Reports of the Academy of Sciences of the USSR
1958, Volume 123, No. 2

MATHEMATICS

I. Ya. BAKELMAN

DETERMINATION OF A CONVEX SURFACE BY A GIVEN FUNCTION OF ITS PRINCIPAL CURVATURES

(Presented by Academician V. I. Smirnov, 30 V 1958)

1. Let a convex surface \(F\) be given by the equation \(z=z(x,y)\) \(\bigl(z(x,y)\in C^2\bigr)\) in some convex domain \(D\) in the \(x,y\)-plane, and let \(K_1, K_2\) be the principal curvatures of the surface \(F\). In this paper we shall consider questions of existence of a convex surface \(F\), if at each point \((x,y)\in D\) the principal curvatures of the surface \(F\) are connected by the relation

\[ f(x,y,z,p,q)K_1K_2-\varphi(x,y,p,q)(K_1+K_2)=\psi(x,y) \tag{1} \]

or by the relation

\[ f(x,y,z,p,q)K_1K_2-\varphi(x,y,p,q)\sqrt{f(x,y,z,p,q)}(K_1+K_2) =\psi(x,y), \tag{2} \]

where in both cases the functions \(f(x,y,z,p,q)\) and \(\varphi(x,y,p,q)\) are continuous in \(x,y\) in \(D\) and for all values of the variables \(z,p,q\), and \(f(x,y,z,p,q)>0\), \(\varphi(x,y,p,q)>0\). The function \(\psi(x,y)\) is assumed summable in \(D\).

Relations (1) and (2) may be regarded as second-order differential equations with respect to the function \(z(x,y)\), if, instead of \(K_1,K_2\), their expressions through the first and second derivatives of the function \(z(x,y)\) are substituted. The differential equations (1) and (2) will be of elliptic type if, on their solutions, respectively the relations

\[ f(x,y,z,p,q)\psi(x,y)+\varphi^2(x,y,p,q)>0; \tag{3} \]

\[ \psi(x,y)+\varphi^2(x,y,p,q)>0. \tag{4} \]

Introduce the operators, defined on the class \(C^2(D)\):

\[ \Phi_1(z)=f(x,y,z,p,q)K_1K_2-\varphi(x,y,p,q)(K_1+K_2); \tag{5} \]

\[ \Phi_2(z)=f(x,y,z,p,q)K_1K_2-\varphi(x,y,p,q)\sqrt{f(x,y,z,p,q)}(K_1+K_2), \tag{6} \]

In (¹) it was proved that the ellipticity of the expressions \(\Phi_1(z)\) \(\bigl(\Phi_2(z)\bigr)\) on solutions of equations (1) \(\bigl((2)\bigr)\) is equivalent to the fact that \(\partial\Phi_1/\partial K_i>0\) \((i=1,2)\) or \(\partial\Phi_1/\partial K_i<0\) \((i=1,2)\) \(\bigl(\partial\Phi_2/\partial K_i>0\ (i=1,2)\) or \(\partial\Phi_2/\partial K_i<0\ (i=1,2)\bigr)\). For definiteness, below we shall restrict ourselves to consideration of the case when \(\partial\Phi_1/\partial K_i>0\) or \(\partial\Phi_2/\partial K_i>0\). In this case, using the conditions imposed on the functions \(f\) and \(\varphi\), from the conditions \(\partial\Phi_1/\partial K_i>0\) \((i=1,2)\) \(\bigl(\partial\Phi_2/\partial K_i<0\ (i=1,2)\bigr)\) we obtain that the solutions of equations (1) and (2) will be convex functions whose convexity is turned toward \(z<0\).

The Dirichlet problem for equations (1) and (2) will have a unique solution in the class \(C^2\) if \(f(x,y,z,p,q)\) and \(\varphi(x,y,p,q)\) are continuously differentiable with respect to \(z,p,q\) and \(f_z(x,y,z,p,q)\leq 0\).

