Cascade Token Pruning

• The paper introduces a cascade-based method that progressively filters tokens to reduce inference cost without major accuracy loss.
• It employs a multi-stage thresholding mechanism to dynamically assess token importance and discard redundant information.
• Empirical evaluations report up to 30% reduction in computational overhead on benchmark tasks while maintaining performance.

Nonvariational Elliptic PDEs with Gradient Dependence are a central class of nonlinear partial differential equations whose source terms depend explicitly on both the unknown function $u(x)$ and its gradient $\nabla u(x)$. Such problems arise in numerous contexts—nonlinear physics, geometric PDEs, control and Hamilton-Jacobi frameworks—and their analysis fundamentally differs from variational equations due to the absence of Euler–Lagrange structure. Recent advances have resolved major open problems in this class, especially concerning existence, multiplicity, gradient estimates, and regularity of solutions, even in settings with unbounded or singular coefficients and without symmetry assumptions.

1. Model Equations and Structural Hypotheses

The canonical nonvariational elliptic PDE with gradient dependence is

$$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$

for a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq 3$, with a Lipschitz boundary. The typical functional setting is $u\in H^1_0(\Omega)$, with the weak formulation:

$$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$

Assumptions on $f$ typically include Carathéodory regularity; precise growth (e.g., $|f(x, t, \xi)| \leq c_1[1 + |t|^{s-1} + |\xi|]$ for $1\leq s < 2^*$ and $\nabla u(x)$0); and Lipschitz continuity in both $\nabla u(x)$1 and $\nabla u(x)$2 (Bahrouni, 24 Dec 2025). Spectral gap conditions relating the constants to the Laplacian's principal eigenvalue $\nabla u(x)$3 ensure certain contraction properties crucial for fixed-point arguments. More general operators include quasilinear forms (e.g., weighted $\nabla u(x)$4-Laplacians, Hessian operators) and fully nonlinear Hamiltonian equations.

2. Methodological Innovations: Truncation and Fixed-Point Schemes

A breakthrough for existence and multiplicity without variational symmetry was delivered via refined truncation methods. For each $\nabla u(x)$5, construct truncated nonlinearities $\nabla u(x)$6 in $\nabla u(x)$7 while retaining full gradient dependence:

• $\nabla u(x)$8 is supported in $\nabla u(x)$9, $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$0 in $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$1, with zeros at endpoints, so the solution $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$2 is naturally confined to the $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$3th "layer" (Bahrouni, 24 Dec 2025). The full scheme couples a variational minimization at frozen gradient (solving $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$4 for $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$5 fixed) with a contraction-based fixed-point iteration

$$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$6

where $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$7 is the associated energy functional. The contraction constant $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$8 allows invocation of Banach's fixed-point theorem, yielding convergence to $$-\Delta u(x) = f(x, u(x), \nabla u(x)), \qquad x \in \Omega, \quad u|_{\partial\Omega} = 0$$9, a nontrivial solution in $\Omega\subset\mathbb{R}^n$0. This is the first existence and structure theorem for infinitely many positive and negative solutions in this setting, absent symmetry (Bahrouni, 24 Dec 2025).

3. Regularity Theory and Gradient Bounds

Recent advances establish optimal Hölder and $\Omega\subset\mathbb{R}^n$1 (even log-Lipschitz and $\Omega\subset\mathbb{R}^n$2 Schauder) bounds for solutions of fully nonlinear nondivergence form PDEs with gradient-dependent Hamiltonians, under minimal integrability and oscillation control (Silva et al., 2020). For problems of the type:

$\Omega\subset\mathbb{R}^n$3

with $\Omega\subset\mathbb{R}^n$4 uniformly elliptic and $\Omega\subset\mathbb{R}^n$5 exhibiting either sublinear $\Omega\subset\mathbb{R}^n$6 or superlinear $\Omega\subset\mathbb{R}^n$7 gradient growth, existence and sharp regularity estimates for $\Omega\subset\mathbb{R}^n$8-viscosity solutions hold. For $\Omega\subset\mathbb{R}^n$9, one shows $n\geq 3$0, and the gradient satisfies

$n\geq 3$1

where $n\geq 3$2 encodes dependence on drift coefficients (Silva et al., 2020). Singular equations with $n\geq 3$3 and nonvariational drift are also covered. BMO bounds for $n\geq 3$4 and $n\geq 3$5 derive from a priori energy estimates and viscosity constructions.

In the entire-space setting, sharp pointwise gradient bounds for nonvariational, quasilinear elliptic equations with arbitrary gradient dependence (including singular/degenerate cases) were established by constructing suitable $n\geq 3$6-functions and applying the Maximum Principle:

$n\geq 3$7

which inverts to explicit upper bounds on $n\geq 3$8 in terms of $n\geq 3$9 (Cavaterra et al., 2019). Such results generalize Modica-type estimates and allow treatment of reaction-drift terms $u\in H^1_0(\Omega)$0.

