ABP–Krylov–Tso Estimate in Nonlinear PDEs

• ABP–Krylov–Tso estimate is a maximum-principle type bound that controls the negative part of a solution using boundary data and Lⁿ or L^(n+1) forcing norms.
• It employs contact sets, convex envelopes, and determinant or Jacobian estimates to translate geometric touch conditions into a priori bounds across elliptic, parabolic, and discrete settings.
• Its adaptations in transmission, geometric, and metric frameworks advance regularity theory by ensuring uniqueness, stability, and convergence in complex PDE problems.

The ABP–Krylov–Tso estimate designates a class of maximum-principle-type a priori bounds in which the negative part or supremum of a solution is controlled by boundary data together with an (L^n)- or (L^{n+1})-type norm of the positive forcing term. In the fully nonlinear parabolic setting, the estimate is associated with the Krylov–Tso extension of the Aleksandrov–Bakelman–Pucci method; in elliptic, geometric, metric, and discrete settings, closely related ABP estimates retain the same contact-set, Jacobian, or envelope philosophy while adapting the underlying structure to singular submanifolds, metric measure spaces, finite element schemes, or transmission interfaces [2507.19277][1411.6036].

1. Defining form and structural features

An ABP–Krylov–Tso estimate is characterized in the supplied literature as a maximum principle-type bound. In the parabolic fully nonlinear case without transmission, it controls the negative part of a subsolution or supersolution by boundary data and the (L^{n+1})-norm of the positive forcing term. In the elliptic non-divergence setting, the classical ABP estimate appears in the form
[
\sup_{\Omega}u^- \le C\left(\int_{\mathcal C^-(u)} |f(x)|^d\,dx\right)^{1/d},
]
where (u^-=\max{-u,0}) and (\mathcal C^-(u)) is the lower contact set of (u) with its convex envelope (\Gamma(u)) [1411.6036].

The same sources distinguish this framework from energy estimates. The characteristic ABP–Krylov–Tso ingredients are the control of a supremum or negative part, the use of contact sets or convex envelopes, and a determinant, Jacobian, or measure estimate on the set of touching points. The parabolic version replaces static convex-envelope arguments by parabolic convex envelopes and time-monotone contact-set constructions. This suggests that the expression “ABP–Krylov–Tso” is best reserved for bounds that preserve the maximum-principle architecture rather than merely any a priori estimate involving forcing terms [2507.19277][2002.12726].

2. Fully nonlinear parabolic transmission estimate

A precise modern instance appears in Theorem 2.1 of “Fully nonlinear parabolic fixed transmission problems” [2507.19277]. The setting is a parabolic transmission problem across a time-dependent interface:
[
\begin{cases}
u_t - F^\pm(D^2u)= f^\pm & \text{ in } \Omega^\pm,\ u_\nu^+ - u_\nu^- = g & \text{ on } \Gamma,
\end{cases}
]
with
[
\Omega^\pm := {(x,t)\in \mathcal C_1 : \pm(x_n-\psi(x',t))>0},\qquad \Gamma := {(x,t)\in \mathcal C_1 : x_n=\psi(x',t)},
]
where (\mathcal C_1=B_1\times(-1,0]). The assumptions are that (\psi \in C^{1,\alpha}(\overline{\mathcal C_1'})), (f\in C(\mathcal C_1\setminus \Gamma)\cap L^\infty(\mathcal C_1)), (g\in L^\infty(\Gamma)), and (F^\pm) are uniformly ((\lambda,\Lambda))-elliptic with (F^\pm(0)=0).