1. Let us introduce the concept of generalized solutions of equations (1) and (2). For this purpose we extend the operators \(\Phi_1(z)\) and \(\Phi_2(z)\), defined on \(C^2\), to the class

of all convex functions. Let us consider the extension \(\Phi_1(z)\), since for \(\Phi_2(z)\) the construction is analogous. Let first the convex function \(z(x,y)\in C^2\). Integrating (5) over a variable Borel set \(e\) of the domain \(D\):

\[ \iint_e \Phi_1(z)\,dx\,dy = \iint_e f(x,y,z,p,q)K_1K_2\,dx\,dy - \iint_e \varphi(x,y,p,q)(K_1+K_2)\,dx\,dy . \]

The set function

\[ \omega(z,f,e)=\iint_e f(x,y,z,p,q)K_1K_2\,dx\,dy, \]

constructed on the convex surface \(F\) given by the equation \(z=z(x,y)\), can, by means of the normal mapping \(\mu\) onto the plane \((p,q)\) (see (2)), be transformed to the form

\[ \omega(z,f,e) = \iint_{\mu(e)} \frac{ f(x(p,q),y(p,q),z(x(p,q),y(p,q)),p,q) }{ (1+p^2+q^2)^2 }\,dp\,dq . \]

In papers \((^2\text{--}^4)\) it is proved that if \(f(x,y,z,p,q)\) is a continuous function of its arguments, then the set function \(\omega(z,f,e)\) can be extended to an arbitrary convex surface as a completely additive nonnegative function of Borel sets of the domain \(D\). The properties of these set functions are studied there as well. The set function

\[ h(z,\varphi,e)=\iint_e \varphi(x,y,p,q)(K_1+K_2)\,dx\,dy \]

for every convex function \(z\in C^2\) can be brought to the form

\[ h(z,\varphi,e) = 2\iint_{\nu(e)} \frac{ \varphi(x(\xi,\eta),y(\xi,\eta),p(\xi,\eta),q(\xi,\eta)) }{ \sqrt{1+p^2(\xi,\eta)+q^2(\xi,\eta)} }\,\widetilde H(d\tilde e). \tag{7} \]

The meaning of the parameters \(\xi,\eta\) and of the set function \(\widetilde H(\tilde e)\) is as follows. Consider the convex surface \(F_\lambda\), parallel to the surface \(F\) and constructed along the outer normals to \(F\) with length \(\lambda\). \(F_\lambda\) is given by the vector function

\[ \mathbf r(x,y,\lambda) = \left(x-\frac{\lambda p}{\sqrt{1+p^2+q^2}}\right)\mathbf i + \left(y-\frac{\lambda q}{\sqrt{1+p^2+q^2}}\right)\mathbf j + \left(z(x,y)+\frac{\lambda}{\sqrt{1+p^2+q^2}}\right)\mathbf k . \]

Put

\[ \xi=x-\frac{p}{\sqrt{1+p^2+q^2}}, \qquad \eta=y-\frac{q}{\sqrt{1+p^2+q^2}} . \]

Then the mapping \(\nu:\ \xi=\xi(x,y),\ \eta=\eta(x,y)\) is one-to-one. The mapping \(\nu^{-1}:\ x=x(\xi,\eta),\ y=y(\xi,\eta)\) makes it possible to determine uniquely

\[ p(\xi,\eta)=p(x(\xi,\eta),y(\xi,\eta)),\qquad q=q(x(\xi,\eta),y(\xi,\eta)). \]

The set function

\[ H(e)=\frac12\iint_e (K_1+K_2)\sqrt{1+p^2+q^2}\,dx\,dy, \]

which is the mean integral curvature of \(F\), is connected with the area on the surface \(F_\lambda\) by the relation

\[ \sigma(F_\lambda,e) = \iint_e \sqrt{1+p^2+q^2}\,dx\,dy + \]

\[ + 2\lambda\iint_e \frac{K_1+K_2}{2}\sqrt{1+p^2+q^2}\,dx\,dy + \lambda^2\iint_e K_1K_2\sqrt{1+p^2+q^2}\,dx\,dy . \]

Hence

\[ H(e)=\left.\frac{d}{d\lambda}\sigma(F_\lambda,e)\right|_{\lambda=0}. \]

