UDC 517.944

MATHEMATICS

A. I. PRILEPKO

AN EXTERNAL INVERSE PROBLEM FOR THE VOLUME POTENTIAL OF VARIABLE DENSITY FOR A BODY CLOSE TO A GIVEN ONE

(Presented by Academician M. A. Lavrent’ev, 28 VI 1968)

1°. In this paper a solution is given of the inverse problem for the metaharmonic potential in the following formulation \(( (^{1,7});\) see also \((^{1,2,6-8})\), where the question of uniqueness and stability of the external problem was investigated).

One seeks a body \(T_1\) such that its external metaharmonic potential \((\chi \geqslant 0)\) of a given variable density \(\mu\) is equal outside the body \(T_1\) to a given metaharmonic function \(H\) \((H\) is a regular solution of the equation \(\Delta H - \chi^2 H = 0)\), \(H\) decreases at infinity as a metaharmonic potential and is close, in the sense of a certain functional metric, to the external metaharmonic potential \(V\) of the given body \(T\) of density \(\mu\). It is assumed that the boundary \(S\) of the body \(T\) belongs to the class \(A^{(2,\lambda)}\).

In the present article, for the metaharmonic potential \((\chi \geqslant 0)\), the existence and uniqueness of the solution of the indicated problem are proved. For the Newtonian potential \((\chi = 0)\), in the case of constant density \(\mu = 1\), a problem of this kind for stellar bodies was solved by V. K. Ivanov \((^{1})\), and under the assumption that the body \(T\) is a sphere, a similar problem was studied by L. N. Sretenskii \((^{8})\).

We note that the main result obtained in this paper is new also for the Newtonian potential \((\chi = 0)\): first, a general class of variable densities is considered; second, there are no restrictions of the “stellar” type on the body \(T\). This remark also applies to the metaharmonic potential \((\chi \geqslant 0)\), for which a similar problem was studied in the case of a constant density of a stellar body \(T\) in the author’s work \((^{7})\).

2°. By virtue of the smoothness conditions on the surface \(S\), for sufficiently small \(v\) \((|v| \leqslant 3\varepsilon_0,\ \varepsilon_0 > 0)\) every point \(y = (y_1, y_2, y_3)\) of the three-dimensional space \(E^3\), by the formula

\[ \mathbf{y}=\mathbf{x}+v\mathbf{n}_x \tag{1} \]

can be defined in a unique way in a neighborhood of the surface \(S\) with the aid of three curvilinear coordinates \((\xi,\eta,v)\), where \((\xi,\eta,0)\) are the curvilinear coordinates of the point \(\mathbf{x}\) of the surface \(S\); \(\mathbf{n}_x\) is the unit vector of the exterior normal to the surface \(S\) at the point \(\mathbf{x}\). Denote the metaharmonic potential \((\chi \geqslant 0)\) of a set \(A,\ A \subset E^3\), with density \(\mu(y) \neq 0\) almost everywhere for \(y \in A\), by

\[ V(x; A,\mu)=\int_A \mu(y)\frac{e^{-\chi r_{xy}}}{r_{xy}}\,dy, \]

where \(r_{xy}=|y-x|\) is the distance between the points \(y\) and \(x\). Consider a bounded simply connected domain \(T\), bounded by a surface \(S\) of class \(A^{(2,\lambda)}\). Let the external metaharmonic potential \(V(x;T,\mu)\) of the body \(T\) of density \(\mu\) be known. In addition, suppose that outside a domain \(T_0\), lying inside \(T\) at a positive distance \(d\) from the boundary \(S\), a metaharmonic function \(H\) is given which at infinity behaves like a metaharmonic potential.

In addition we shall assume that:

1) \(\mu(y)\) is a given real analytic function in the domain \(D'\) \((D' \supset T+S\), the boundary of \(D'\) is at a positive distance, greater than \(3\varepsilon_0\), from the surface \(S\)) and \(\mu(y)\) is nowhere equal to zero on the surface \(S\).

2) Each of the quantities
\[ \left\|\frac{\partial H}{\partial \nu}-\frac{\partial V(T,\mu)}{\partial \nu}\right\|,\qquad \left\|\frac{\partial^2 H}{\partial \nu^2}-\frac{\partial^2 V(T,\mu)}{\partial \nu^2}\right\| \]
does not exceed \(\omega C\), where \(0<\omega<d\), \(C=C(T)\), \(\omega=\omega(T,\mu,\varepsilon_0,d)\), and the norm \(\|\cdot\|\) of the limiting, from outside \(T\), values of
\[ \frac{\partial}{\partial \nu}(H-V),\qquad \frac{\partial^2}{\partial \nu^2}(H-V) \]
is equivalent to the norm in the space \(C^{(1,\lambda)}(s)\).

