Abstract
Full Text
UDC 517.944
MATHEMATICS
A. I. PRILEPKO
CONTACT INVERSE PROBLEMS OF GENERALIZED MAGNETIC POTENTIALS
(Presented by Academician M. A. Lavrent’ev on 2 XII 1967)
We consider the problem of determining the shape of a contact body from the values of the external generalized magnetic potential, which is represented as the sum of the potential of volume masses and of a simple layer. This problem is a typical example of ill-posed problems, methods for solving which have been developed in works \((^{1,3,5,10,11})\) and others. Thus, in \((^{11})\), using the example of the logarithmic potential of volume masses of constant density, an effective method was proposed for solving the external contact problem by reducing this problem to an equivalent nonlinear equation of the first kind. The question of uniqueness of the solution of the external contact problem was studied in \((^{1,6,9,12})\) for harmonic potentials of constant densities under a number of restrictions on the boundary of the bodies. In the author’s paper \((^{7})\), the uniqueness of the solution of the indicated problem was studied for harmonic and metaharmonic potentials of volume masses of variable densities.
In the present article the uniqueness of the solution of the indicated problem is studied for generalized magnetic potentials of a general linear elliptic operator of the second order with variable coefficients, and arbitrary contact bodies with variable densities are considered. We note that in the case of the Laplace operator the results obtained are new also for harmonic potentials \((n \geqslant 2)\).
\(1^\circ\). Consider the uniformly elliptic operator
\[ Lu=\sum_{i,k=1}^{n} a_{ik}\frac{\partial^{2}u}{\partial x_i \partial x_k} +\sum_{k=1}^{n} b_k \frac{\partial u}{\partial x_k}+cu, \tag{1} \]
where
\[ a_{ik}=a_{ki}\in C^{(2,\lambda)}(D_0'),\qquad b_k\in C^{(1,\lambda)}(D_0'),\qquad c\in C^{(0,\lambda)}(D_0')\quad (0<\lambda<1) \]
are functions of the point \(x=(x_1,\ldots,x_n)\in D_0'\); \(D_0'\) is a bounded domain of Euclidean space \(E^n\) \((n\geqslant 2)\). Suppose, moreover, that the coefficients of the operator (1) satisfy the condition
\[ c(x)\leqslant 0,\qquad c^*(x)\leqslant 0\quad \text{for } x\in D_0', \tag{2} \]
where \(c^*\) is the coefficient of the function \(v\) in the operator \(L^*v\) \((L^*\) is the operator adjoint to \(L\) in the sense of Lagrange), i.e.
\[ c^*(x)=c-\sum_{k=1}^{n}\frac{\partial e_k}{\partial x_k},\qquad e_k=b_k-\sum_{i=1}^{n}\frac{\partial a_{ki}}{\partial x_i}. \]
Let \(A_\alpha\) \((\alpha=1,2)\) be open sets with boundaries \(\Gamma_\alpha\); \(A_\alpha\) is the union of a finite number of domains \(T_\alpha^j\) with boundaries \(S_\alpha^j\) of class \(A^{(1,\lambda)}\) (see \((^{7})\)); \(\overline{A}_\alpha\subset D_0\), \(D_0\) is an arbitrary domain, \(\overline{D}_0\subset D_0'\). We introduce generalized potentials of volume masses and generalized potentials of sim-
of the simple layer
\[ U^\alpha(x)=U(x; A_\alpha,z_\alpha)=\int_{A_\alpha}\Omega(x,y)z_\alpha(y)\,dy; \tag{3} \]
\[ V^\alpha(x)=V(x;\Gamma_\alpha,\xi_\alpha)=\int_{\Gamma_\alpha}\Omega(x,y)\xi_\alpha(y)\,d_yS, \tag{4} \]
where \(\Omega(x,y)\) is the principal elementary solution (see (2)); moreover, if \(z_\alpha\in C^{(0,\lambda)}(\bar A_\alpha)\), then
\[ L[U^\alpha(x)]= \begin{cases} -z_\alpha(x), & x\in A_\alpha,\\ 0, & x\in D_0\setminus \bar A_\alpha; \end{cases} \]
\(z_\alpha(y)\ne 0\) \((\xi_\alpha(y)\ne 0)\) almost everywhere for \(y\in A_\alpha\) \((y\in \Gamma_\alpha)\).
