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A. I. PRILEPKO
ON AN INVERSE PROBLEM OF THE METAHARMONIC POTENTIAL
(Presented by Academician M. A. Lavrent'ev on VIII 19, 1963)
1°. In the present article a solution is given of the inverse problem of the metaharmonic potential in the following formulation.
One seeks a body \(T_1\), if its exterior metaharmonic potential \(V_1\) is known, close in the sense of a certain functional metric to the exterior metaharmonic potential \(V\) of a given body \(T\). It is assumed that the body \(T\) is star-shaped with respect to some interior point, and that the boundary \(S\) of the body \(T\) is such that the functions of its parametric representation are twice differentiable and their second derivatives satisfy the Hölder condition with exponent \(\lambda < 1\) (a surface of class \(A^{(2,\lambda)}\)). For the Newtonian potential, a problem of this kind was completely solved by V. K. Ivanov \((^{1})\) (under the assumption that the body \(T\) is a sphere, a similar problem was studied earlier by L. N. Sretenskii \((^{2})\)).
2°. Let \(x\) be a point of the surface \(S\) with radius vector
\[ \mathbf{R}(x)=\mathbf{R}_x(\xi,\eta). \]
Define the point \(y \in E^3\) by the radius vector
\[ \mathbf{R}_y=\mathbf{R}_x+\nu \mathbf{n}_x, \]
where \(\mathbf{n}_x\) is the exterior normal to the surface \(S\) at the point \(x\); \(\nu\) is a given number. The triple \((\xi,\eta,\nu)\) may be regarded as curvilinear coordinates of the point \(y\). It is known that, for sufficiently small \(\nu\) (\(|\nu| \leqslant \varepsilon_0\)), distinct points \(y\) correspond to distinct triples of numbers, and conversely.
Denote by \(R_2\) the set of functions
\[ \zeta(x)\equiv \zeta(\xi,\eta), \]
defined on the surface \(S\) and belonging to the class \(C^{(1,\lambda)}\) (the first derivatives of the function \(\zeta(x)\) satisfy the Hölder condition with exponent \(0<\lambda<1\)). On the set \(R_2\) introduce a norm, taking it equal to the largest of the numbers
\[ 4\max |\zeta(x)|,\quad 4\max |\zeta_\xi(x)|,\quad 4\max |\zeta_\eta(x)|, \]
\[ 4\sup \frac{|\zeta_\xi(y)-\zeta_\xi(x)|}{|y-x|^\lambda},\quad 4\sup \frac{|\zeta_\eta(y)-\zeta_\eta(x)|}{|y-x|^\lambda}. \]
The space with this norm is a Banach ring.
The function
\[ V(x)=\int_T \frac{e^{-\varkappa r_{xy}}}{r_{xy}}\,dy \]
we shall call the metaharmonic potential of the body \(T\) of unit density, where \(\varkappa=\mathrm{const}>0\), \(r_{xy}=|y-x|\) is the distance between the points \(y\) and \(x\). If the surface \(S\in A^{(2,\lambda)}\), then the boundary values outside of \(\partial V/\partial \nu\), \(\partial^2 V/\partial \nu^2\) belong to the space \(R_2\).
Consider a body \(T\), bounded by a surface \(S\in A^{(2,\lambda)}\) and star-shaped with respect to some interior point. Suppose the metahar-
metaharmonic potential \(V(x)\) of the body \(T\) of unit density. Moreover, suppose that in the domain exterior to the surface \(S\) there is defined a metaharmonic function \(V_1\) (i.e., a regular solution of the equation \(\Delta V_1-\varkappa^2 V_1=0\)), which at infinity behaves like a metaharmonic potential.
We shall additionally assume that:
1) \(V_1\) admits metaharmonic continuation through \(S\) into the body \(T\) for some positive distance \(d\);
2) each of the quantities
\[ \left\|V_1-V\right\|,\quad \left\|\frac{\partial V_1}{\partial \nu}-\frac{\partial V}{\partial \nu}\right\|,\quad \left\|\frac{\partial^2 V_1}{\partial \nu^2}-\frac{\partial^2 V}{\partial \nu^2}\right\| \]
does not exceed \(\omega C\), where \(C=C(T)\), \(0<\omega<d\), \(\omega=\omega(T,V,V_1,\varepsilon_0)\).