Let \(\{S_1\}\) be the class of surfaces whose equation in the curvilinear system of coordinates (1) has the form
\[ \{v=\zeta(\xi,\eta)\},\qquad |v|\leqslant \varepsilon_0,\qquad \zeta\in C^{(1,\lambda)}. \]

Under these conditions, for the body \(T\), the surfaces \(S\) and \(S_1\), and the functions \(V\) and \(H\), the following holds.

Theorem. There exists, and moreover is unique, a surface \(S_1\) bounding a body \(T_1\), satisfying the condition \(\|\xi\|<d\), such that the external metaharmonic potential \(V(x;T_1,\mu)\) of the body \(T_1\) of the given density \(\mu\) is equal to the given metaharmonic function \(H\) in the domain exterior to the surface \(S_1\), i.e.
\[ H(x)=V(x;T_1,\mu)\quad \text{for } x\in E^3\setminus \overline{T}_1 . \]

\(3^0\). The derivation of the nonlinear integro-differential equation determining the boundary of the sought body \(T_1\) is based on the ideas of V. K. Ivanov \((^1)\), L. Lichtenstein \((^3)\), and A. M. Lyapunov \((^4)\). It should be noted that the arguments used in the theory of nonhomogeneous equilibrium figures \((^{3,4})\), for our problem, lead to an integral equation of the first kind, which cannot be investigated even in the case \(\mu=1\), as is noted in \((^1)\). Therefore, following the method of V. K. Ivanov \((^1)\), we use the normal derivative of the potential, although for our case, owing to the variable density, the derivation of the principal nonlinear equation is carried out somewhat differently than in \((^1)\).

Denote by \(D\) the domain bounded by the surface \(S_{2\varepsilon_0}\), which in the curvilinear system of coordinates (1) is defined by the equation
\[ v=2\varepsilon_0. \]

The derivation of the equation and its investigation are based on the following

Lemma. Let \(T\) be a simply connected domain bounded by a surface \(S\in A^{(2,\lambda)}\); let \(\hat T\) be a simply connected domain bounded by a surface \(\hat S\), defined in the curvilinear system of coordinates (1) by the equation
\[ v=\zeta(\xi,\eta),\qquad |v|\leqslant \varepsilon_0; \]
\(L\) be the domain with boundary \(S_{2\varepsilon_0}\), \(D\) contain \(T+S\) and \(\hat T+\hat S\). There exist positive numbers \(\Omega\) and \(K\), depending only on the shape of \(T\), such that for \(\|\xi\|\leqslant\Omega\) one can construct a continuous one-to-one mapping of the form
\[ \hat y_k=y_k+\alpha_k(y_1,y_2,y_3)\qquad (k=1,2,3) \]
of the domain \(D\) onto some domain \(\hat D\), which maps \(T+S\) onto \(\hat T+\hat S\), and moreover the Jacobian \(\hat J(y)\) of the mapping
\[ \hat J(y)=\partial(\hat y_1,\hat y_2,\hat y_3)/\partial(y_1,y_2,y_3)\ne 0\qquad \text{for } y\in D; \]
the functions \(\alpha_k(y)\in C^{(1,\lambda)}(D)\) \((k=1,2,3)\) and satisfy the condition
\[ |\alpha_k|,\ |\partial\alpha_k/\partial y_i|,\ |\partial\alpha_k/\partial y_i|_\lambda\leqslant \Pi, \]
where \(\Pi\) is a positive number that may be taken equal to
\[ K\|\xi\|=\Pi, \]

and, moreover, for the functions \(a_k\) the conditions are satisfied

\[ a_k\big|_{\nu=0}=a_k(x)=\zeta(x)a_k(x),\qquad x\in S;\quad \mathbf n_x=\{a_1,a_2,a_3\};\qquad \partial a_k/\partial \nu\big|_{\nu=0}=0. \]

\(4^\circ\). Introduce the function

\[ F_t^\varepsilon=\int_{T_t'}\mu(y_t)\frac{\partial}{\partial \nu} \left(\frac{e^{-\chi |y_t-z|}}{|y_t-z|}\right)\,dy_t \bigg|_{\nu=t\zeta+\varepsilon}, \]

where \(0<\varepsilon\le \varepsilon_0\), the point \(z\) has curvilinear coordinates \((\xi,\eta,\nu)\) \(( (\xi,\eta,0)\) are the curvilinear coordinates of the point \(x\in S)\); \(T_t\) is the domain bounded by the surface \(S_t\), depending on the parameter \(t\), \(S_t\) being defined by the equation

\[ \nu=t\zeta(\xi,\eta),\qquad 0\le t\le 1. \]

