Abstract Generated abstract
This paper studies the unique determination of a piecewise regular ovaloid of positive Gaussian curvature with boundary, allowing discontinuities of second derivatives along an internal gluing curve. The authors formulate boundary conditions on the increments of normal curvature and geodesic torsion, transform the Gauss and Peterson-Codazzi equations into a complex boundary-value problem for generalized analytic functions, and analyze the associated transmission and boundary indices. They prove that, when the prescribed boundary expression vanishes and the index of the boundary coefficient satisfies Ind of mu plus i lambda less than 2, the surface is uniquely determined. A modification is also given for the case in which the internal nonregular set consists of open arcs, adding auxiliary index contributions from the corresponding Riemann problems.
Full Text
MATHEMATICS
G. S. BARKHIN, V. T. FOMENKO
ON THE UNIQUE DETERMINACY OF A PIECEWISE REGULAR SURFACE OF POSITIVE CURVATURE WITH A BOUNDARY CONDITION
(Presented by Academician I. N. Vekua, May 11, 1963)
Consider a regular ovaloid \(S\) of class \(D_{3,p}\) \((p>2)\) (the radius vector of the surface \(\bar r(u,v)\) admits three generalized Sobolev derivatives summable to the power \(p\)), with boundary \(\Gamma \in C_\mu^1\) \((0<\mu<1)\), containing a certain linear set \(\gamma\) of nonregular points, where discontinuities of the second derivatives are allowed. Suppose, moreover, that the line \(\gamma\), belonging to the class \(C_\mu^1\), divides the surface \(S\) into two pieces: a simply connected \(S^+\) and a doubly connected \(S^- \supset \Gamma\), glued along \(\gamma\) in such a way that the latter is their line of contact (the angle between the corresponding strips of the curve \(\gamma\) is zero). The Gaussian curvature of \(S\) is positive up to the boundary \(\Gamma\). In the present note we give boundary conditions, analogous to \((^1)\), imposed on the normal curvature and the geodesic torsion of the contour \(\Gamma\), under which the surface \(S\) is uniquely determined.
- Introduce on \(S\) a single isothermally conjugate parametrization, homeomorphically mapping it onto a domain \(D=D^+ + D^-\) of the plane of the complex variable \(z=x+iy\), where \(D^+\) is situated inside the curve \(l\) \((x=x(s), y=y(s))\), into which \(\gamma\) is mapped, and \(D^-\) is the curvilinear annulus between \(l\) and the image of \(\Gamma\), the curve \(L\). The curves \(l\) and \(L\) belong to \(C_\mu^1\).
Without loss of generality one may assume that \(L\) is the unit circle \(x=\cos\varphi,\ y=\sin\varphi\). This can be achieved by a conformal transformation preserving the introduced parametrization.
The fundamental forms of the surface \(S\) take the form:
\[ ds^2=E\,dx^2+2F\,dx\,dy+G\,dy^2;\qquad II=b_0(dx^2+dy^2), \]
where \(E,F,G,b_0\) are, in general, piecewise-continuous functions given in the domain \(D\), and \(b_0\) is assumed \(>0\). In what follows, a function \(f(z)\) given in \(D\) will be denoted by \(f^+(z)\) if \(z\in D^+\), and by \(f^-(z)\) if \(z\in D^-\). The limiting values of \(f(z)\) on the contour will be denoted respectively by \(f^+(t)\) and \(f^-(t)\).
