Geodesic
Determination of density of elliptic potential
Eurasian mathematical journal, Tome 12 (2021) no. 4, pp. 43-52

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In this paper, using techniques of finding boundary conditions for the volume (Newton) potential, we obtain the boundary conditions for the volume potential $$ u(x)=\int_\Omega\varepsilon(x,\xi)\rho(\xi)d\xi, $$ where $\varepsilon(x,\xi)$ is the fundamental solution of the following elliptic equation $$ L(x,D)\varepsilon(x,\xi)=-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}a_{ij}(x)\frac{\partial}{\partial x_j}\varepsilon(x,\xi)+a(x)\varepsilon(x,\xi)=\delta(x,\xi). $$ Using the explicit boundary conditions for the potential $u(x)$, the density $\rho(x)$ of this potential is uniquely determined. Also, the inverse Sommerfeld problem for the Helmholtz equation is considered. 
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     author = {T. Sh. Kalmenov and A. K. Les and U. A. Iskakova},
     title = {Determination of density of elliptic potential},
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     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2021_12_4_a3/}
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T. Sh. Kalmenov; A. K. Les; U. A. Iskakova. Determination of density of elliptic potential. Eurasian mathematical journal, Tome 12 (2021) no. 4, pp. 43-52. http://geodesic.mathdoc.fr/item/EMJ_2021_12_4_a3/