Dynamic Semi-Convexity Condition

• Dynamic semi-convexity is a parameter-dependent lower-curvature condition that adjusts the admissible semi-convexity based on governing PDE parameters.
• In special Lagrangian equations, the dynamic threshold κ = tan(θ) ensures that phase-lowering rotations preserve viscosity subsolutions and supersolutions.
• Broader formulations in DC decompositions and online optimization demonstrate how dynamic curvature bounds yield quantitative regularity and dynamic regret control.

Dynamic semi-convexity is a parameter-dependent lower-curvature condition in which the admissible amount of semi-convexity is not fixed a priori, but varies with the governing parameters of a problem. In the special Lagrangian equation, this notion is realized by a phase- and dimension-dependent threshold

$$u+\frac{1}{2}\tan(\theta)\,|x|^2\;\text{convex},\qquad \theta=\frac{\tfrac{\pi}{2}-\Theta}{n-1},$$

for viscosity solutions of

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$

where $\lambda_i$ are the Hessian eigenvalues and $\Theta$ is the phase. In that setting, the condition is “dynamic” because the threshold $\kappa=\tan\theta$ tightens or relaxes as $\Theta$ and $n$ vary, and because it is exactly the threshold needed for phase-lowering rotations to preserve viscosity subsolutions and supersolutions. This mechanism yields analyticity, interior derivative estimates, sharp counterexamples, and a Liouville theorem in the subcritical almost-negative regime (Mooney et al., 20 Oct 2025).

1. General meaning of dynamic semi-convexity

Semi-convexity, in its standard form, requires that

$x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$

be convex on a convex domain $D$. When $f$ is twice differentiable, this is equivalent to the Hessian lower bound

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$0

A dynamic or parametric variant replaces the constant $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$1 by a parameter-dependent function, for example requiring that for each $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$2,

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$3

be convex in $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$4. This yields a time-dependent DC decomposition

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$5

whenever $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$6 uniformly (Proudnikov, 2017).

In the special Lagrangian setting, the same structural idea becomes quantitatively sharper. The lower-curvature allowance is not arbitrary: it is locked to the phase $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$7 and the dimension $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$8 through

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$9

The term “dynamic” therefore refers to a threshold that must be adjusted as the ambient PDE parameters change, rather than to time evolution (Mooney et al., 20 Oct 2025).

A related but distinct usage appears in online optimization. There, a practical dynamic semi-convexity condition is formulated through per-round smoothness and an error-bound condition, namely

$\lambda_i$0

together with slowly drifting solution sets. This condition supports contraction toward $\lambda_i$1 and dynamic regret bounds depending on path-length or squared path-length (Zhang et al., 2016).

Context	Dynamic semi-convexity form	Main role
Special Lagrangian PDE	$\lambda_i$2 convex	Preserves viscosity structure under rotation
DC representation	$\lambda_i$3 convex in $\lambda_i$4	Produces dynamic DC decompositions
Dynamic regret	Error-bound/semi-strong convexity with $\lambda_i$5	Gives contraction toward time-varying minimizers

2. Special Lagrangian formulation and the phase-dependent threshold

For a potential $\lambda_i$6 on $\lambda_i$7, the special Lagrangian equation is

$\lambda_i$8

with phase $\lambda_i$9. The paper isolates the subcritical phase range

$\Theta$0

and defines

$\Theta$1

This parameter distributes the gap from $\Theta$2 uniformly across the $\Theta$3 non-maximal eigen-directions (Mooney et al., 20 Oct 2025).

The associated dynamic semi-convexity condition is

$\Theta$4

equivalently

$\Theta$5

Its dependence on $\Theta$6 is monotone. As $\Theta$7, one has $\Theta$8, so the admissible semi-convexity becomes increasingly restrictive. As $\Theta$9, one has $\kappa=\tan\theta$0, and the allowable semi-convexity becomes much larger. The threshold is therefore neither a purely local regularity assumption nor a generic lower Hessian bound; it is tied to the phase geometry of the equation itself (Mooney et al., 20 Oct 2025).

