Two-Sided Quadratic Gradient Growth (QGG)

• Two-Sided QGG is defined by a balanced quadratic growth in gradient terms of nonlinear PDEs, ensuring structured behavior and critical solution properties.
• Analytical methods including exponential transformations, maximum principles, and degree theory establish uniform a priori bounds and strong regularity.
• The dichotomy between coercive and noncoercive regimes reveals unique solutions in the former and bifurcation with multiple branches in the latter.

Two-Sided Quadratic Gradient Growth (QGG) denotes a structural property of nonlinear operators, functionals, or systems whereby the growth in the gradient is balanced in both directions—above and below—by quadratic expressions, often arising in elliptic and parabolic PDEs, stochastic differential equations, and optimization. This critical regime is distinguished from subquadratic or superquadratic cases due to its profound implications for existence, uniqueness, multiplicity of solutions, regularity, and performance of iterative algorithms.

1. Precise Equation Formulation and QGG Definition

Two-Sided QGG appears in PDEs with terms that grow precisely quadratically in the gradient. A canonical model is: $$- F(x, u, Du, D^2u) = \lambda c(x) u + \langle M(x) Du, Du \rangle + h(x) \;\; \text{in } \Omega, \; u = 0 \text{ on } \partial\Omega$$ with $\lambda \in \mathbb{R}$; $c(x), h(x)$ in $L^p(\Omega)$, $p > n$; and $M(x)$ satisfying $0<\mu_1 I \leq M(x) \leq \mu_2 I$.

The operator $F$ is a general fully nonlinear, uniformly elliptic operator, encompassing Isaacs or Hamilton–Jacobi–Bellman types. Its structure is formalized by comparison inequalities with Pucci extremal operators: $$\mathcal{M}^-(X-Y) - b(x)|p-q| \leq F(x, r, p, X) - F(x, s, q, Y) \leq \mathcal{M}^+(X-Y) + b(x)|p-q|$$ The quadratic nonlinearity is encoded in $\langle M(x) D u, D u \rangle$. In several sections, an exponential change of variables, $v = (e^{m u} - 1)/m$, is employed to analyze the role of the gradient term and facilitate variational arguments.

2. Uniform A Priori Bounds and Regularity

A foundational result is the establishment of uniform $L^\infty$ bounds for solutions: $$\|u^+\|_\infty \leq C \quad\text{and}\quad \|u^-\|_\infty \leq C$$ for all $\lambda$ in appropriate ranges, with $C$ depending only on data, ellipticity, domain geometry, and norm bounds for $c, h$. Classic direct energy methods fail in the natural growth regime; instead, the approach combines:

• Global $C^{1,\alpha}$ regularity (interior and boundary)
• Half-Harnack inequalities and boundary maximum principles
• A strong maximum principle (Vázquez type), adapted to quadratic gradient terms

These enable the deployment of degree theory, via a completely continuous fixed-point operator in $C^1(\overline{\Omega})$, to analyze the solution set topology.

3. Solution Multiplicity and Structure

A key insight is the dichotomy between the coercive ($\lambda c \leq 0$) and noncoercive ($\lambda c > 0$) cases:

• Coercive regime: Uniqueness holds, supported by classical maximum principles.
• Noncoercive regime: Multiplicity arises; precisely, for $\lambda > 0$ there exist at least two $L^p$-viscosity solutions:
  • The minimal branch $u_1(\lambda)$ converges to the unique solution at $\lambda=0$ as $\lambda \to 0^+$.
  • The secondary branch $u_2(\lambda)$ has its positive part diverging $(\|u_2(\lambda)\|_\infty \to +\infty)$ as $\lambda \to 0^+$.
  • The parameter-solution continuum exhibits bifurcation phenomena from $u_0$ at $\lambda=0$ and may extend for all $\lambda > 0$ or originate from infinity.

Ordering of solutions is established (using the relation $\ll$ in $C^1(\overline{\Omega})$), and topological degree arguments confirm the existence and separation of solution branches.

4. Analytical Techniques and Principles

Several analytical tools underpin the results:

• Vázquez-Type Maximum Principle: Adapts comparison theorems for equations involving quadratic gradient terms.
• Alexandrov–Bakelman–Pucci (ABP) Inequality: Modified for natural (quadratic) growth to extract uniform bounds.
• Exponential Transformation: Converts $\langle M(x) Du, Du \rangle$ into a form amenable to standard techniques.
• Degree Theory: Application via a fixed-point operator on the solution space, relying on the compactness induced by a priori $L^\infty$ bounds.
• Regularity Theory: Use of global $C^{1,\alpha}$ and $W^{2,p}$ estimates to achieve compactness.

These tools collectively ensure solvability, regularity, and precise characterization of solution sets under QGG.

5. Comparison with Divergence Form Problems and Extensions

Previous work (Arcoya, de Coster, Jeanjean, Tanaka) dealt with divergence form equations, exploiting integral representations to paper critical gradient growth. In contrast, the present analysis extends to fully nonlinear, nondivergence form operators—where energy and integration-by-parts are unavailable. Instead, viscosity solutions, half-Harnack techniques, and maximum principles are the backbone. Extension to Isaacs and HJB operators substantially broadens applicability beyond earlier frameworks.

6. Applications and Broader Implications

Natural gradient growth equations appear in:

• Hamilton–Jacobi–Bellman, Isaacs equations in stochastic control, differential games, mean-field models
• Nonlinear PDEs in geometric analysis (e.g., prescribed Gaussian curvature)
• Nonlinear elasticity and reaction–diffusion in physics and biology

The existence and multiplicity results obtained rigorously justify model predictions (e.g., risk-sensitive stochastic control) and reveal that small noncoercive perturbations yield rich solution sets, a phenomenon of substantial interest for bifurcation theory and nonlinear analysis.

Possible future directions include treating equations with more singular coefficients, exploring non-variational settings where transformation techniques fail, and further refining uniqueness and bifurcation diagrams for solutions.

7. Summary Table: Main Structural Properties

Property	Coercive Case ($\lambda c \leq 0$)	Noncoercive Case ($\lambda c > 0$)
Uniform a priori bounds	Yes	Yes
Uniqueness	Yes	No (Multiplicity)
Change of variable useful?	Sometimes	Crucial (for variational approach)
Solution set structure	Single branch	Continuum, bifurcation, ordering

In summary, Two-Sided Quadratic Gradient Growth governs the analytic and topological structure of fully nonlinear equations where gradient terms exhibit natural quadratic growth, leading to a rich interplay between regularity, multiplicity, and variational geometry that extends classical maximum principles and energy methods. This perspective has profound consequences for nonlinear PDE theory and applications across stochastic control, geometry, and physical modeling (Nornberg et al., 2018).