Expanding the function \(F_t^\varepsilon\) in a series in powers of \(t\), putting \(t=1\), and letting \(\varepsilon\) tend to zero, we obtain for the function \(\zeta(\xi,\eta)\), which determines the boundary of the sought body \(T_1\), the nonlinear integro-differential equation

\[ A(\zeta\mu)= \frac{\partial}{\partial \nu}V(T,\mu)\bigg|_{\nu=0} +\zeta\frac{\partial^2 V(T,\mu)}{\partial \nu^2}\bigg|_{\nu=0} -\frac{\partial H}{\partial \nu}\bigg|_{\nu=\zeta} +\Psi(\zeta), \tag{2} \]

where \(\dfrac{\partial}{\partial \nu}V(T,\mu)\), \(\dfrac{\partial^2 V(T,\mu)}{\partial \nu^2}\) are the exterior limiting values on the surface \(S\) of the metaharmonic potential \(V(T,\mu)=V(x;T,\mu)\) of the body \(T\) with density \(\mu\),

\[ A(\zeta\mu)=2\pi \zeta(x)\mu(x) -\int_S \frac{\partial}{\partial n_x}\frac{e^{-\chi r_{xy}}}{r_{xy}}\, \mu(y)\zeta(y)\,dS_y,\qquad x\in S, \]

\[ \Psi(\zeta)=\sum_{n=2}^{\infty} \left[ \lim_{\varepsilon\to 0}\frac{1}{n!}\frac{d^n}{dt^n}F_t^\varepsilon \bigg|_{t=0} \right]. \]

We note that every sufficiently small solution of equation (2) is a solution of the problem.

For the operator \(\Psi(\zeta)\) estimates hold that are similar to those proved in theorem 3 of the author’s paper \((7a)\) (see also \((1, 7b)\)). In view of the indicated estimates, under the conditions of the main theorem the existence and uniqueness of a solution of equation (1) are proved, and this solution can be found by the method of successive approximations.

Remark. The posed problem has, moreover, a unique solution under the assumption that the body \(T\) consists of a finite number of simply connected finite domains \(T_1,\ldots,T_m\) with boundaries \(S_1,\ldots,S_m\) of class \(A^{(2,\lambda)}\) (with \(T_k+S_k\ne T_j+S_j,\ j\ne k\)); by \(S\), the boundary of the body \(T\), we denote the union of the boundaries \(S_1,\ldots,S_m\). Instead of the domain \(T_0\), one considers a body \(T^0\) consisting of simply connected domains \(T_1^0,\ldots,T_m^0\), lying respectively inside \(T_1,\ldots,T_m\) at a positive distance \(d\) from \(S_1,\ldots,S_m\). The function \(H\) is a regular solution of the metaharmonic equation in the domain \(E^3\setminus \overline{T}\) with the corresponding decrease at infinity. It is assumed that \(\mu(y)\) is the collection of given functions \(\mu_k\), analytic in the domains \(D_k'=T_k+S_k\) \((k=1,2,\ldots,m)\), each function \(\mu_k(y)\) being nowhere zero on the surface \(S_k\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
31 V 1968

CITED LITERATURE

\({}^{1}\) V. K. Ivanov, Izv. AN SSSR, ser. matem., 20, 793 (1956).
\({}^{2}\) M. M. Lavrent’ev, On Some Incorrect Problems of Mathematical Physics, 1962.
\({}^{3}\) L. Lichtenstein, Figures of Equilibrium of a Rotating Fluid, 1966.
\({}^{4}\) A. M. Lyapunov, Collected Works, 5, 1965.
\({}^{5}\) I. P. Nedyalkov, Izv. Geofiz. inst., 1, 75 (1960); DAN, 144, No. 4 (1962).
\({}^{6}\) P. S. Novikov, DAN, 28, No. 3, 465 (1938).
\({}^{7}\) A. I. Prilepko, a) DAN, 154, No. 3, 534 (1964); b) Sibirsk. matem. zhurn., 6, No. 6, 1332 (1965).
\({}^{8}\) L. N. Sretenskii, Izv. AN SSSR, ser. matem., 2, 551 (1938).
\({}^{9}\) A. N. Tikhonov, DAN, 39, 195 (1943).