Let \(\{S_1\}\) be a family of surfaces whose equation in the curvilinear coordinate system is given in the form
\[ \{\nu=\zeta(x)\},\qquad |\nu|\leqslant \varepsilon_0,\qquad \zeta\in C^{(1,\lambda)} . \]
Under these conditions on the body \(T\), the surfaces \(S\) and \(S_1\), and the functions \(V\) and \(V_1\), the following holds.
Theorem (main). There exists, and moreover is unique, a surface \(S_1\), bounding a body \(T_1\), satisfying the condition \(\|\zeta\|<d\), such that the exterior metaharmonic potential of the body \(T_1\) of unit density is equal to the prescribed metaharmonic function \(V_1\) in the domain exterior to the surface \(S_1\).
3°. The function determining the boundary of the sought body is a solution of a nonlinear integro-differential equation. The derivation of this equation is based on ideas of V. K. Ivanov \((^1)\) and L. Lichtenstein \((^{3,4})\).
Let the equation of the sought surface \(S_1\) in curvilinear coordinates have the form
\[ \nu=\zeta(\xi,\eta). \]
Introduce between \(S\) and \(S_1\) a one-parameter family of surfaces \(S_t\), defined by the equation
\[ \nu=t\zeta(\xi,\eta)\qquad (0\leq t\leq 1). \]
Denote by \(T_t\) the domain bounded by the surface \(S_t\), and introduce the points \(x(\xi,\eta,0)\), \(x_t(\xi,\eta,t\zeta)\), \(z_t(\xi,\eta,t\zeta+\varepsilon)\) (\(\varepsilon>0\), curvilinear coordinates).
Let \(\mathbf r_t=\mathbf R(y)-\mathbf R(x_t)\), where \(\mathbf R(x_t)=\mathbf R(x)+t\zeta\mathbf n_x\). In these notations, the metaharmonic potential of the body \(T_t\) at the point \(z_t\) of unit density is written in the form
\[ V_t^\varepsilon(z_t)=\int_{T_t^\varepsilon} \frac{e^{-\varkappa|r_t-\varepsilon n_x|}}{|r_t-\varepsilon n_x|}\,dy . \]
Theorem 1. The solution \(\zeta(x)\) of the stated problem, under the hypotheses of the main theorem, satisfies the nonlinear integro-differential equation
\[ 2\pi\zeta(x)-\int_S \frac{\partial}{\partial n_x}\left(\frac{e^{-\varkappa r_{xy}}}{r_{xy}}\right) \zeta(y)\,d_y\sigma = f+g\zeta+\Phi(\zeta)+\Psi(\zeta), \tag{1} \]
where
\[ f=-\left.\frac{\partial}{\partial \nu}(V_1-V)\right|_{\nu=0}, \qquad g=-\left.\frac{\partial^2}{\partial \nu^2}(V_1-V)\right|_{\nu=0}, \]
\[ \Phi(\zeta)= \left.\frac{\partial V_1}{\partial \nu}\right|_{\nu=0} + \left.\zeta\frac{\partial^2 V}{\partial \nu^2}\right|_{\nu=0} - \left.\frac{\partial V_1}{\partial \nu}\right|_{\nu=\zeta}, \]
\[ \Psi(\zeta)=\sum_{n=2}^{\infty} Z_n, \]
where
\[ Z_n=\lim_{\varepsilon\to0}\frac1{n!}\left\{\left[\frac{\partial^n}{\partial t^n}\left(\frac{\partial V_t^\xi}{\partial \nu}\right)\right]_{t=0}\right\},\qquad n=2,3,\ldots, \]
is an integro-power form with respect to \(\xi,\xi_\xi,\xi_\eta\).
We shall now prove the converse: every sufficiently small solution of equation (1) is a solution of the problem.
Let \(\xi\) be a function prescribed on the surface \(S\) and satisfying equation (1). Lay off the values \(\xi\) along the outward normal at each point \(x\) of the surface \(S\), and denote by \(\hat S_1\) the surface thus obtained; denote by \(\hat T_1\) the body bounded by this surface, and by \(\hat V_1\) its metaharmonic potential of unit density.
