Let $ D $ be a bounded domain in $ {{\mathbb R}}^3 $ with a closed, smooth, connected boundary $ S $, $ N $ be the outer unit normal to $ S $, $ k > 0 $ be a constant, $ u_{N^{\pm}} $ are the limiting values of the normal derivative of $ u $ on $ S $ from $ D $, respectively $ D': = {{\mathbb R}}^3\setminus \bar{D} $; $ g(x, y) = \frac{e^{ik|x-y|}}{4\pi |x-y|} $, $ w: = w(x, \mu): = \int_S g_{N}(x, s)\mu(s)ds $ be the double-layer potential, $ u: = u(x, \sigma): = \int_S g(x, s)\sigma(s)ds $ 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' $.
Citation: Alexander G. Ramm. When does a double-layer potential equal to a single-layer one?[J]. AIMS Mathematics, 2022, 7(10): 19287-19291. doi: 10.3934/math.20221058
Let $ D $ be a bounded domain in $ {{\mathbb R}}^3 $ with a closed, smooth, connected boundary $ S $, $ N $ be the outer unit normal to $ S $, $ k > 0 $ be a constant, $ u_{N^{\pm}} $ are the limiting values of the normal derivative of $ u $ on $ S $ from $ D $, respectively $ D': = {{\mathbb R}}^3\setminus \bar{D} $; $ g(x, y) = \frac{e^{ik|x-y|}}{4\pi |x-y|} $, $ w: = w(x, \mu): = \int_S g_{N}(x, s)\mu(s)ds $ be the double-layer potential, $ u: = u(x, \sigma): = \int_S g(x, s)\sigma(s)ds $ 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' $.
| [1] | D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, Berlin, 1983. |
| [2] | A. Kirillov, A. Gvishiani, Theorems and problems in functional analysis, Springer Verlag, Berlin, 1982. https://doi.org/10.1007/978-1-4613-8153-2 |
| [3] | A. G. Ramm, Scattering of Acoustic and Electromagnetic Waves by Small Bodies of Arbitrary Shapes, Applications to Creating New Engineered Materials, Momentum Press, New York, 2013. |
| [4] |
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572–611. https://doi.org/10.1016/0022-1236(84)90066-1 doi: 10.1016/0022-1236(84)90066-1
|
Alexander G. Ramm. When does a double-layer potential equal to a single-layer one?[J]. AIMS Mathematics, 2022, 7(10): 19287-19291. doi: 10.3934/math.20221058
DownLoad: