacle problem methods (Korvenpaa et al., 2016).

This variational viewpoint generalizes beyond the order-theoretic and maximum-principle techniques, covering non-irreducible matrices, operators lacking monotonicity, and domains or boundary data for which pointwise maximum-minimum characterizations are not available.

2. Minimax Principles and the Role of Sion’s Theorem

A major technical ingredient is the minimax identity underpinning the variational characterization of principal values and solution operators. In the finite-dimensional matrix setting, for noncompact cones, Sion’s minimax theorem is applied to show:

$$\sup_{u\in C} \inf_{v\in C^\circ} \lambda(u,v) = \inf_{v\in C^\circ} \sup_{u\in C} \lambda(u,v)$$

while ensuring the coherence of extremal values even in the absence of compactness or direct attainment (Il'yasov et al., 9 Feb 2024). The necessary quasiconvexity/quasiconcavity properties of $\lambda(u,v)$ with respect to its arguments are rigorously established, ensuring applicability of Sion’s theorem.

In PDE and variational settings (e.g., the Dirichlet principle for elliptic equations), convexity and lower semicontinuity of the energy functionals play the analogous roles, providing the functional-analytic framework for existence, uniqueness, and comparison (Arendt et al., 2023, Björn et al., 2020).

3. Generalizations to Operator Theory and Nonlinear Analysis

The variational Perron method is not confined to the setting of positive matrices or operators. For arbitrary matrices, including those that are not irreducible or lack a positive cone preservation property, the method introduces the concept of a quasi-eigenvalue, determined variationally and invariant under orthogonal transformations. This quantity encodes the extremal real parts of the spectrum:

$\lambda_C(A) = \max \Re \operatorname{spec}(A)$

when $C=S_+$, for example, and coincides with the classical Perron root in the nonnegative-irreducible case (Il'yasov et al., 9 Feb 2024).

Stability estimates generalizing Weyl’s inequalities to this nonlinear-minimax context are established for these quasi-eigenvalues, with explicit Lipschitz constants quantifying sensitivity under perturbations:

$$|\lambda_{S_+}(A+D) - \lambda_{S_+}(A)| \leq c_0(A) \|D\|_m$$

Extensions to nonlinear equations, such as degenerate or fractional PDEs (e.g., driven by the fractional $p$-Laplacian), also fall within the reach of the variational framework. The energy minimization is recast in terms of the appropriate function spaces (e.g., $W^{s,p}$) and nonlocal interaction forms, with regularizing Poisson modifications and super-/subharmonic envelope constructions (Korvenpaa et al., 2016).

4. Resolutivity, Invariance, and Uniqueness

A central aspect of the variational Perron method is its treatment of resolutivity—the property that the upper and lower Perron envelopes coincide—and invariance under negligible perturbations of the data.

For divergence-form degenerate elliptic equations with $p$-admissible weights, the Perron solution is unique for continuous boundary data and is invariant under changes on sets of $(p,w)$-capacity zero:

$P(f + h) = P(f)$

if $h=0$ quasi-everywhere on the boundary (Björn et al., 2020).

For fractional and fine-topology contexts, resolutivity is established for broad classes of boundary data (Sobolev, uniformly continuous), with the Perron envelopes shown to coincide quasi-everywhere, and the capacity (in the appropriate sense) playing a key role in formulating the negligible sets (Björn et al., 2022, Korvenpaa et al., 2016).

The method provides canonical uniqueness: among all bounded solutions matching the data outside a negligible set, there is exactly one that is a solution in the variational or energy sense.

5. Numerical and Algorithmic Aspects

The variational structure facilitates robust and flexible numerical methodology. For linear algebraic contexts (principal quasi-eigenvalue computation), block-coordinate ascent/descent and nonsmooth Newton-type schemes can be applied to the saddle-point system:

$$\partial_u \lambda(u_0, v_0) = 0,\qquad \partial_v \lambda(u_0, v_0) = 0$$

yielding the variational maximizers/minimizers and corresponding quasi-eigenpairs (Il'yasov et al., 9 Feb 2024).

In operator and PDE contexts, the variational Perron approach integrates naturally with neural-network-based discretizations. Residual-based variational physics-informed neural networks (RVPINNs) are employed to minimize variational residuals discretized over trial and test spaces (Udomworarat et al., 8 May 2025). The minimization:

$$\mathcal{L}_{\mathrm{RVPINN}}(\theta) = \max_i |a(u_\theta, v_i) - \ell(v_i)|$$

is performed over neural ansatz functions, with provable a priori error bounds quantifying the expressivity of the network, the discretization quality of test spaces, and the optimization error.

This framework achieves mesh-free, high-expressivity approximations, outperforming classical fixed-grid methods in benchmark examples, including both smooth and singular solutions.

6. Fine Topology and Nonlinear Metric Extensions

In metric measure spaces with a doubling measure and supporting a $p$-Poincaré inequality, the variational Perron method enables the development of fine potential theory, including concepts of fine topology, fine $p$-harmonic functions, and fine boundary resolutivity (Björn et al., 2022).

In this context, the method is constructed using Newtonian–Sobolev function spaces, variational energies given by integrals of minimal $p$-weak upper gradients, and envelope minimizers over classes of fine superminimizers and subminimizers. It is established that, under natural conditions, the various upper Perron solutions (Perron, S, Q, R) coincide quasi-everywhere, yielding a fine $p$-minimizer, and that for uniformly continuous or Sobolev boundary data, all Perron solutions agree and are finely continuous.

A key novelty is the systematic use of the fine boundary and finely open sets, in contrast to merely the metric boundary—these are the correct objects for the fine potential theory (Björn et al., 2022).

7. Illustrative Examples, Applications, and Impact

The variational Perron approach underpins crucial results including new proofs of the Perron–Frobenius theorem, robust stability results under arbitrary perturbations, energy and trace criteria for solution regularity, and the complete resolution of Dirichlet problems for degenerate, weighted, nonlocal, or metric boundary regularity settings.

Algorithmically, the method enables the use of modern neural-network solvers for Perron–Frobenius-type power series problems, with rigorous guarantees and favorable convergence in both low- and high-dimensional domains (Udomworarat et al., 8 May 2025).

In summary, the variational Perron method unifies and extends existence, uniqueness, and regularity theory for eigenvalue problems and boundary value problems in linear, nonlinear, and nonlocal settings. Its applicability ranges from matrix theory (classical and beyond) to general divergence-form and nonlocal PDEs, and into the field of computational mathematics and neural approximation. Its foundational reliance on minimax and variational principles brings a robustness and generality unavailable to solely order- or maximum-principle-based techniques.