Theorem 2. If \(\xi\) is a sufficiently small solution of equation (1), then the equality
\[ \hat V_1 \equiv V_1 \]
holds, i.e. the body \(T_1\), bounded by the surface \(\nu=\xi(\xi,\eta)\), has \(V_1\) as its exterior potential.
4°. The solution of equation (1) and the investigation of the properties of the operator \(\Psi(\xi)\) have been carried out by us on the basis of the method of V. K. Ivanov \((^1)\).
To obtain the properties of the operator \(\Psi(\xi)\), estimates are derived and used for the boundary values of metaharmonic potentials and their derivatives. In doing so, the theory of differentiable mappings developed by L. Lichtenstein \((^{3,4})\), by means of introducing a complex parameter, is used.
For the operator \(\Psi(\xi)\) the following is valid.
Theorem 3. If the function \(\xi\in R_2\), defined on the surface \(S\), has a sufficiently small norm, then \(\Psi(\xi)\) belongs to the space \(R_2\), and for any \(\xi,\hat \xi\) from \(R_2\) satisfying the conditions
\[ \|\xi\|\leq \omega,\qquad \|\hat \xi\|\leq \omega, \]
the inequalities
\[ \|\Psi(\xi)\|\leq a_1\omega^2, \]
\[ \|\Psi(\xi)-\Psi(\hat \xi)\|\leq b_1\omega\|\xi-\hat \xi\| \]
hold, where \(a_1,b_1,d\) are positive constants, \(0<\omega\leq d\).
Write equation (1) in the form
\[ A\xi=f+g\xi+P(\xi), \tag{2} \]
where
\[ A\xi=2\pi\xi(x)-\int_S \frac{\partial}{\partial n_x}\left(\frac{e^{-\varkappa r_{xy}}}{r_{xy}}\right)\xi(y)\,d_y\sigma,\qquad P(\xi)=\Phi(\xi)+\Psi(\xi). \]
If \(\xi\in R_2\) and the norm of \(\xi\) is sufficiently small, then all terms in (2) belong to \(R_2\), and the operator \(A\) has in \(R_2\) a continuous inverse \(A^{-1}\). Consequently, (2) can be replaced by the equivalent equation
\[ \xi=A^{-1}f+A^{-1}g\xi+A^{-1}P(\xi). \tag{3} \]
Let \(\omega\) be a positive number satisfying the inequalities
\[ \omega\leq d,\qquad \omega(1+b\|A^{-1}\|)<1,\qquad \omega^2-2(1+2a\|A^{-1}\|)\omega+1>0, \]
where \(a,b,d\) are positive constants depending on the form of the domain \(T\).
Theorem 4. If the norms of the functions \(f\) and \(g\) satisfy the estimates
\[ \|f\|\leq \frac{\omega}{\|A^{-1}\|},\qquad \|g\|\leq \frac{\omega}{\|A^{-1}\|}, \]
then equation (3) has a unique solution satisfying the condition \(\|\xi\|<d\), and this solution can be found by the method of successive-
successive approximations
\[ \xi_1 = A^{-1} f, \]
\[ \xi_{n+1} = A^{-1} f + A^{-1} g \xi_n + A^{-1} P(\xi_n), \qquad n = 1, 2, \ldots \]
Theorems 3 and 4, together with Theorems 1 and 2, lead to the conclusion that was formulated in the form of the main theorem.
In conclusion, I express my sincere gratitude to A. V. Bitsadze, under whose supervision this work was carried out.
Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR
Received
13 VIII 1963
References
- V. K. Ivanov, Izv. AN SSSR, Ser. Mat., 20, 793 (1956).
- L. N. Sretenskii, Izv. AN SSSR, Ser. Mat., 2, 551 (1938).
- L. Lichtenstein, Gleichgewichtsfiguren der rotierenden Flüssigkeiten, Berlin, 1933.
- L. Lichtenstein, Vorlesungen über einige Klassen nichtlinearer Integralgleichungen, Berlin, 1931.