@article{Ramm2022, 
author = {Alexander G. Ramm},
title = {When does a double-layer potential equal to a single-layer one?},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {10},
pages = {19287-19291},
keywords = {potential theory},
url = {https://www.sciopen.com/article/10.3934/math.20221058},
doi = {10.3934/math.20221058},
abstract = {Let    D be a bounded domain in                                R                      3   with a closed, smooth, connected boundary    S,    N be the outer unit normal to    S,    k  >  0 be a constant,        u                  N                  ±                     are the limiting values of the normal derivative of    u on    S from    D, respectively        D    ′    :=                              R                      3    ∖            D      ¯      ;    g  (  x  ,  y  )  =            e              i        k                  |                x        −        y                  |                            4      π              |            x      −      y              |            ,    w  :=  w  (  x  ,  μ  )  :=      ∫    S        g          N        (  x  ,  s  )  μ  (  s  )  d  s be the double-layer potential,    u  :=  u  (  x  ,  σ  )  :=      ∫    S    g  (  x  ,  s  )  σ  (  s  )  d  s be the single-layer potential.In this paper it is proved that for every    w there is a unique    u, such that    w  =  u in    D and vice versa. This result is new, although the potential theory has more than 150 years of history.Necessary and sufficient conditions are given for the existence of    u and the relation    w  =  u in        D    ′  , given    w in        D    ′  , and for the existence of    w and the relation    w  =  u in        D    ′  , given    u in        D    ′  .}
}