Suppose that under an isometric transformation of the surface \(S\) the coefficients of the second form receive increments \(\Delta L,\Delta M,\Delta N\), and the normal curvature and geodesic torsion of the contours receive increments \(\Delta k_n\) and \(\Delta \tau_g\). The former then satisfy the Gauss and Peterson–Codazzi equations:
\[ (b_0+\Delta L)(b_0+\Delta N)-\Delta M^2=Ka,\qquad a=EG-F^2, \]
\[ \Delta L_y-\Delta M_x-\Gamma_{11}^1\Delta L+(\Gamma_{11}^1-\Gamma_{12}^2)\Delta M+\Gamma_{11}^2\Delta N=0, \tag{1} \]
\[ \Delta N_x-\Delta M_y+\Gamma_{22}^1\Delta L+(\Gamma_{22}^2-\Gamma_{12}^1)\Delta M-\Gamma_{12}^2\Delta N=0. \]
To obtain a relation between the increments of the coefficients of the second form, we take into account the conditions of conjugacy on the gluing line. From the formulas for the rotation of a strip \((^2)\), taking the gluing angle \(\theta\) to be zero, we obtain \(\Delta\theta=0\), and, consequently, the conjugacy conditions on \(l\) have the form:
\[ \Delta k_n^+(t)=\Delta k_n^-(t);\qquad \Delta \tau_g^+(t)=\Delta \tau_g^-(t)\quad (t\in l). \tag{2} \]
Set \(\Delta M=U,\ \Delta L=\Pi+V,\ \Delta N=\Pi-V\).
Gauss’ equation gives, since \(K>0\):
\[ \Pi=-b_0+\sqrt{b_0^2+U^2+V^2}. \]
Representing the function \(\Pi\) in the form
\[ \Pi=q_1(b_0,U,V)U+q_2(b_0,U,V)V, \tag{3} \]
where
\[ q_1=\frac{U}{b_0+\sqrt{b_0^2+U^2+V^2}},\qquad q_2=\frac{V}{b_0+\sqrt{b_0^2+U^2+V^2}}, \]
and introducing the notation
\[ I^{\pm}=E^{\pm}\dot{x}^{\,2}+2F^{\pm}\dot{x}\dot{y}+G^{\pm}\dot{y}^{\,2}, \]
\[ \sqrt{a^{\pm}}\,I^{\pm}\alpha^{\pm} = E^{\pm}\dot{x}^{\,2}-G^{\pm}\dot{y}^{\,2} + \left[-F^{\pm}(\dot{x}^{\,2}-\dot{y}^{\,2})+(E^{\pm}-G^{\pm})\dot{x}\dot{y}\right]q_1^{\pm}, \]
\[ \sqrt{a^{\pm}}\,I^{\pm}\beta^{\pm} = -F^{\pm}(\dot{x}^{\,2}+\dot{y}^{\,2})-(E^{\pm}+G^{\pm})\dot{x}\dot{y} + \left[-F^{\pm}(\dot{x}^{\,2}-\dot{y}^{\,2})+(E^{\pm}-G^{\pm})\dot{x}\dot{y}\right]q_2^{\pm}, \]
\[ I^{\pm}\gamma^{\pm}=2\dot{x}\dot{y}+(\dot{x}^{\,2}+\dot{y}^{\,2})q_1^{\pm}, \]
\[ I^{\pm}\delta^{\pm}=(\dot{x}^{\,2}-\dot{y}^{\,2})+(\dot{x}^{\,2}+\dot{y}^{\,2})q_2^{\pm}, \tag{4} \]
we write condition (2) in the following form:
\[ \alpha^+U^+ + \beta^+V^+ = \alpha^-U^- + \beta^-V^-, \]
\[ \gamma^+U^+ + \delta^+V^+ = \gamma^-U^- + \delta^-V^-. \qquad (t\in l) \tag{5} \]
On the contour \(L\), analogously to (1), we obtain the relation
\[ \left(\mu\sqrt{\frac{a^-}{e^-}}\alpha^-+\lambda\gamma^-\right)U^- + \left(\mu\sqrt{\frac{a^-}{e^-}}\beta^-+\lambda\delta^-\right)V^- = \lambda\Delta k_n^-+\mu\frac{\sqrt{a^-}}{e^-}\Delta\tau_g^-=\sigma. \tag{6} \]
Here \(\lambda\) and \(\mu\) are certain prescribed functions of points of the contour \(L\), belonging to the class \(C_\mu^1(L)\); the coefficients \(\alpha,\beta,\gamma,\delta\) have the same form as in (4), but refer to the contour \(L\).
We shall now consider the boundary-value problem (1), (5), (6), regarding \(U,V\) as the unknown functions, and \(\sigma\) as prescribed on \(L\).