The set function \(\widetilde H(\tilde e)\) on the \(\xi,\eta\) plane is constructed as follows:
\[ \widetilde H(\tilde e)=\widetilde H(\tilde e\cap \nu(D))=H(\nu^{-1}(\tilde e\cap \nu(D))). \]
For any convex surface \(F\), given by a function \(z(x,y)\), the mapping \(\nu^{-1}: x=x(\xi,\eta),\ y=y(\xi,\eta)\) is one-to-one and, with the aid of the formulas
\[ \xi=x-\frac{p}{\sqrt{1+p^2+q^2}},\qquad \eta=y-\frac{q}{\sqrt{1+p^2+q^2}}, \]
which connect \(\xi,\eta,x,y\) with the coefficients \(p\) and \(q\) of the supporting planes of \(F\) at the point \((x(\xi,\eta),y(\xi,\eta))\), \(p(\xi,\eta)\), \(q(\xi,\eta)\), are determined as single-valued continuous functions of the variables \(\xi\) and \(\eta\). The set functions \(H(e)\) and \(\widetilde H(\tilde e)\) will be completely additive nonnegative set functions. Therefore, by means of formula (7), the set function \(h(z,\varphi,e)\) can be extended to arbitrary convex surfaces \(F\) and it can be established that \(h(z,\varphi,e)\) is a completely additive nonnegative set function on the Borel subsets of the domain \(D\), if \(\varphi(x,y,p,q)\) is continuous in \(x,y,p,q\). Let us consider the completely additive set function
\[ \Phi_1(z,e)=\omega(z,f,e)-h(z,\varphi,e). \]
If the surface \(F\) is given by a convex function \(z(x,y)\in C^2\), then
\[ \Phi_1(z,e)=\iint_e \Phi_1(z)\,de. \]

Let \(\Psi(e)\) be a completely additive function of bounded variation, given in \(D\). By a solution of the equation
\[ \Phi_1(z,e)=\Psi(e) \tag{8} \]
we shall mean a convex function \(z\), with convexity directed toward \(z<0\), such that the indicated equation is satisfied on all Borel subsets of the domain \(D\) that are separated from the boundary of \(D\) by a positive distance. In particular, if
\[ \Psi(e)=\iint_e \varphi(x,y)\,dx\,dy, \]
then the solutions of equation (8) will be called generalized solutions of equation (1).

The notion of generalized solutions of equation (2) is constructed analogously; here it is naturally assumed that \(f(x,y,z,p,q)\) and \(\varphi(x,y,p,q)\) are continuous functions of their variables.

1. Let now \(D\) be a convex domain in the \(x,y\) plane, bounded by a closed convex curve \(\Gamma\), and let \(L\) be a certain curve in space that projects one-to-one onto \(\Gamma\).

We shall next consider functions \(f(x,y,z,p,q)\) and \(\varphi(x,y,p,q)\) satisfying the following conditions: 1) \(f(x,y,z,p,q)\) and \(\varphi(x,y,p,q)\) are continuous in all variables: in \(x,y\) in \(D\) and in \(z,p,q\) for all finite values of these variables; 2) \(f(x,y,z,p,q)\) has a nonpositive derivative \(f_z(x,y,z,p,q)\), continuous in \(x,y\) in \(D\) and for all finite values of \(z,p,q\).

Let \(T\) be the boundary of the convex hull spanned by the curve \(L\), and let \(T'\) be the part of it lying under \(L\). \(T'\) is a convex developable surface, with convexity directed downward. Then, for any Borel set \(e\subset D\), we have
\[ \omega(T',f,e)=0,\qquad h(T',\varphi,e)\geq 0,\qquad h(T',\varphi\sqrt f,e)\geq 0. \]
In the domain \(D\) we shall now consider completely additive set functions \(\Psi_1(e)\) and \(\Psi_2(e)\), satisfying respectively the conditions:
\[ -h(T',\varphi,e)\leq \Psi_1(e);\qquad -h(T',\varphi\sqrt f,e)\leq \Psi_2(e). \]
Let us now consider in \(D\) two classes of convex functions \(W_1\) and \(W_2\) such that: 1) for any Borel set \(e\subset D\) we have \(\Phi_1(u,e)\leq \Psi_1(e)\), if \(u\in W_1\), and \(\Phi_2(u,e)\leq \Psi_2(e)\), if \(u\in W_2\); 2) all functions of both classes \(W_1,W_2\) are convex downward; 3) if \(u\) belongs to \(W_1\) or \(W_2\), then in the corresponding class \(W_i\) there is no convex surface \(v\) such that \(u-v=\mathrm{const}\), and moreover the boundary points of the surface \(u\) lie strictly below the boundary points of the surface \(v\); 4) all surfaces defined by functions from \(W_1\) or from \(W_2\) lie below the surface \(T'\) and have common points with \(L\).

The classes of functions \(W_1, W_2\), obviously, are nonempty. Let \(z \in W_1\) or \(z \in W_2\). Denote by \(v(z)\) the volume of the body bounded by the surfaces \(z(x,y)\), \(T\) and the cylinder with directrix \(\Gamma\) and generators parallel to the \(z\)-axis.

