Hessian estimates for non-divergence form elliptic equations arising from composite materials

TL;DR

This paper establishes local $W^{2, ext{infinity}}$ regularity and piecewise $C^{2}$ smoothness for solutions to non-divergence form elliptic equations with piecewise Dini mean oscillation coefficients, including global weak-type estimates.

Contribution

It provides new regularity results for elliptic equations with discontinuous coefficients and minimal boundary smoothness assumptions, extending previous theories.

Findings

Solutions are locally $W^{2, ext{infinity}}$ and piecewise $C^{2}$.

Global weak-type $(1,1)$ estimates are derived.

Results are independent of the distance between coefficient discontinuity surfaces.

Abstract

In this paper, we prove that any $W^{2,1}$ strong solution to second-order non-divergence form elliptic equations is locally $W^{2,\infty}$ and piecewise $C^{2}$ when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be $C^{1,\text{Dini}}$. We also derive global weak-type $(1,1)$ estimates with respect to $A_{1}$ Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these surfaces of discontinuity of the coefficients.