Full Text
MATHEMATICS
V. T. FOMENKO
ON THE RIGIDITY OF SURFACES WITH BOUNDARY IN A RIEMANNIAN SPACE
(Presented by Academician I. N. Vekua on 16 I 1969)
Let \(S\) be a surface of positive exterior curvature \(K \ge k_0 > 0\), with boundary \(\mathscr L\), in a Riemannian space \(R_3\). Denote by \(\Lambda\) the set of unit vector fields \(l\) along \(\mathscr L\) that are nowhere tangent to the surface \(S\) and to the boundary \(\mathscr L\). We shall study infinitesimal bendings of the surface \(S\) subject on the boundary to the condition of generalized sliding:
\[ Ul\big|_{\mathscr L}=\sigma(s), \]
where \(U\) is the bending field of \(S\); \(l\in\Lambda\); \(\sigma(s)\) is a prescribed function.
Let \(\Lambda_0\) be the set of unit vector fields \(l_0\) along \(\mathscr L\) that belong to the surface and are nowhere tangent to \(\mathscr L\) at any point of the boundary. For each field \(l_0,\ l_0\in\Lambda_0\), form the set \(\Lambda(l_0)\) of vector fields \(l_\alpha\) from \(\Lambda\) whose tangential component is collinear with \(l_0\). We shall call \(\Lambda(l_0)\) the normal section of the set \(\Lambda\) in the direction of the field \(l_0\). It is evident that each vector field \(l_\alpha\) in the section \(\Lambda(l_0)\) is uniquely determined by specifying the angle
\[ \alpha(s)=\widehat{\,l_0,l_\alpha\,}, \]
measured in the positive direction.*
Theorem. Let \(S\in C^{3,\mu}\), \(\mathscr L\in C^{1,\mu}\), \(l_0,l_\alpha\in C^{1,\mu}\), \(\sigma\in C^{1,\mu}\), \(0<\mu<1\). Then for every normal section \(\Lambda(l_0)\), \(l_0\in\Lambda_0\), one can indicate an \(\alpha_0,\ \alpha_0>0\), such that for all vector fields \(l_\alpha\) from \(\Lambda(l_0)\) satisfying the condition
\[ \pi-\alpha_0<\alpha(s)<\pi, \]
the boundary condition \(Ul_0=\sigma(s)\) is quasi-correct and almost rigid, with three degrees of freedom.
Proof. Introducing on the surface an isothermally conjugate parametrization \((u,v)\), we reduce the investigation of infinitesimal bendings with the boundary condition of generalized sliding to the investigation of solvability, in a domain \(D\) of the plane \(z=u+iv\), of the following boundary-value problem:
\[ \partial_z w+B(z)\overline w=0,\qquad z\in D, \]
\[ \operatorname{Re}\left\{\left(\partial_t w+\partial_t\ln \sqrt{\,g\sqrt K^{\,3}\,}\, w\right)\sin\alpha-\cos\alpha\,\overline{\lambda(t)}\,w\right\}=\sigma,\qquad t\in\Gamma, \tag{1} \]
where, without loss of generality, the domain \(D\) may be regarded as the unit disk; \(\Gamma\) is the boundary of \(D\); \(B,K,\lambda,g\) are known functions of class \(C^{1,\mu}\), \(0<\mu<1\); \(\operatorname{Ind}\lambda(t)=+1\). To prove the theorem it is enough to establish that one can indicate an \(\alpha_0,\ \alpha_0>0\), such that for all functions \(\alpha(s)\) of class \(C^{1,\mu}\), \(0<\mu<1\), satisfying the condition
\[ \pi-\alpha_0<\alpha(s)<\pi, \tag{2} \]
the boundary-value problem (1) is solvable for any function \(\sigma\) of class \(C^{1,\mu}\), \(0<\mu<1\), and the solution depends on three parameters.
* In what follows we use the terminology of the book (¹), Ch. 5, § 10.
