Abstract
A construction of numerically implementable explicit expressions for the solutions of the two- and three-dimensional equations \(\operatorname{div}(\alpha(w)\nabla w)=0\) and \(\operatorname{div}(\beta\nabla w)=0\) with Cauchy data on an analytic boundary is presented.
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Notes
Due to the orthogonality of the curve \(\Gamma_\lambda\) to the plane \(\{x_3=0\}\), the normal vector to \(\mathcal T=\bigcup_{\lambda\in\mathbb{T}} \Gamma_\lambda\) is also normal to \(\Gamma_\lambda\).
For the reader’s convenience, we give a very short proof, which is preferable to the original one [1]. In view of (12) and (13), \(A\) and \(B\) are harmonic functions in \(V_{\mathbb{T}}\). The Cauchy–Riemann conditions \(\partial A/\partial\rho=(1/\rho)\partial B/\partial\theta\) and \(\partial B/\partial\rho=-(1/\rho)\partial A/\partial\theta\) clearly hold, and hence \(A+iB\) is an analytic function in \(V_{{\mathbb{T}}}\). We have \(A+iB\overset{(14)}{=}\ln(dz/d\zeta)\) and \(|dz/d\zeta|_{\zeta\in{\mathbb{T}}}=e^A|_{\rho=1}\overset{(13)}{=}1\). Next, \(1=|dz/d\zeta|_{\rho=1}=ds(\theta)/d\theta\). It can be assumed that \(s(\theta) =\theta\). We have \(B|_{\rho=1}\overset{(11)-(13)}{=} Q(s(\theta))\overset{(11)}{=}N(P_{s(\theta)})-s(\theta)= N(P_{s(\theta)})-\theta\), and so \(B(\rho,\theta)|_{\rho=1} =\arg(dz/d\zeta)|_{\zeta=e^{i\theta}}\). Since \(e^A|_{\rho=1}\overset{(13)}{=}1\) and in view the geometric meaning of \(\arg(dz/d\zeta)\), we see that \(\zeta=\rho e^{i\theta}\mapsto z(\zeta)\) is an isometry of \(V_{{\mathbb{T}}}\) onto \(V_{\Gamma}\).
For example, as in Lemma 1.
See, for example, https://mathworld.wolfram.com/ToroidalCoordinates.html.
References
A. S. Demidov, “Functional geometric method for solving free boundary problems for harmonic functions”, Uspekhi Mat. Nauk, 65:1(391) (2010), 3–96; English transl.: Russian Math. Surveys, 65:1 (2010), 1–94.
V. I. Arnol’d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, 250 Springer-Verlag, New York, 1983.
A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, CRC Press, Boca Raton, FL, 2012.
P. G. Grinevich and R. G. Novikov, “Moutard transforms for the conductivity equation”, Lett. Math. Phys., 109:10 (2019), 2209–2222.
Funding
This work was supported in part by RFBR grant no. 20-01-00469.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 65–72 https://doi.org/10.4213/faa3823.
Translated by A. R. Alimov
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Demidov, A.S. On Numerically Implementable Explicit Formulas for the Solutions to the 2D and 3D Equations \(\operatorname{div}(\alpha(w)\nabla w)=0\) and \(\operatorname{div}(\beta\nabla w)=0\) with Cauchy Data on an Analytic Boundary. Funct Anal Its Appl 55, 52–58 (2021). https://doi.org/10.1134/S0016266321010068
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DOI: https://doi.org/10.1134/S0016266321010068