4. Topological, Nonvariational, and Index Methods in Systems

Systems with gradient-dependent nonlinearities—for example,

$u\in H^1_0(\Omega)$1

or more generally, systems with boundary conditions depending on functionals of both $u\in H^1_0(\Omega)$2 and $u\in H^1_0(\Omega)$3—are handled via topological fixed-point theory. On bounded domains and cones in appropriate Banach spaces, compactness of relevant integral operators (Green's functions) and explicit comparison principles allow computation of fixed-point indices (index zero on small cones, index one on large cones), leading to existence, multiplicity, and nonexistence results (Biagi et al., 2019). Key preparatory lemmas provide gradient estimates and barrier function constructions to control the nonlinearities.

For Neumann problems on annular domains, the use of radial symmetry and Green's functions produces multiple positive solutions under suitable box-type growth conditions in the nonlinear terms involving gradients (Cianciaruso et al., 2017).

5. Inclusion and Variational Inequality Frameworks

Gradient-dependent terms naturally arise in variational inequality (especially with unilateral constraints), inclusion, and obstacle problems. Existence and regularity of strong solutions for nonlinear Neumann inclusions of the form

$u\in H^1_0(\Omega)$4

with $u\in H^1_0(\Omega)$5 (convex subdifferential), $u\in H^1_0(\Omega)$6 multivalued in $u\in H^1_0(\Omega)$7, are established via Moreau–Yosida regularization of the subdifferential, a topological fixed-point alternative, and uniform $u\in H^1_0(\Omega)$8 a priori bounds (affirmed via energy and Nagumo–Hartman-type estimates). Solutions exist in the strong sense $u\in H^1_0(\Omega)$9 with measurable selections for reaction terms, even under noncoercive growth regimes (Papageorgiou et al., 2018).

6. Transformations and Rigidity for Gradient Terms

Introducing natural quadratic gradient terms aligned to the operator structure allows the use of Kazdan–Kramer-type changes of variables $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$0, explicitly removing the gradient dependence and transforming the original PDE to one with only zero-order terms (after a nonuniform reweighting of $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$1) (Oliveira, 2 Nov 2025). This framework recovers and unifies the theory for Laplacian, $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$2-Laplacian, Hessian, and infinity-Laplacian equations with natural gradient terms. For example,

$$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$3

transforms under $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$4 to a PDE without the gradient term, enabling classical regularity and rigidity results. The Aronsson-type theorem is extended: any $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$5 solution is either constant or has everywhere nonzero gradient, confirmed via the change-of-variables method. Viscosity solution existence and uniqueness in Dirichlet problems for the perturbed infinity-Laplacian are established under mild growth and monotonicity conditions.

7. Gradient Estimates and Liouville-Type Theorems

Gradient estimates for positive solutions to equations of the form

$$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$6

are derived without restriction on the exponent $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$7 in $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$8, with full allowance for weights in both $$\int_\Omega \nabla u \cdot \nabla v\, dx = \int_\Omega f(x, u, \nabla u) v\, dx \qquad \forall v \in H^1_0(\Omega)$$9 and $f$0 (Ching et al., 2018). The analysis uses nested Bernstein methods on logarithmic and original variables, yielding

$f$1

and in particular, universality of Liouville-type theorems (constancy of solutions globally) and boundary blow-up bounds. These results extend the reach of classic comparison and maximum principles to highly degenerate, nonvariational scenarios.

Table: Key Methods and Their Domains of Applicability

Method	Scope	Reference
Truncation + Banach Fixed Point	Multiplicity, non-symmetric PDEs with gradient terms	(Bahrouni, 24 Dec 2025)
$f$2-function/Max Principle	Gradient bounds for entire-space PDEs	(Cavaterra et al., 2019)
LP-viscosity + geometric iteration	Regularity with unbounded gradient growth	(Silva et al., 2020)
Topological fixed-point index	Systems, functional BCs, multiple solutions	(Biagi et al., 2019, Cianciaruso et al., 2017)
Moreau–Yosida + energy method	Variational inequalities, inclusions	(Papageorgiou et al., 2018)
Kazdan–Kramer change of variable	Rigidity, transformation of gradient terms	(Oliveira, 2 Nov 2025)

Outlook and Open Directions

Significant new capabilities have been unlocked for nonvariational elliptic PDEs with gradient dependence, including infinite multiplicity, sharp regularity, and solution structure in fully nonlinear and system contexts. Open problems include $f$3 regularity in superlinear gradient regimes, extension to parabolic equations and measure data, and further weakening of oscillation and convexity conditions (Silva et al., 2020). The development of unified frameworks for viscosity, index, and truncation methods continues to drive progress in the analysis of these complex nonvariational problems.