2. Following (3) and putting \(w=U+iV,\ \partial_{\bar z}\equiv \frac12(\partial_x+i\partial_y)\), \(\partial_z\equiv \frac12(\partial_x-i\partial_y)\), we write our boundary-value problem in complex form
\[ \partial_{\bar z}w+A(z)w+B(z)\overline{w}=C(z)\Pi+i\partial_z\Pi; \tag{7} \]
\[ w^+(t)=G_1(t,w^+,w^-)w^-(t)+G_2(t,w^+,w^-)\overline{w^-(t)} \qquad (t\in l); \tag{8} \]
\[ \operatorname{Re}\left[\mu\frac{\sqrt{a^-}}{e^-}\alpha^-+\lambda\gamma^- -i\left(\mu\frac{\sqrt{a^-}}{e^-}\beta^-+\lambda\delta^-\right)\right]w^-(t) =\sigma(t) \qquad (t\in L). \tag{9} \]
Here \(A,B,C\) are known functions of class \(L_{p,2}\), expressible in terms of the Christoffel symbols,
\[ 2(\alpha^+\delta^+-\beta^+\gamma^+)G_1 = \alpha^-\delta^+-\beta^+\gamma^-+\alpha^+\delta^- -\beta^-\gamma^+ \]
\[ \qquad\qquad +i(\alpha^+\gamma^- -\alpha^-\gamma^+ +\beta^+\delta^- -\beta^-\delta^+), \tag{10} \]
\[ 2(\alpha^+\delta^+-\beta^+\gamma^+)G_2 = \alpha^-\delta^+-\beta^+\gamma^- -\alpha^+\delta^-+\beta^-\gamma^+ \]
\[ \qquad\qquad +i(\alpha^+\gamma^- -\alpha^-\gamma^+ -\beta^+\delta^-+\beta^-\delta^+). \]
Lemma 1. The coefficients \(G_1\) and \(G_2\) of the boundary condition (8), uniformly in \(w^+(t),w^-(t)\subset C_\mu(l)\), satisfy the conditions:
\[ 1.\quad |G_1(t,w^+,w^-)|>|G_2(t,w^+,w^-)|. \tag{11} \]
\[ 2.\quad \operatorname{Ind}G_1(t,w^+,w^-)=0 \qquad (t\in l). \tag{12} \]
The first assertion follows from the relation
\[ |G_1|^2-|G_2|^2=\frac{\alpha^{-}\delta^{-}}{\alpha^{+}\delta^{+}}-\frac{\beta^{-}\gamma^{-}}{\beta^{+}\gamma^{+}} \]
and the obvious inequality
\[ [2\dot{x}\dot{y}V-(\dot{x}^2-\dot{y}^2)U]^2>0. \]
The second—from the fact that
\[ \operatorname{Re} G_1(t,w^{+},w^{-})>0 \tag{13} \]
uniformly in \(w^{+},w^{-}\).
Lemma 2. The index \(\varkappa^{-}\) of the boundary condition (9) is connected with \(\operatorname{Ind}(\mu+i\lambda)\) by the relation
\[ \varkappa^{-}=-2+\operatorname{Ind}(\mu+i\lambda) \tag{14} \]
uniformly in \(w^{+},w^{-}\).
3. Using the basic integral representation (4) of solutions of system (1), we reduce problem (7), (8), (9) to a boundary-value problem for a piecewise analytic function. We have
\[ w^{\pm}(z)=f^{\pm}(z)e^{\omega_{\pm}(z)+\omega_{\pm}(z)}, \]
where \(f^{\pm}(z)\) is a function analytic in \(D^{\pm}\), \(\omega(z)\in C_{\alpha}(D)\).