The function
\[ Z^\alpha(x)=Z(x;A_\alpha,z_\alpha;\Gamma_\alpha,\xi_\alpha) =\beta U^\alpha(x)+\gamma V^\alpha(x), \tag{5} \]
where \(\beta,\gamma\) are real numbers, \(\beta^2+\gamma^2\ne 0\), will be called a generalized magnetic potential (see [4], the case \(\beta=\gamma=1\), \(L\) is the Laplace operator, \(n=3\)).
In the case where \(L\) is the Laplace operator, the functions (3), (4), and (5) will be called, respectively, harmonic potentials \((n\ge 2)\) of volume masses, of a simple layer, and harmonic potentials.
\(2^\circ\). We investigate uniqueness of the solution of the indicated problem for generalized magnetic potentials of variable densities of constant sign. Denote
\[ B=(A_1\cup A_2)\setminus \bar A_0,\qquad \bar A_0=\bar A_1\cap \bar A_2. \tag{6} \]
Theorem 1. If, for the set \(A_\alpha\) \((\alpha=1,2)\) of nonnegative functions \(z_\alpha\in C^{(1)}(\bar A_\alpha)\), \(\xi_\alpha\in C^{(0)}(\Gamma_\alpha)\), and generalized magnetic potentials \(Z^\alpha(x)\) \((\alpha=1,2)\), defined by formula (5) (where \(\beta,\gamma\) are nonnegative numbers), the condition
\[ Z^1(x)=Z^2(x)\quad \text{for } x\in D_0\setminus \bar B, \]
is satisfied, then
\[ A_1=A_2 \]
and, moreover, if \(\beta\ne 0\), then \(z_1(x)=z_2(x)\), \(x\in A_1\); if \(\gamma\ne 0\), then \(\xi_1(x)=\xi_2(x)\), \(x\in \Gamma_1\).
We shall call the sets \(A_1\) and \(A_2\) externally contact if every component of the set \(A_0\) \((A_0\ne 0)\) has a part \(\Gamma_*\) \((\operatorname{mes}\Gamma_*\ne 0)\) of the common boundary of the \((n-1)\)-dimensional space with one of the components of the set \(D_0\setminus(\bar A_1\cup \bar A_2)\) (see [7]).
Theorem 2. If, for externally contact sets \(A_\alpha\) \((\alpha=1,2)\), nonnegative functions \(z\in C^{(1)}(\bar A_\alpha)\), \(\xi\in C^{(0)}(\Gamma_\alpha)\), and also nonnegative numbers \(\beta\) and \(\gamma\), equality of the external generalized magnetic potentials holds, i.e.,
\[ Z(x;A_1,z;\Gamma_1,\xi)=Z(x;A_2,z;\Gamma_2,\xi) \quad \text{for } x\in D_0\setminus(\bar A_1\cup \bar A_2), \]
then
\[ A_1=A_2. \]
\(3^\circ\). We investigate uniqueness of the solution of the posed problem for generalized potentials of volume masses of variable densities not of constant sign. Let the coefficients of the operator (1) satisfy condition (2) and, in addition,
\[ \frac{\partial}{\partial y_n}L^*h(y)=L^*\frac{\partial}{\partial y_n}h(y),\qquad y=(y_1,\ldots,y_n). \]
Let the function \(z(y)\in C^{(1)}(D_0')\) satisfy the condition
\[ z(y)=\delta(y)\eta(y), \tag{7} \]
where
a) \(\eta(y)\in C^{(1)}(D_0')\) is a nonnegative function, monotone with respect to \(y_n\),
\[ \tag{8} \]
b) the function \(\delta(y)\), generally speaking not of constant sign, satisfies the condition
\[ \partial\delta/\partial y_n=0. \tag{9} \]
Theorem 3. If, for the external contact sets \(A_\alpha\) \((\alpha=1,2)\) of generalized potentials of volume masses \(U(x;A_\alpha,z)\) with density \(z\) of class (7)—(9), the equality
\[ U(x;A_1,z)=U(x;A_2,z)\quad \text{for } x\in D_0\setminus(\overline{A}_1\cup\overline{A}_2), \]
holds, then
\[ A_1=A_2. \]
Let \((\rho,\theta)\) denote the spherical coordinates of a point \(y\) of the space \(E^n\) \((n\ge 2)\).