A central algebraic identity is

$\kappa=\tan\theta$1

This quantitatively links the maximal Hessian eigenvalue to the remaining eigenvalues shifted by $\kappa=\tan\theta$2. In the later analysis, that identity provides the eigenvalue rigidity needed to rule out blow-up of the largest rotated eigenvalue (Mooney et al., 20 Oct 2025).

The viscosity framework is standard: if a quadratic polynomial $\kappa=\tan\theta$3 touches $\kappa=\tan\theta$4 from above near $\kappa=\tan\theta$5, then $\kappa=\tan\theta$6; if it touches from below, then $\kappa=\tan\theta$7. Ellipticity is expressed by

$\kappa=\tan\theta$8

At smooth points, the linearization agrees with the Laplace–Beltrami operator on the gradient graph $\kappa=\tan\theta$9 with metric $\Theta$0, so directional second derivatives $\Theta$1 are subsolutions of the linearized operator (Mooney et al., 20 Oct 2025).

3. Rotation, touching preservation, and phase descent

The decisive mechanism is a Lewy–Yuan-type rotation of the gradient graph. For $\Theta$2,

$\Theta$3

with $\Theta$4 and $\Theta$5. The rotated potential is defined by the Legendre-transform formula

$\Theta$6

The rotation preserves touching, and the eigen-angles satisfy

$\Theta$7

This allows the phase to be lowered while retaining viscosity control (Mooney et al., 20 Oct 2025).

Under the phase range and dynamic semi-convexity hypothesis, both subsolutions and supersolutions are preserved: $\Theta$8

$\Theta$9

The paper emphasizes that this preservation is sharp in the phase/semi-convexity range considered; outside that regime, rotations can fail to preserve subsolutions (Mooney et al., 20 Oct 2025).

A specific choice of angle drives the problem into a negative supercritical regime: $n$0 so that

$n$1

For this rotated phase, $n$2 solves a concave, uniformly elliptic PDE with convex superlevel set. Evans–Krylov then yields $n$3 regularity, and Morrey’s theorem upgrades the rotated solution to analyticity. Because the rotation map is distance-increasing and locally bi-Lipschitz, the regularity can be transferred back to the original function $n$4 (Mooney et al., 20 Oct 2025).

4. Eigenvalue rigidity, analyticity, and quantitative interior estimates

The dynamic semi-convexity hypothesis does more than provide a lower Hessian bound. Combined with the phase splitting in equation (1), it enforces a rigidity phenomenon at any putative blow-up point of the maximal rotated eigenvalue. If

$n$5

at some point, then the remaining rotated eigenvalues must saturate the lower bound: $n$6 By convexity of level sets for the rotated equation of $n$7, sums of the largest $n$8 eigenvalues are viscosity subsolutions of the linearized operator; the strong maximum principle then forces constancy of the smaller eigenvalues, and the equation forces constancy of $n$9. This contradicts Alexandrov’s theorem that $x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$0 is finite almost everywhere. Consequently,

$x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$1

everywhere (Mooney et al., 20 Oct 2025).

The inverse Hessian transform is

$x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$2

Since the right-hand side is locally bounded once $x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$3 is controlled, the original potential acquires a locally bounded Hessian and hence analyticity. This is the content of the regularity theorem: if

$x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$4

and $x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$5 is a viscosity solution in $x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$6 with

$x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$7

then $x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$8 is analytic. Moreover, for $x\mapsto f(x)+\frac{\mu}{2}\|x\|^2$9,

$D$0

The dependence is exponential in the Lipschitz norm and depends only on $D$1 (Mooney et al., 20 Oct 2025).

The analysis also yields pointwise control of the non-maximal eigenvalues: $D$2 This estimate feeds into a volume bound for the rotated image: $D$3 A chain-of-balls argument in the rotated domain, combined with the weak Harnack inequality, gives for nonnegative supersolutions $D$4 of the linearized equation at $D$5,

$D$6

for universal $D$7. Applying this to

$D$8

produces the exponential Hessian control (Mooney et al., 20 Oct 2025).

