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

Hongjie Dong; Longjuan Xu

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arXiv · 1904.10950

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

Hongjie Dong·Longjuan Xu

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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.

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BibTeX RIS

Hongjie Dong, Longjuan Xu. 2019-04-24. Hessian estimates for non-divergence form elliptic equations arising from composite materials. https://arxiv.org/abs/1904.10950

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