Consider the nonhomogeneous problem
\[ \partial_{\bar z}\varphi = F(z), \qquad z \in D; \tag{3} \]
\[ \operatorname{Re}\{(\partial_t\varphi(t)+\partial_t\ln \sqrt{g\sqrt{K^3}}\,\varphi(t)) -\operatorname{ctg}\alpha\,\overline{\lambda(t)}\,\varphi(t)\}=0, \qquad t\in\Gamma, \]
where \(F(z)\) is a given function of class \(C^{1,\mu}(D)\), \(0<\mu<1\). According to results of B. V. Boyarskii \((^2)\), the number \(l\) of solutions of the homogeneous problem (3) is related to the number \(l^*\) of solutions of the adjoint problem by the formula
\[ l=l^*+3, \tag{4} \]
and the nonhomogeneous problem (3) is solvable if and only if the orthogonality conditions for the free term of the nonhomogeneous problem (3) with respect to all solutions of the adjoint problem are satisfied. As shown in \((^3)\), one can specify an \(a_1\), \(a_1>0\), such that, under the condition \(\alpha(s)\in(\pi-a_1,\pi)\), the homogeneous problem (3) has no more than three linearly independent solutions, and therefore from formula (4) we find \(l^*=0\). Consequently, for \(\alpha\in(\pi-a_1,\pi)\), problem (3) admits a three-parameter family of solutions for any function \(F(z)\) of class \(C^{1,\mu}\), \(0<\mu<1\), which can be represented in the form
\[ \varphi(z)=T_\alpha F+\sum_{k=1}^{3} c_k\varphi_\alpha^k, \tag{5} \]
where the operator \(T_\alpha\) depends on \(\alpha\); \(\varphi_\alpha^k\) are linearly independent solutions of the homogeneous problem (3); \(c_k\) are arbitrary real constants.
Consider the family of problems
\[ \partial_z w+B\bar w=0, \qquad z\in D; \tag{6} \]
\[ \operatorname{tg}\alpha\,\operatorname{Re}\{\partial_t w+\partial_t\ln \sqrt{g\sqrt{K^3}}\,w\} -\operatorname{Re}\{\overline{\lambda(t)}\,w\}=0, \qquad t\in\Gamma; \]
regarding \(\alpha\) as an arbitrary function of class \(C^{1,\mu}\), \(0<\mu<1\). We shall establish the existence of an \(a_0>0\) such that problem (6) has three linearly independent solutions for any function \(\alpha(s)\) satisfying the condition
\[ \pi-a_0<\alpha(s)<\pi. \tag{7} \]
The constant \(a_0\) is constructed from the coefficients of problem (6) and does not depend on the function \(\alpha(s)\). To prove this fact, we reduce problem (6), with the aid of formula (5), to the integral equation
\[ w+T_\alpha(B\bar w)=\sum_{k=1}^{3} c_k w_\alpha^k, \tag{8} \]
assuming that \(\alpha(s)\in(\pi-a_1,\pi)\). We shall seek solutions of problem (3) satisfying the conditions
\[ \varphi(0)=0, \qquad \operatorname{Re}\{\varphi(0)/\chi(0)\}=\operatorname{Im}\{\varphi(0)/\chi(0)\}. \tag{9} \]
Under these conditions the homogeneous problem (3) has only the zero solution if \(\pi-a_1<\alpha<\pi\). The nonhomogeneous problem (3) under conditions (9) is always uniquely solvable. Indeed, from formula (8) it follows that at the point \(z=0\) the following conditions must be satisfied:
\[ T_\alpha F(0)+\sum_{k=1}^{3}c_k\varphi_\alpha^k(0)=0; \tag{10} \]
\[ \operatorname{Re}\left\{\frac{1}{\chi(0)} \left[T_\alpha F(0)+\sum_{k=1}^{3}c_k\varphi_\alpha^k(0)\right]\right\} = \operatorname{Im}\left\{\frac{1}{\chi(0)} \left[T_\alpha F(0)+\sum_{k=1}^{3}c_k\varphi_\alpha^k(0)\right]\right\}. \]
These conditions may be regarded as a linear system of three equations with respect to the real constants \(c_1,c_2,c_3\). Since the determi-
