Three sphere inequality for second order elliptic equations with coefficients with jump discontinuity

Abstract

This is a short note to complete the paper appeared in J. Differential Equations 261 (2016), no. 10, pp. 5306-5323, where a rough version of the classical well known Hadamard three-circle theorem for solution of an elliptic PDE in divergence form has been proved. Precisely, instead of circles, the authors obtain a similar inequality in a more complicated geometry. In this paper we clean the geometry and obtain a generalized version of the three-circle inequality for elliptic equation with coefficients with discontinuity of jump type.

Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

“…We begin with deriving a three balls inequality, which is a direct consequence of Theorem 3.1. We would like to remark that a version of three balls inequality for the second order elliptic equation with jump-type discontinuous coefficients was obtained in [4]. However, the estimate in [4] does not fit what we need.…”

“…We would like to remark that a version of three balls inequality for the second order elliptic equation with jump-type discontinuous coefficients was obtained in [4]. However, the estimate in [4] does not fit what we need. So we derive our own three balls inequality here to serve a building block in the proof of the main theorem.…”