The theorem states
[
\sup_{\mathcal C_1} u_- \leq \sup_{\partial_p \mathcal C_1} u_- + C\Big(\max_\Gamma g_+ + |f_+|{L^{n+1}(\mathcal C_1)}\Big),
]
where (u-=\max{-u,0}), (g_+=\max{g,0}), and (C) depends only on
[
n,\ \lambda,\ \Lambda,\ \alpha,\ |\psi|{C^{1,\alpha}(\overline{\mathcal C_1'})}.
]
For a flat interface (\Gamma=\mathcal C_1'), the paper records the sharper contact-set form
[
\sup{\mathcal C_1}u_- \leq \sup_{\partial_p\mathcal C_1}u_- + C\left(\max_{\Gamma}g_+ + |f_+|{L^{n+1}(\mathcal C_1\cap{u=C_u})}\right),
]
where (C_u) is the parabolic convex envelope of (-u-) [2507.19277].

The transmission term is the genuinely new feature. The interface condition (u_\nu^+-u_\nu^-\le g) enters the estimate through (\max_\Gamma g_+), so the theorem is not merely a phase-by-phase application of a classical ABP bound. The paper uses the estimate to prove a maximum principle, a comparison principle for flat-interface problems, uniqueness of viscosity solutions, existence via Perron’s method, and later Harnack and stability arguments. It is also identified there as a key tool for the regularity theory leading to piecewise (C^{1,\alpha}) regularity up to the interface [2507.19277].

3. Geometric, metric, and low-dimensional variants

The ABP mechanism extends far beyond uniformly elliptic PDE on smooth Euclidean domains. In “ABP inequalities for singular submanifolds of bounded mean curvature” [1809.02198], the estimate is reformulated for White’s ((m,h))-sets, a class of singular submanifolds of arbitrary codimension. For a relatively closed ((m,h))-subset (T\subset C_1(0)), a closed base set (C\subset \mathbf B^n(0,1)), and the contact set (A_a(T;C)) defined through upward-opening paraboloids, Theorem 1.2 gives
[
\mathscr H^n(C) \le \gamma\,(1+a+h a^{-1})^{m}\,(1+a+2a^{-1})^{\,n-m} \, \mathscr H^n\bigl(A_a(T;C)\bigr),
]
with
[
\gamma=4^{\,1-m}(2m+4h)^m.
]
The proof replaces PDE structure by geometric measure theory: a vertex map
[
Y(w,\nu)=w'+a^{-1}\nu'
]
is defined on the generalized normal bundle, curvature bounds are read from the touching paraboloids, and an area formula on the normal bundle yields the measure estimate. In codimension (1), Corollary 1.3 sharpens the estimate to a projected contact set (A_a'(T;C)), and Corollary 1.4 recovers Savin’s ABP estimate for viscosity subsolutions of the minimal surface equation [1809.02198].

A different non-smooth extension appears in “ABP estimate on metric measure spaces via optimal transport” [2408.10725]. There the ambient object is an (\mathrm{RCD}(K,N)) metric measure space, and the contact sets are defined through distance-based barriers:
[
R_2(D,\Omega,u,t)
\quad\text{and}\quad
R_1(D,\Omega,u,t).
]
Theorem 1.4 controls the measure of a compact set (D) by the measure of the corresponding contact set, with explicit curvature-dimension distortion factors depending on (K), (N), the transport scale (t), and the geometry of (D). In the zero-curvature case, the theorem yields the clean form
[
m(D)\le m(R_i)\exp!\bigl(t\,|(\Delta u)+|{L^\infty}\bigr),\qquad i=1,2.
]
The paper emphasizes that this estimate is sharp and dimension-dependent, and that optimal transport replaces classical Jacobi-field computations. It also proves a functional ABP estimate via entropy along Wasserstein geodesics, with the geometric estimate obtained as a corollary [2408.10725].

At the opposite end of generality, “A Classical Analysis Counterpart of Viterbo's Symplectic Geometry Proof of ABP in the Plane” [2605.12712] isolates the two-dimensional analytic core of ABP. For compactly supported (C^2) functions on (\mathbb R^2), it proves
[
\bigl(\sup_K f - \inf_K f\bigr)^2 \le 16\,\diam(K)^2 \int_K |\det D^2f|\,dx,
]
and then removes the compact support hypothesis to obtain the boundary-term version
[
\bigl(\sup_K f - \inf_K f - (\sup_{\partial K} f - \inf_{\partial K} f)\bigr)^2 \le 16\,\diam(K)^2 \int_K |\det D^2 f|\, d x.
]
The proof explicitly avoids convexity and contact sets. Instead it uses a co-area decomposition of (\det D^2f), one-dimensional oscillation estimates along level curves, and a path construction through alternating sign regions. The paper presents this as an ABP inequality rather than a full PDE Krylov–Tso theorem [2605.12712].