Let the function \(z(y)\) satisfy the conditions
\[ z(y)=\delta(y)\eta(y), \tag{10} \]
where
a) the function \(\delta(y)\), generally speaking not of constant sign, satisfies the condition
\[ \partial\delta/\partial\rho=0; \tag{11} \]
b) \(\rho^n\eta(y)=\psi(y)\) is a function monotone with respect to \(\rho\).
\[ \tag{12} \]
Theorem 4. If, for the external contact sets \(A_\alpha\) \((\alpha=1,2)\) of harmonic potentials of volume masses \(U(x;A_\alpha,z)\) \((n\ge 2)\) with density \(z(y)\) of class (10)—(12), the equality
\[ U(x;A_1,z)=U(x;A_2,z)\quad \text{for } x\in D_0\setminus(\overline{A}_1\cup\overline{A}_2) \]
holds, then
\[ A_1=A_2. \]
4°. We investigate the uniqueness of the solution of the external contact inverse problem for generalized potentials of a simple layer of variable densities, generally speaking not of constant sign.
Let the coefficients of operator (1) satisfy condition (2).
Theorem 5. If, for the external contact sets \(A_\alpha\) \((\alpha=1,2)\) of generalized potentials of a simple layer \(V(x;\Gamma_\alpha,\zeta)\) with density \(\zeta\in C^{(0)}(\Gamma_\alpha)\), generally speaking of variable sign, the equality
\[ V(x;\Gamma_1,\zeta)=V(x;\Gamma_2,\zeta)\quad \text{for } x\in D_0\setminus(\overline{A}_1\cup\overline{A}_2), \]
holds, then
\[ \Gamma_1=\Gamma_2. \]
We note that if, in the conditions of Theorem 5, one does not assume the external contactness of the bodies \(A_\alpha\), then, even in the case of constant density, in work (8) an example is given of two different star-shaped bodies having equal external logarithmic potentials of a simple layer.
For generalized potentials of a simple layer of operator (1), the coefficients of which satisfy condition (2), the following theorems hold.
Theorem 6. Let \(A_\alpha\) be open sets consisting of a finite number of convex domains. If the external generalized potentials of a simple layer \(V(x;\Gamma_\alpha,1)\) of constant density equal to unity coincide, i.e.
\[ V(x;\Gamma_1,1)=V(x;\Gamma_2,1)\quad \text{for } x\in D_0\setminus(\overline{A}_1\cup\overline{A}_2), \]
\[ \Gamma_1 = \Gamma_2. \]
Introduce the notation (see (7)). \(\Gamma^l\) is the boundary of the set \(\bar A_1 \cup \bar A_2\). If \(T_1^j \ne T_2^k\), then we denote
\[
\Gamma_1^i = \Gamma_1 \cap (\bar A_1 \cap \bar A_2), \qquad
\Gamma_1^l = \Gamma_1 \setminus \Gamma_1^i,
\]
\[
\Gamma_2^l = \Gamma_2 \cap \Gamma^i, \qquad
\Gamma_2^i = \Gamma_2 \setminus \Gamma_2^l.
\]
If \(T_1^i = T_2^k\), then we put \(S_1^j = (S_1^j)^l,\ S_2^k = (S_2^k)^l\).
Theorem 7. Let the bounded sets \(A_\alpha\) with boundaries \(\Gamma_\alpha\) and the functions \(\zeta_\alpha(y) \in C^{(0)}(\Gamma_\alpha)\) satisfy the condition
\[ \int_{\Gamma_1^i} |\zeta_1(y)|\,dyS + \int_{\Gamma_2^i} |\zeta_2(y)|\,dyS < \int_{\Gamma_1^l} |\zeta_1(y)|\,dyS + \int_{\Gamma_2^l} |\zeta_2(y)|\,dyS. \]
If \(A_1 \ne A_2\), then the exterior generalized simple-layer potentials \(V(x;\Gamma_1,\zeta_1)\) and \(V(x;\Gamma_2,\zeta_2)\) do not coincide, i.e., if \(A_1 \ne A_2\), then there exists a point \(\hat x \in D_0 \setminus (\bar A_1 \cup \bar A_2)\) such that
\[ V(\hat x;\Gamma_1,\zeta_1) \ne V(\hat x;\Gamma_2,\zeta_2). \]
Remark 1. Under the hypotheses of Theorems 1–7, the restrictions imposed on the smoothness of the boundaries \(\Gamma_\alpha\), as well as of the functions \(z_\alpha, \zeta_\alpha\), can be weakened (see (7)).
Remark 2. If \(\Gamma D_0\), the boundary of the domain \(D_0\), belongs to the class \(A^{(1,\lambda)}\) and the set \(D_0 \setminus (\bar A_1 \cup \bar A_2)\) consists of one component, then the assertions of Theorems 2–6 remain valid if everywhere in the hypotheses of these theorems, instead of the assumption \(x \in D_0 \setminus (\bar A_1 \cup \bar A_2)\), we write \(x \in \Gamma D_0\).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
14 XI 1967
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