2019/12/20 by Bruno Poggi, Poggi, Bruno · 1 citation
Mathematics · #35J15 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J15
paper · pdf · doi:10.48550/arxiv.1912.10115
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arxiv created 2019/12/20 · arxiv updated 2019/12/24
In this note, we explore some consequences of the Modica-Mortola construction of a singular elliptic measure, as regards the link between the quantitative absolute continuity (A∞) of their approximations and the suitability of a well-known tool, the so-called Kenig-Pipher condition (KP). The Kenig-Pipher condition is used to ascertain absolute continuity in the presence of some mild regularity of the coefficient matrix. We perform some modifications of the Modica-Mortola example to show the following two statements: (a) There are sequences of matrices for which both KP and the A∞ condition break down in the limit. (b) There are sequences of matrices for which KP breaks down but A∞ is preserved in the limit.