UDC 519.33

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. V. POGORELOV

ONE VARIATIONAL PROBLEM IN PARAMETRIC FORM

Let \(H(x_1,x_2,x_3)\) be a positive homogeneous function of the first degree that is strictly convex. This means that for any \(\lambda \geq 0\)
\[ H(\lambda x_1,\lambda x_2,\lambda x_3)=\lambda H(x_1,x_2,x_3) \]
and the second differential \(d^2H\) is a positive definite form.

We consider the following variational problem. Find a regular surface \(F: r=r(u,v)\) with a given spherical image \(\omega\), with a given support function along the boundary \((rn=\psi)\), which gives the functional
\[ I(F)=\iint_F H(A_1,A_2,A_3)\,du\,dv \]
a stationary value \((\delta I=0)\). Here \(A_1,A_2,A_3\) are the components of the vector product \(r_u\times r_v\). By a regular surface we mean the image of an arbitrary domain of the \(uv\)-plane by means of a regular vector function \(r(u,v)\). The usual requirement \(r_u\times r_v\neq 0\) is not assumed, since geometric singularities on the surface are not excluded.

Theorem. Whatever the function \(H\) of class \(C^{2,\lambda}\), the domain \(\omega\) of class \(C^{2,\lambda}\) lying entirely on one hemisphere of the unit sphere, and the function \(\psi\), prescribed on the boundary \(\gamma\) of the domain \(\omega\), the variational problem stated above is always solvable in the class of surfaces \(C^{2,\lambda}\).

Proof. We shall seek the solution of our variational problem among surfaces \(F\) with nonpositive Gaussian curvature and one-to-one spherical image. Such a surface can be specified by its support function \(\Phi(n)=rn\), where \(r\) is the vector of a point of the surface and \(n\) is the unit normal vector at this point.

By the condition of the theorem, the domain \(\omega\) (the spherical image of the sought surface) lies entirely on one hemisphere. Without loss of generality, one may assume that this hemisphere is
\[ x_1^2+x_2^2+x_3^2=1,\quad x_3>0. \]
The domain \(\omega\) is projected from the origin of coordinates onto a certain domain \(\bar\omega\) in the plane \(x_3=1\), bounded by a curve \(\bar\gamma\) of class \(C^{2,\lambda}\). To each point \(n\) of the domain \(\omega\) under this projection there corresponds a certain point \((x_1,x_2)\) of the domain \(\bar\omega\) in the plane \(x_3=1\). In our assumption the numbers \(x_1,x_2\) may be regarded as coordinates on the surface \(F\).

By virtue of the homogeneity of the function \(H\), we have
\[ H(A_1,A_2,A_3)\,du\,dv = H(n_1,n_2,n_3)\lvert r_u\times r_v\rvert\,du\,dv = H(n_1,n_2,n_3)\,d\sigma, \]
where \(n_1,n_2,n_3\) are the components of the vector \(n\) (the unit normal to the surface \(F\)), and \(d\sigma\) is the area element of the surface \(F\).

Put \(\Phi(x)=(rx)\), where \(x\) is any nonzero vector directed into the domain \(\omega\), and \(r\) is the vector of a point of the surface \(F\) at which the normal \((n)\) has the direction of the vector \(x\). The function \(\Phi(x)\) is positively homogeneous of the first degree. The coordinates of points of the surface \(F\) are expressed through the derivatives of the function \(\Phi\). Namely,
\[ x_1'=\partial\Phi/\partial x_1,\qquad x_2'=\partial\Phi/\partial x_2,\qquad x_3'=\partial\Phi/\partial x_3. \]

Setting \(\Phi(x_1,x_2,1)=\varphi(x_1,x_2)\), we obtain

\[ x_1'=\varphi_1,\qquad x_2'=\varphi_2,\qquad x_3'=\varphi-x_1\varphi_2-x_2\varphi_2, \]

where \(\varphi_1\) and \(\varphi_2\) are the first derivatives of the function \(\varphi\) with respect to the variables \(x_1\) and \(x_2\), respectively. With the aid of these formulas for the coordinates of points of the surface \(F\), we find the expression for the area element

\[ d\sigma=-(\varphi_{11}\varphi_{22}-\varphi_{12}^2)\sqrt{1+x_1^2+x_2^2}, \]

where \(\varphi_{ij}\) are the second derivatives of the function \(\varphi\).

Noting that

\[ n_1=\frac{x_1}{\sqrt{1+x_1^2+x_2^2}},\qquad n_2=\frac{x_2}{\sqrt{1+x_1^2+x_2^2}},\qquad n_3=\frac{1}{\sqrt{1+x_1^2+x_2^2}}, \]

we obtain

\[ I(F)=-\iint_{\bar{\omega}} h(x_1,x_2)(\varphi_{11}\varphi_{22}-\varphi_{12}^2)\,dx_1\,dx_2, \]

where, for brevity, we have denoted \(H(x_1,x_2,1)\equiv h(x_1,x_2)\).

Let us form the Euler equation for the functional \(I\). We obtain

\[ h_{22}\varphi_{11}-2h_{12}\varphi_{12}+h_{11}\varphi_{22}=0. \tag{*} \]

From the positive definiteness of the form \(d^2H\) follows the positive definiteness of the form \(d^2h\) and, consequently, the ellipticity of the equation for the function \(\varphi\).

The finding of the surface \(F\) solving our variational problem is reduced to solving equation \((*)\) in the domain \(\bar{\omega}\) with the boundary condition

\[ \varphi\big|_{\gamma}=\psi\sqrt{1+x_1^2+x_2^2}. \]

As is known, this boundary-value problem is always solvable in the class \(C^{2,\lambda}\), if the coefficients of the equation \((h_{ij})\) belong to \(C^{0,\lambda}\), and the curve \(\gamma\) bounding the domain is of class \(C^{2,\lambda}\) \((^1)\). In our case these conditions are satisfied.

If \(\varphi(x_1,x_2)\) is a solution of the boundary-value problem for equation \((*)\), then the surface \(F\) solving our variational problem is given by the equations

\[ x_1'=\partial\varphi/\partial x_1,\qquad x_2'=\partial\varphi/\partial x_2,\qquad x_3'=\varphi-x_1\partial\varphi/\partial x_1-x_2\partial\varphi/\partial x_2. \]

We note that the curvature of the surface \(F\) has the sign of the discriminant \(\varphi_{11}\varphi_{22}-\varphi_{12}^2\) and, consequently, by virtue of the ellipticity of equation \((*)\), is nonpositive, as was assumed. The theorem is proved.

Physical-Technical Institute
of Low Temperatures
Academy of Sciences of the Ukrainian SSR

Received
19 II 1968

CITED LITERATURE

\(^1\) K. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.