Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate

Abstract

In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work [5]. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.

Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients

“…When we are at one side of the interface, we can use the usual propagation of smallness for equations with Lipschitz coefficients. When we near the interface, we then use the three-region inequality derived in [5]. The three-region inequality of [5] is used to propagate the smallness across the interface.…”

Stable determination of an inclusion in a layered medium

“…Before proving Proposition , let us briefly recall this result contained in Francini et al, Theorem 3.1. Based on some suitable Carleman estimate (see Di Cristo et al 4, Theorem 2.1 ), the following 3 region inequality in the L 2 norm across the interface y =0 holds.…”

“…Here, we will use the same approach. As for the quantitative estimates of unique continuation, we will take advantage of the recent paper . In Alessandrini and Di Cristo, the smallness information provided by the boundary measurements where dragged inside the domain up to the inclusion using 3 sphere inequality techniques.…”

“…This approach cannot be taken any longer as such an inequality is not available in the present geometry where there are different layers with different conductivity, and the location of the layers is not known. In Di Cristo et al, it has been derived a 3 region inequality for second order elliptic equations with jump discontinuous coefficient. Using such tool, that is based on a specific Carleman estimate developed in Di Cristo, it is possible to estimate in a precise manner how the boundary smallness propagates as we move towards the inclusion (see Proposition below).…”

Size estimates for the weighted p-Laplace equation with one measurement

“…The size estimate problem for linear equations or systems has been extensively studied. We refer to the nice survey article [3] for some early results and to the recent paper [17] for the case where the background medium is discontinuous. The idea is to use the power gap to derive upper and lower bounds of |D|.…”