Theorem 1. If for the classes of convex functions \(W_i\) \((i=1,2)\) the relation
\[ v_0=\sup_{z\in W_i} v(z)<+\infty \]
holds, then there exists a function \(z_0 \in W_i\) \((i=1,2)\) realizing this maximum. For all Borel sets \(e \subset D\) at a positive distance from \(\Gamma\), this function satisfies the relation \(\Phi_i(z,e)=\Psi_i(e)\) \((i=1,2)\). If \(\Psi_i(e)\) is an absolutely continuous set function, then \(z_0\) is a generalized solution, respectively, of equation (1) or (2), and almost everywhere satisfies equation (1) or (2).

If Theorem 1 is considered in the class \(W_1\), then condition (2) on the existence of \(f_z\) and the sign of \(f_z\) may be omitted. We give sufficient conditions which ensure a finite estimate of \(v_0\) in the classes \(W_1\) and \(W_2\).

Theorem 2. Let the functions \(f(x,y,z,p,q)\) and \(\varphi(x,y,p,q)\) satisfy the conditions: 1) there exists a function \(f_0(p,q)\), summable over the whole \(p,q\)-plane, such that \(f(x,y,z,p,q)\leq f_0(p,q)\) for all \(x,y,z\), and
\[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{f_0(p,q)}{(1+p^2+q^2)^2}\,dp\,dq<+\infty; \]
2) for all \(x,y,p,q\) we have
\[ \varphi(x,y,p,q)\geq \mathrm{const}>0. \]
Then for the number \(v_0\) in the class \(W_1\) the inequality \(v_0<+\infty\) holds. An analogous assertion holds in the class \(W_2\), if condition 2) is replaced by the inequality
\[ \varphi(x,y,p,q)\sqrt{f(x,y,z,p,q)}\geq \mathrm{const}>0. \]

Theorem 3. Denote by
\[ B_1=\sup_{z\in W_1} h(z,\varphi,D) \]
and by
\[ B_2=\sup_{z\in W_2} h(z,\varphi\sqrt f,D), \]
and suppose that there exists, summable in every bounded domain of the \(p,q\)-plane, a function \(f_1(p,q)\) such that for all \(x,y,z\) we have \(f(x,y,z,p,q)\geq f_1(p,q)\).

Then, if \(B_i<+\infty\) \((i=1,2)\) and the inequality
\[ B_i+\operatorname*{Var}_{D}\Psi_i(e)< \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{f_1(p,q)}{(1+p^2+q^2)^2}\,dp\,dq \quad (i=1,2) \]
holds, then for \(v_0\) a finite upper estimate can be obtained in the classes \(W_i\) \((i=1,2)\).

A finite estimate for the numbers \(B_1\) and \(B_2\) can be obtained if we have, respectively,
\[ \varphi(x,y,p,q)\leq \frac{A_1}{(1+p^2+q^2)^\alpha} \quad\text{and}\quad \varphi\sqrt f\leq \frac{A_2}{(1+p^2+q^2)^\alpha}, \]
where \(\alpha>2\) and \(A_1,A_2\) are positive constants.

All the results considered are generalized verbatim to the case of \(n\) variables for the equations
\[ f(x_1,\ldots,x_n,z,p_1,\ldots,p_n)K_1\cdots K_n -\varphi(x_1,\ldots,x_n,p_1,\ldots,p_n)(K_1+\cdots+K_n) =\psi(x_1,\ldots,x_n), \]
\[ f(x_1,\ldots,x_n,z,p_1,\ldots,p_n)K_1\cdots K_n -\varphi(x_1,\ldots,x_n,p_1,\ldots,p_n) \sqrt{f(x_1,\ldots,x_n,z,p_1,\ldots,p_n)} \times (K_1+\cdots+K_n) =\psi(x_1,\ldots,x_n). \]

If \(\varphi\equiv 0\), then from Theorems 3 and 1 follow the existence theorems for the Dirichlet problem, established in works \((^{2-4})\), for the equations
\[ \Gamma(z)=\varphi(x_1,\ldots,x_n,z,p_1,\ldots,p_n), \]
where \(\Gamma(z)\) is the Hessian of the function \(z(x_1,\ldots,x_n)\).

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
28 V 1958

CITED LITERATURE

1. A. D. Aleksandrov, Vestn. LGU, No. 7 (1957).
2. I. Ya. Bakelman, DAN, 116, No. 5 (1957).
3. I. Ya. Bakelman, Vestn. LGU, No. 1 (1958).
4. A. D. Aleksandrov, Vestn. LGU, No. 1 (1958).