4. Discrete and probabilistic analogues

The ABP framework also has a discrete finite element incarnation. “Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form” [1411.6036] studies
[
A(x):D^2u(x)=f(x)\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,
]
under uniform ellipticity, and constructs a two-scale finite element method with fine scale (h) and coarse scale (\epsilon). The discrete operator is
[
L_h^\epsilon u_h^\epsilon(x_i) :=\frac{\lambda}{2}\Delta_h u_h^\epsilon(x_i)+I_\epsilon u_h^\epsilon(x_i)=f_i,
]
and the mesh is required to be face weakly acute. The discrete ABP estimate is then
[
\sup_\Omega v_h^- \le C\left(\sum_{x_i\in\mathcal C_h^-(v_h)} |f_i^+|^d\,|\omega_i|\right)^{1/d},
]
where (\mathcal C_h^-(v_h)) is the discrete nodal contact set and (|\omega_i|) is the nodal star volume. Its proof combines a discrete Alexandroff estimate with a geometric control of subdifferentials by jumps of the gradient across mesh faces. This estimate is the stability mechanism behind the max-norm convergence rate
[
| u - u^{\epsilon}_h |{L\infty(\Omega)} \leq \, C(A,u) \, h^{2\alpha /(2 + \alpha)} \big| \ln h \big|,
\qquad
\epsilon \approx h^{2/(2+\alpha)},
]
and, in the smoother case, the quasi-linear rate (h|\ln h|) [1411.6036].

A probabilistic and game-theoretic link appears in “Hölder estimate for a tug-of-war game with (1<p<2) from Krylov-Safonov regularity theory” [2212.10807]. The paper introduces a dynamic programming principle for the singular range (1<p<2),
[
u(x) = \frac12 \left( \sup_{|z|=1} I^z_\varepsilon u(x) + \inf_{|z|=1} I^z_\varepsilon u(x) \right) +\varepsilon^2 f(x),
]
and proves that its solutions satisfy extremal inequalities
[
\mathcal L^+ u + f \ge 0, \qquad \mathcal L^- u + f \le 0.
]
These are the discrete analogues of Pucci-type inequalities needed by the abstract Krylov–Safonov theory developed in the cited earlier ABP work. The resulting Hölder estimate has the form
[
|u(x)-u(y)| \le C \left( \sup_{B_R}|u| + R^2\sup_{B_R}|f| \right) \left(\frac{|x-y|}{R}+\varepsilon\right)^\gamma, \quad x,y\in B_{R/2}.
]
The paper does not prove a new ABP theorem; rather, it shows how an appropriately designed DPP can be inserted into an existing ABP/Krylov–Safonov regularity engine [2212.10807].

5. Proof architecture and recurring mechanisms

Across these settings, the ABP–Krylov–Tso mechanism is organized around contact geometry. In the transmission problem of [2507.19277], the proof begins with a barrier (w) satisfying
[
0\le w\le1,\qquad w_\nu\ge c_0>0 \quad \text{on }\Gamma\cap\mathcal C_1,
]
obtained using a Hopf lemma for fully nonlinear parabolic equations. One then subtracts a multiple of (w),
[
v = u - \frac{1}{c_0}\big(\max_\Gamma g_+ + \varepsilon\big)w,
]
so that
[
v_\nu^+-v_\nu^- \le -\varepsilon \quad\text{on }\Gamma.
]
This strict jump inequality excludes interface points from the parabolic contact set after the last zero time. The argument then passes to the parabolic convex envelope (C_v) and derives the determinant bound
[
(-\partial_t C_v)\det D^2C_v \le C f_+^{n+1}
]
on the contact set, which is the parabolic Monge–Ampère-type core of the estimate [2507.19277].