of this system is different from zero (otherwise the homogeneous problem (3) would have a nonzero solution), then the constants \(c_k\) are determined from (10) always and uniquely. Therefore the solution of the nonhomogeneous problem (3) under conditions (9) can be represented in the form:
\[ \varphi=\widetilde T_\alpha F, \tag{11} \]
where \(\widetilde T_\alpha\) is a homogeneous additive operator. We shall show that the operator \(\widetilde T_\alpha\) admits the representation
\[ \widetilde T_\alpha=\widetilde T_0+\widetilde T_{1\alpha}, \]
where \(\widetilde T_0\) is a completely continuous operator in \(C^{1,\mu}\), \(0<\mu<1\), independent of \(\alpha\), while the norm of the operator \(\widetilde T_{1\alpha}\), which depends on \(\alpha\), tends to zero as \(\alpha\to0\) in some specially chosen Banach space, for example in \(L_2(D)\). Indeed, consider the problem:
\[ \partial_z\varphi_0=F;\qquad \varphi_0(0)=0;\qquad \operatorname{Re}\{\chi^{-1}(0)\varphi_0(0)\}=\operatorname{Im}\{\chi^{-1}(0)\varphi_0(0)\}; \tag{12} \]
\[ \operatorname{Re}\{\lambda(t)\varphi_0\}=0,\qquad t\in\Gamma. \]
The solution of this problem exists and is unique for any function \(F\) of class \(C^{1,\mu}\), \(0<\mu<1\), and is given by the formula
\[ \varphi_0=\widetilde T_0F, \]
where \(\widetilde T_0F\in C^{2,\mu}\), \(0<\mu<1\).
Let us now establish that the norm of the operator \(\widetilde T_{1\alpha}\) in the space \(L_2(D)\) tends to zero as \(\alpha\to0\). For this purpose consider the difference
\[ \psi_\alpha=\varphi_\alpha-\varphi_0, \]
where \(\varphi_\alpha\) and \(\varphi_0\) are the solutions, respectively, of problems (3), (9) and (12). The function \(\psi_\alpha\) is analytic in \(\bar D\) and is a solution of the boundary-value problem
\[ \partial_{\bar z}\psi_\alpha=0,\qquad \psi_\alpha(0)=0,\qquad \operatorname{Re}\{\chi^{-1}(0)\psi_\alpha(0)\}=\operatorname{Im}\{\chi^{-1}(0)\psi_\alpha(0)\}; \]
\[ \operatorname{Re}\{(\partial_t\psi_\alpha+\partial_t\ln\sqrt{g\sqrt{K^3}}\ \psi_\alpha)-\operatorname{ctg}\alpha\,\overline{\lambda(t)}\psi_\alpha\} \]
\[ =-\operatorname{Re}\{\partial_t\varphi_0+\partial_t\ln\sqrt{g\sqrt{K^3}}\ \varphi_0\};\qquad t\in\Gamma. \]
It follows from this that the function \(\psi_{1\alpha}=\psi_\alpha\chi^{-1}\) is a solution of the boundary-value problem
\[ \partial_{\bar z}\psi_{1\alpha}=0;\qquad \operatorname{Re}\psi_{1\alpha}(0)=\operatorname{Im}\psi_{1\alpha}(0); \]
\[ \operatorname{Re}\{(\partial_t\psi_{1\alpha}\cdot\chi(t)+[\partial_t\chi(t)+\partial_t\ln\sqrt{g\sqrt{K^3}}\chi]\psi_{1\alpha}\} \]
\[ -\operatorname{ctg}\alpha\,\operatorname{Re}\{\overline{\lambda(t)}\chi\psi_{1\alpha}\} =-\operatorname{Re}\{\partial_t\varphi_0+\partial_t\ln\sqrt{g\sqrt{K^3}}\varphi_0\},\qquad t\in\Gamma. \]
But then the functions \(u_{1\alpha}=\operatorname{Re}\psi_{1\alpha}\) and \(v_{1\alpha}=\operatorname{Im}\psi_{1\alpha}\) satisfy the relation
\[ \iint_D(\nabla u_{1\alpha})^2\,dx\,dy+\operatorname{ctg}\alpha\oint A_1^2u_{1\alpha}^2\,ds =\oint B_1u_{1\alpha}v_{1\alpha}\,ds+ \]
\[ +\frac12\oint \partial_s\operatorname{tg}\psi\,u_{1\alpha}^2\,ds+\oint u_{1\alpha}\gamma\,ds, \]
where \(\gamma\) is a known function independent of \(\alpha\).
