Elliptic · We need ABP

An entropy approach to elliptic ABP

Astra’s proof of the kinetic Krylov–Safonov theorem is based on a novel approach to the ABP estimate. It derives an entropy estimate for the adjoint equation by regularising along an auxiliary flow and exploiting entropy dissipation.

Pucci’s conjecture

Pucci’s conjecture concerns the optimal integrability of the forcing term in the Aleksandrov–Bakelman–Pucci maximum estimate for elliptic equations with merely measurable coefficients. Pucci formulated the conjecture in Operatori ellittici estremanti (Ann. Mat. Pura Appl., 1966, DOI).

Conjecture: Fix an integer \(n\ge2\), \(0<\delta<1/n\), and \(1\le p<\infty\). Let \(B_1=\{x\in\mathbb R^n:|x|<1\}\), where \(|\cdot|\) is the Euclidean norm. Consider all measurable symmetric matrix fields \(A=(a_{ij}):B_1\to\mathbb R^{n\times n}\) satisfying, for almost every \(x\),

\operatorname{tr}A(x)=1, \qquad \xi^T A(x)\xi\ge\delta|\xi|^2 \quad\text{for every }\xi\in\mathbb R^n,

and define

L_Au=\sum_{i,j=1}^n a_{ij}(x)\,\partial_{ij}^{2}u.

There exists a finite constant \(C=C(n,\delta,p)>0\) such that

\sup_{x\in\overline{B_1}}|u(x)| \le C\left(\int_{B_1}|L_Au(x)|^p\,\mathrm{d}x\right)^{1/p}

holds for every solution \(u\) if and only if

p>\max\left\{\frac n2,\;n\bigl[1-(n-1)\delta\bigr]\right\}.

This conjecture was solved by Astra, and we are preparing an exposition of the mathematical ideas in its proof.

An ABP estimate for nonlocal equations

Local diffusion depends on how a function changes near a point. Nonlocal diffusion also compares its values at distant points, through an integral describing possible jumps. The question is whether the adjoint approach can give a maximum estimate in this setting as well.

The supplied AI manuscript proves an Aleksandrov estimate for a specific class of even kernels. Their coefficients admit a representation using nonnegative spatial densities and bounded jump profiles, with a uniform moment bound; integrable corrections are allowed. An additional feature is that the estimates remain uniform as the order of the equation approaches two. The kernel representation is a substantive hypothesis, and the manuscript includes an example showing that ellipticity alone does not imply it.