In the arbitrary-codimension geometric theory of [1809.02198], the same role is played by a vertex map on the generalized normal bundle. The contact geometry of paraboloids yields curvature inequalities such as
[
\kappa_i(w,\nu)\ge -a\,\nu_{n+1}^{-1},
\qquad
\sum_{i=1}^m \kappa_i(w,\nu)=\operatorname{trace}Q_T(w,\nu)\le h,
]
and the area formula converts those curvature bounds into a measure estimate for the image of the vertex map. In the metric measure setting of [2408.10725], optimal transport replaces differential geometry: Kantorovich potentials, entropy convexity, and distortion coefficients (C_{K/N}) and (S_{K/N}) act as the non-smooth analogue of the Jacobian determinant estimate. In the two-dimensional analytic proof of [2605.12712], the determinant is converted by co-area into one-dimensional total variation,
[
\int_K |\det D^2 f(x)|\,dx = \int_{-\infty}^{\infty} \TV_z(f_{x_1})\,dz,
]
and the role of the contact set is replaced by level curves and admissible path constructions.

These variations show that the ABP–Krylov–Tso estimate is less a single formula than a method for converting geometric information about touching structures into supremum bounds. A plausible implication is that the robustness of the method lies in the portability of three elements: a notion of contact set, a Jacobian- or determinant-type control, and a comparison principle tying those quantities back to the forcing term [2507.19277][1809.02198][2408.10725][2605.12712].

6. Scope, limitations, and common misidentifications

The supplied literature explicitly separates ABP–Krylov–Tso estimates from other a priori inequalities that do not have maximum-principle form. The clearest non-example is “Estimates of solutions to the linear Navier-Stokes equation” [2002.12726]. That paper studies the linear incompressible Stokes/Navier–Stokes system in a bounded domain with zero initial and boundary data, writes the Green function as (G=Z+V), uses (\partial_{x_i}Z=-\partial_{\xi_i}Z) together with (\operatorname{div}\mathbf u=0), derives an explicit representation for the pressure, and proves the (L^2) estimate
[
\int_0^t \sum_{i=1}^3 \left|\frac{\partial p(x,\tau)}{\partial x_i}\right|{L_2(\Omega)}^2\,d\tau
< c\int_0^t \sum{i=1}^3 |w_i(x,\tau)|{L_2(\Omega)}^2\,d\tau.
]
It also states
[
|u|{W^{2,1}(Q_t)} \le c\,|w|_{L^2(Q_t)}.
]
The paper is not an ABP–Krylov–Tso paper in the usual sense: it treats a linear system, uses Green-function representation, and derives an (L^2) pressure-gradient bound rather than a supremum estimate based on contact sets or convex envelopes [2002.12726].

Related scope limitations appear elsewhere. The metric-space paper [2408.10725] develops a sharp elliptic ABP estimate but does not prove a parabolic Krylov–Tso theorem. The two-dimensional Viterbo-style analysis paper [2605.12712] proves a Hessian-determinant ABP inequality but does not treat lower-order terms, measurable coefficients, or the full PDE estimate. The tug-of-war paper [2212.10807] uses ABP/Krylov–Safonov regularity theory as an external tool rather than establishing a new ABP estimate. These distinctions matter because the phrase “ABP–Krylov–Tso estimate” properly refers to a specific maximum-principle paradigm, not to every bound that is merely adjacent to it.

In this sense, the contemporary literature exhibits both breadth and precision. The estimate survives transmission interfaces, singular submanifolds, (\mathrm{RCD}(K,N)) spaces, finite element discretizations, and game-theoretic schemes, but the defining criterion remains stable: control of extrema through boundary data and forcing, mediated by contact geometry and a Jacobian, determinant, or measure estimate on the set of touching points [2507.19277][1411.6036].