5. Sharpness, counterexamples, and Liouville rigidity

The phase range and the dynamic threshold are both sharp. First, if

$D$9

and $f$0, there exist singular viscosity solutions of $f$1 such that $f$2 is convex. Thus, once the phase crosses $f$3, no negative uniform lower bound on $f$4 is sufficient to force regularity. Second, if

$f$5

and $f$6, there exist singular viscosity solutions such that

$f$7

that is,

$f$8

Hence the threshold $f$9 cannot be relaxed. In both regimes, the examples are Lipschitz but not $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$00 and have non-minimal gradient graphs (Mooney et al., 20 Oct 2025).

The construction begins from a rank-$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$01 model

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$02

for which $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$03 has a nondegenerate local minimum at the origin. One then solves the constant-phase equation near a small convex set $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$04, glues analytically to a function $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$05 with one-sided sign on $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$06, and applies the Legendre transform to define $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$07. Away from the image hypersurface, $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$08 is analytic and solves

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$09

with one Hessian eigenvalue tending to $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$10 and the others controlled (Mooney et al., 20 Oct 2025).

The exponential dependence in equation (5) is also optimal. In dimension $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$11 and phase $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$12, the explicit solution

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$13

satisfies

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$14

Therefore any interior Hessian estimate must be at least exponential in $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$15. For $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$16 in $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$17, a partial Legendre–Lewy–Wang–Yuan transform yields a rotated potential solving

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$18

and the exponential behavior persists. Adding quadratic directions with coefficient $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$19 or $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$20 embeds this optimality into higher dimensions (Mooney et al., 20 Oct 2025).

The same dynamic mechanism yields a Liouville theorem. If

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$21

and $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$22 is an entire viscosity solution of the special Lagrangian equation, then $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$23 is a quadratic polynomial. After rotation to a negative supercritical phase, Evans–Krylov and Schauder imply decay of the Hölder seminorm of $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$24 on large balls, forcing $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$25 to be quadratic; the inverse rotation then gives the same conclusion for $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$26 (Mooney et al., 20 Oct 2025).

6. Broader formulations beyond the special Lagrangian equation

Outside special Lagrangian geometry, dynamic semi-convexity appears as a reusable structural pattern. In the DC setting, one formulation requires that for each parameter value $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$27,

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$28

be convex. Uniform control $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$29 produces a time-uniform decomposition into convex components. In the smooth case, if

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$30

then

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$31

is convex, so

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$32

is a DC decomposition. The same paper characterizes DC functions in terms of bounded variation of directional derivatives along circle or plane arcs, which serves as a nonsmooth analogue of curvature control (Proudnikov, 2017).

In online learning, the analogous role is played by semi-strong convexity, defined by

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$33

This is weaker than strong convexity because the minimizer set may be non-singleton and the curvature is only enforced relative to distance from the solution set. With $F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$34-smooth losses and one gradient query per round, dynamic regret scales with the path-length of the comparator sequence. With multiple gradient queries per round,

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$35

the paper derives bounds of order

$F(D^2u):=\sum_{i=1}^n \tan^{-1}(\lambda_i)=\Theta,$36

under strong convexity, and under semi-strong convexity when the aggregate gradients at minimizers are suitably small. In the self-concordant case, a damped Newton scheme achieves an analogous dependence in local Hessian norms (Zhang et al., 2016).

These broader formulations indicate that dynamic semi-convexity is not a single invariant definition across all fields. Rather, it denotes a family of parameter-dependent lower-curvature conditions whose exact form is dictated by the surrounding analytic mechanism. In special Lagrangian theory, the threshold is phase-driven and rotation-compatible; in DC analysis, it is tied to quadratic regularization and variation bounds; in dynamic regret, it appears as an error-bound condition that contracts iterates toward moving solution sets. A plausible implication is that the common mathematical content is not merely “semi-convexity with varying constants,” but semi-convexity calibrated to the transformation or stability principle that a problem requires (Mooney et al., 20 Oct 2025, Proudnikov, 2017, Zhang et al., 2016).