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Brief communications On explicit numerically realizable formulae for Poincaré–Steklov operators A. S. Demidov Lomonosov Moscow State University
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 6(474), DOI: https://doi.org/10.4213/rm10118 
Bibliographic databases:
    1.Let $u$ be a solution of the Dirichlet problem for the Laplace equation $\Delta u=0$ in a simply connected domain $\Omega\subset\mathbb{R}^2$ with $C^1$-smooth boundary $\Gamma$ of length $2\pi$. To be more precise: $\partial^2 u/\partial x^2+\partial^2 u/\partial y^2=0$ in $\Omega$, $u(P_s)=F(s)$, $P_s\in \Gamma$, where $s\in \mathbb{T}\overset{\rm def}{=}{\mathbb{R}}/{2\pi}$ is the natural parameter on $\Gamma$ (the arc length $\smile\! P_0P_s$ on the curve $\Gamma$ measured in the positive direction from some point $P_0$ to the point $P_s$), and the function $F\in C^1(\mathbb{T})$ is given by the Fourier series
$$
\begin{equation}
F\colon\mathbb{T}\ni s\mapsto \sum_{k\geqslant 0}(a_k\cos ks+b_k\sin ks).
\end{equation}
\tag{1}
$$
Let us begin with numerically realizable formulae for the Dirichlet–Robin operator (a particular case of the Poincaré–Steklov operator [1]) given by
$$
\begin{equation}
\mathcal D\mathcal R(F)\overset{\rm def}{=}\biggl(\alpha u+ \beta\,\frac{\partial u}{\partial\nu}\biggr)\bigg|_{P_s\in\Gamma}\,,\qquad |\alpha|+|\beta|\ne 0,
\end{equation}
\tag{2}
$$
where $\nu$ is the unit outward normal vector to the curve $\Gamma$. If the domain $\Omega_\varepsilon$ differs from $\Omega$ only in that its boundary $\Gamma_\varepsilon=\partial\Omega_\varepsilon$ is analytic and its Hausdorff $C^1$-distance differs from that of $\Gamma$ by a quantity of order $\varepsilon>0$, then the operators $\mathcal D\mathcal R$ for $\Omega_\varepsilon$ and $\Omega$ differ in the $C^1$-metric by the same order. So we can assume that the curve $\Gamma$ is analytic. Under this assumption explicit numerically realizable formulae for the univalent mapping
$$
\begin{equation*}
z\colon V_{\mathbb{T}}\ni\zeta=\rho e^{i\theta}\mapsto z(\zeta)= x(\rho,\theta)+iy(\rho,\theta)\in V_{\Gamma}
\end{equation*}
\notag
$$
of a neighbourhood $V_{\mathbb{T}}$ of the unit circle $\mathbb{T}=\{\rho=1,\,\theta\in\mathbb{R}/2\pi\}$ to a neighbourhood $V_{\Gamma}\ni \zeta=\rho e^{i\theta}$ of the curve $\Gamma\subset\mathbb{R}^2\simeq\mathbb C\ni z=x+iy$ that is an isometry on $\mathbb T$ are presented in [2]. The isometry property
$$
\begin{equation}
\biggl|\frac{dz(\zeta)}{d\zeta}\biggr|\equiv 1\quad \text{for}\ \ \zeta\in\mathbb{T}
\end{equation}
\tag{3}
$$
plays a key role in the construction of formulae for the operator (2). Consider a curve $\gamma=z(\mathcal C)\subset V_{\Gamma}$, where $\mathcal C=\{0<|\zeta|=r\}$, and $r$ is small. Let $g$ be the trace on $\gamma$ of the solution $u$ of the original Dirichlet problem. We set $f\colon\mathbb{T}\ni\theta\mapsto f(\theta)= g(z)\big|_{z=z(re^{i\theta})\in\gamma}$. The Fourier series of this periodic function has the form
$$
\begin{equation}
f\colon\mathbb{T}\ni \theta\mapsto \sum_{k\geqslant 0}(c_k\cos k\theta+d_k\sin k\theta).