From this formula it follows that (3)
\[ \mu_3(\alpha)\|u_{1\alpha}\|_{L_2(\Gamma)}^2 \le(\mu_1+\mu_2)\|u_{1\alpha}\|_{L_2(\Gamma)}^2 +c\|u_{1\alpha}\|_{L_2(\Gamma)}, \]
where \(c\) is a constant independent of \(\alpha\). Consequently,
\[ \mu_3(\alpha)\le(\mu_1+\mu_2)+c\|u_{1\alpha}\|_{L_2(\Gamma)}^{-1}. \]
Since as \(\|a\|_C \to 0\) the constant \(\mu_3(a)\) tends to infinity \({}^{(3)}\), it follows from the last inequality that \(\|u_{1\alpha}\|_{L_2(\Gamma)} \to 0\) as \(\|a\|_C \to 0\). Further, since \(\|u_{1\alpha}\|_{L_2(\Gamma)}=\|v_{1\alpha}\|_{L_2(\Gamma)}\), we have \(\|\psi_{1\alpha}\|_{L_2(\Gamma)} \to 0\) as \(\|a\|_C \to 0\). Taking into account that \(\psi_\alpha=\chi\psi_{1\alpha}\) and \(\chi \ne 0\) on \(\Gamma\), hence we obtain that \(\|\psi_\alpha\|_{L_2(\Gamma)} \to 0\) as \(\|a\|_C \to 0\). Let us estimate the norm of \(\psi_\alpha\) in \(L_2(D)\). We have
\[ \|\psi_\alpha\|_{L_2(D)}^2 = \iint_D |\psi_\alpha|^2\,dx\,dy = \int_0^{2\pi}\int_0^1 |\psi_\alpha(re^{i\varphi})|^2 r\,dr\,d\varphi \le \int_0^1\left(\int_0^{2\pi}|\psi_\alpha(re^{i\varphi})|^2\,d\varphi\right)dr. \]
From the theory of analytic functions it is known that, for any function \(\psi_\alpha\) analytic in the disk \(|z|<1\), the \(L_p\)-norms
\[ \int_0^{2\pi} |\psi_\alpha|^p d\varphi \]
increase monotonically with \(r\). By virtue of the continuity of \(\psi_\alpha\) in the closed disk we have
\[ \int_0^{2\pi} |\psi_\alpha(re^{i\varphi})|^2\,d\varphi \le \int_0^{2\pi} |\psi_\alpha(e^{i\varphi})|^2\,d\varphi = \|\psi_\alpha\|_{L_2(\Gamma)}^2. \]
Thus, we obtain the estimate
\[ \|\psi_\alpha\|_{L_2(D)}\le \|\psi_\alpha\|_{L_2(\Gamma)}, \]
whence it follows that \(\|\psi_\alpha\|_{L_2(D)} \to 0\) as \(\|a\|_C \to 0\).
Let us now consider the equation:
\[ w+\widetilde T_0(B\overline w)+\widetilde T_{1\alpha}(Bw)=0. \tag{13} \]
Putting \(P_0w\equiv \widetilde T_0B\overline w,\ P_{1\alpha}w\equiv -\widetilde T_{1\alpha}B\overline w\), we rewrite it in the form:
\[ w+P_0w=P_{1\alpha}w. \tag{14} \]
By what has been said, the operator \(P_0\) is completely continuous in \(C^{1,\mu}\), \(0<\mu<1\), and \(\|P_{1\alpha}\|_{L_1(D)}\to 0\) as \(\|a\|_C\to 0\).
It can be shown that the operator \(P_0\) has an inverse, and then equation (14) is equivalent to the equation \(w=\Gamma_0P_{1\alpha}w\), where the norm of the operator \(\Gamma_0\) does not depend on \(\alpha\). Choosing \(\alpha_0\) sufficiently small for all \(\alpha\in(\pi-\alpha_0,\pi)\), we may assume \(\|\Gamma_0P_{1\alpha}\|_{L_2(D)}\le q<1\). But then equation (14) has only the zero solution, whence the validity of the theorem follows.
Rostov State University
Received
18 XII 1968
REFERENCES
- I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
- B. V. Boyarskii, DAN, 102, No. 2 (1955).
- V. T. Fomenko, DAN, 181, No. 6 (1969).