Such a representation is possible, since \(e^{\omega^{-}}\) is analytic in \(D^{+}\), while \(e^{\omega^{+}}\) is analytic in \(D^{-}\). Our problem has been reduced to the following:
\[ f^{+}(t)=G_1(t)f^{-}(t)+g_2(t)\overline{f^{-}(t)} \quad (t\in l); \tag{15} \]
\[ \operatorname{Re}[\overline{p(t)}f(t)]=\sigma(t) \quad (t\in L), \tag{16} \]
where \(g_2(t)=G_2(t)e^{\overline{\omega}-\omega}\). In this case
\[ |G_1|>|g_2|,\qquad \operatorname{Ind}p(t)=\varkappa^{-}. \tag{17} \]
Lemma 3. Let
\[ \sup_{t\in l}\left|\frac{g_2(t)}{G_1(t)}\right|<\frac{2}{1+M_p}, \]
where \(M_p\) is the norm of the singular operator \(Sv=\dfrac{1}{\pi i}\int \dfrac{v(\tau)}{\tau-t}\,d\tau\) \({}^{(5)}\) in \(L_p\). Then, for \(\varkappa^{-}\geqslant 0\), problem (15), (16) has a solution depending on \(2\varkappa^{-}+1\) real constants. For \(\varkappa^{-}<0\), the problem may have a solution if \(2|\varkappa^{-}|+1\) additional conditions are satisfied, and the homogeneous problem \((\sigma(t)=0)\) has only the zero solution. The latter is true under the weaker assumption \(|G_1|>|g_2|\).
Proof. Let \(\varkappa^{-}\geqslant 0\) and let \(F(z)\) be a solution of problem (15), obtained by the method \({}^{(5)}\) with the use of the canonical function \(X(z)\), satisfying the conditions:
\[ X^{+}(t)=G_1(t)X^{-}(t)\quad (t\in l);\qquad \operatorname{Im}X^{-}(t)=q\,\Pi \quad (t\in L). \]
Then \(f(z)\) is constructed with the help of \(F(z)\) and of a certain function analytic in \(D\) with prescribed real part on \(L\), determined up to an imaginary constant. For \(\varkappa^{-}<0\), in order for the solution to be analytic, \(\sigma(t)\) must satisfy \(2|\varkappa^{-}|+1\) integral conditions. The last assertion of the lemma is proved by contradiction: using the canonical
function
\[ X^{+}(t)=\left[1+\frac{g_{2}(t)}{G_{1}(t)}\,\frac{\overline{f^{-}(t)}}{f^{-}(t)}\right]X^{-}(t)\quad (t\in l);\qquad \operatorname{Im}X^{-}(t)=0\quad (t\in L), \]
and also, using (12) and (17), we reduce (15), (16), for \(\sigma=0\), to \(f(z)=\psi(z)X(z)\):
\[ \psi^{+}(t)=G_{1}(t)\psi^{-}(t)\quad (t\in l);\qquad \operatorname{Re}\psi^{-}(t)=0\quad (t\in L) \]
and thence to \(f(z)\equiv 0\).
From Lemmas 2 and 3 it follows:
Theorem. Under the condition
\[ \sigma=0,\qquad \operatorname{Ind}(\mu+i\lambda)<2 \tag{18} \]
the surface \(S=S^{+}+S^{-}\) is uniquely determined.
- The contour \(\gamma\) may consist of open arcs \(\gamma_k\): \(\gamma=\sum \gamma_k\). In this case condition (18) is modified as follows:
\[ \operatorname{Ind}(\mu+i\lambda)<2+\sum \chi_k, \]
where \(\chi_k\) is the index of the auxiliary Riemann problem for the corresponding arc \(l_k\) with endpoints \(a_k\) and \(b_k\). As in the case of infinitesimal bendings, in consequence of (13), it behaves in the following way:
\[ \begin{aligned} &\text{if }\operatorname{sign}\operatorname{Im}G_{1}(a_k)=-\operatorname{sign}\operatorname{Im}G_{1}(b_k),\qquad &&\chi_k=\operatorname{sign}\operatorname{Im}G_{1}(a_k);\\ &\text{if }\operatorname{sign}\operatorname{Im}G_{1}(a_k)=\operatorname{sign}\operatorname{Im}G_{1}(b_k),\qquad &&\chi_k=0. \end{aligned} \]
Rostov-on-Don State University
Received
7 V 1963
References Cited
- G. S. Barkhin, V. T. Fomenko, DAN, 140, No. 5 (1961).
- V. Blaschke, Differential Geometry, 1935.
- I. N. Vekua, Generalized Analytic Functions, 1959.
- V. T. Fomenko, DAN, 144, No. 1 (1962).
- L. G. Mikhailov, Izv. Academy of Sciences of the Tajik SSR, Division of Geological, Chemical, and Technical Sciences, issue 3 (5) (1961).