\end{equation}
\tag{4}
$$
Taking (1) and (4) into account we set $\lambda_0=c_0$, and for $k\geqslant 1$ we define
$$
\begin{equation*}
\begin{gathered} \, \lambda_k=a_k-\frac{c_k r^k-a_k r^{2k}}{1-r^{2k}}\,,\quad \mu_k=b_k-\frac{d_k r^k-b_k r^{2k}}{1-r^{2k}}\,, \\ \varphi_k=\frac{c_k-a_kr^k}{1-r^{2k}}\,, \quad\text{and}\quad \psi_k=\frac{d_k-b_kr^k}{1-r^{2k}}\,. \end{gathered}
\end{equation*}
\notag
$$
Hence the function
$$
\begin{equation}
\begin{aligned} \, \zeta&=\rho e^{i\theta}\mapsto U(\rho,\theta) \nonumber\\ &=\lambda_0+\sum_{k\geqslant 1}\biggl\{\rho^k[\lambda_k\cos k\theta+\mu_k\sin k\theta] +\biggl(\frac{r}{\rho}\biggr)^k[\varphi_k\cos k\theta+\psi_k\sin k\theta]\biggr\}, \end{aligned}
\end{equation}
\tag{5}
$$
which is harmonic in the annulus $V_{\mathbb{T}}=\{r<\rho<1;\ \theta\in\mathbb{T}\}$, satisfies the boundary conditions $U(1,\theta)=F(\theta)$ and $U(r,\theta)=f(\theta)$. Theorem 1. For the Dirichlet–Neumann operator $\mathcal D{\mathcal N}\bigl(u\big|_\Gamma\bigr)\overset{\rm def}{=} (\partial u/\partial\nu)\big|_\Gamma$, Now we present numerically realizable formulae for $u\big|_\gamma$, where $u$ is the solution of the original Dirichlet problem. According to [3],
$$
\begin{equation}
u(x,y)=\int_0^{2\pi}\!\mu(t)K(x(t)-x,y(t)-y)\,dt,\quad\text{where}\quad K(\xi(t),\eta(t))=\frac{\eta'(t)\xi-\eta\xi'}{\xi^2+\eta^2}\,,
\end{equation}
\tag{7}
$$
that is, $K(\xi(t),\eta(t))=\dfrac{d}{dt}\arctan\dfrac{\eta(t)}{\xi(t)}$ , and $\mu$ is the solution of the equation
$$
\begin{equation}
\mu(\tau)+\frac{1}{\pi}\int_0^{2\pi}\!\mu(t)\,\frac{d}{dt}\arctan \frac{y(t)-y(\tau)}{x(t)-x(\tau)}\,dt=\frac{1}{\pi}F(\tau)\,.
\end{equation}
\tag{8}
$$
Approximating the integral in (8) by a sum (for details, see [4]) and setting, in succession, $\tau=t_1,\dots,t_n$, we obtain a system of linear algebraic equations, from which we find an approximation to $\mu$, and, correspondingly, an approximation to the solution of the original Dirichlet problem.
2.A numerical realization of the Robin1–Robin2 operator, that is, the operator
$$
\begin{equation*}
\biggl(\alpha_1u+\beta_1\frac{\partial u}{\partial\nu}\biggr)\bigg|_{\Gamma} \mapsto\biggl(\alpha_2u+ \beta_2\frac{\partial u}{\partial\nu}\biggr)\bigg|_{\Gamma}\,,\qquad |\alpha_k|+|\beta_k|\ne 0,
\end{equation*}
\notag
$$
can be found from that for the above Dirichlet–Robin2 problem by means the construction (see, for example, [5]) of the solution of the $\gamma$-Robin1–Dirichlet problem $(\alpha_1u+\beta_1\,\partial u/\partial\nu)\big|_{\Gamma}\mapsto u\big|_{\gamma}$.
3.Another particular case of Poincaré–Steklov operators is the operator $\mathcal G_{\mathcal M}\mathcal R$, where the Grinberg–Mayergoiz operator $ \mathcal G_{\mathcal M}$ [6], [7] is defined for $-1\leqslant\sigma<1$ by the conditions
$$
\begin{equation}
(1-\sigma)u(P_s^-)-(1+\sigma)u(P_s^+)=2F(P_s)\quad\text{and}\quad \frac{\partial u}{\partial \nu}(P_s^-)= \frac{\partial u}{\partial\nu}(P_s^+).
\end{equation}
\tag{9}
$$
Here the value at the point $P_s^+$ (or at $P_s^-$) is defined as the limit value at the point $P_s\in \Gamma$ along the outward (inward) normal vector $\nu$ to the curve $\Gamma$. Explicit formulae for the traces $u\big|_{\gamma}$ and $u\big|_{\Gamma}$ of the solution of problem (9) have been indicated above for the Dirichlet problem, that is, for $\sigma=-1$; for $|\sigma|<1$ formulae of the same type in terms of potentials with density satisfying an integral equation of the second kind can be found in [7]. Proceeding as in Theorem 1 we obtain numerically realizable formulae for $\mathcal G_{\mathcal M}\mathcal R(F)$.
Citation in format AMSBIB
\Bibitem{Dem23} |
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