UDC 517.944

MATHEMATICS

I. Ya. Bakelman

QUASILINEAR ELLIPTIC EQUATIONS AND THE CONSTRUCTION OF A HYPERSURFACE WITH PRESCRIBED MEAN CURVATURE

(Presented by Academician V. I. Smirnov on 15 IV 1967)

1. One of the most interesting problems of geometry leading to boundary-value problems for quasilinear elliptic equations is the problem of constructing a surface from its mean curvature.

We give the formulation of this problem. Let \(E^{n+1}\) be \((n+1)\)-dimensional Euclidean space with Cartesian coordinate system \(x_1, x_2, \ldots, x_n; z\). We denote the plane \(z=0\) by \(E^n\). Points of \(E^{n+1}\) will be denoted briefly by \((x,z)\), and points of \(E^n\) by \(x\). Let \(\Omega\) be a bounded domain in \(E^n\), homeomorphic to an \(n\)-dimensional closed ball, and let \(\Gamma=\partial\Omega\). Suppose a function \(H(x)\) is prescribed in \(\Omega\), and a continuous function \(h(x)\) on \(\Gamma\). The problem of constructing a hypersurface from its mean curvature is reduced to constructing in \(\Omega\) a function \(z(x)\) such that \(z|_{\Gamma}=h(x)\) and the mean curvature of the surface \(\Phi_z\)—the graph of the function \(z(x,y)\)—coincides with the function \(H(x)\). Since for the mean curvature \(H\) of the surface \(\Phi_z\) the formula

\[ H=\sum_{i,k=1}^{n} G_{ik}(Dz) z_{ik}/n \left(1+\sum_{i=1}^{n} z_i^2\right)^{3/2}, \tag{1} \]

holds, where

\[ G_{ii}=1+\sum_{j\ne i} z_j^2,\quad G_{ik}=-z_i z_k,\quad z_i=\partial z/\partial x_i,\quad z_{ik}=\partial^2 z/\partial x_i\partial x_k, \]

the geometric problem formulated above is reduced to the solution of the following boundary-value problem:

\[ \sum_{i,k=1}^{n} G_{ik}(Dz) z_{ik} = nH(x)\left(1+\sum_{i=1}^{n} z_i^2\right)^{3/2}, \tag{2} \]

\[ z|_{\Gamma}=h(x). \tag{3} \]

Denote by \(P\) the \(n\)-dimensional Euclidean space with Cartesian coordinates \(p_1,p_2,\ldots,p_n\). A point (vector) of this space will be denoted by \(p\). As usual, \(|p|\) denotes the length of the vector \(p\), i.e.

\[ |p|=\left(\sum_{i=1}^{n} p_i^2\right)^{1/2}. \]

It is easily verified that for all \(p\in P\) and all real numbers \(\xi_1,\xi_2,\ldots,\xi_n\) the exact inequalities

\[ \sum_{i=1}^{n} \xi_i^2 \le \sum_{i,k=1}^{n} G_{ik}(p)\xi_i\xi_k \le (1+|p|^2)\sum_{i=1}^{n} \xi_i^2 \tag{4} \]

hold.

It follows from inequalities (4) that equation (2) is elliptic, but not uniformly elliptic. Furthermore, the right-hand side of equation (2), with respect to \(\sum_{i=1}^{n} z_i^2\), as \(\sum_{i=1}^{n} z_i^2\to+\infty\), has a higher order of growth than the expres-

\(\sum G_{ik}(Dz)\xi_i\xi_j,\ \sum G_{ik}z_i z_k\). These circumstances exclude equation (2) from the classes of quasilinear elliptic equations for which, in \((^2,^3)\), various boundary-value problems have been thoroughly studied. Let us note that the above-indicated properties of the functions \(G_{ik}(p)\) lead to adjoining necessary and sufficient conditions for solvability of the boundary-value problem (2)—(3), which are expressed by inequalities between the geometric characteristics of the domain \(\Omega\) and the properties of the function \(H(x)\) (see §§ 3, 4, 5, 6 of the present paper). These circumstances lead to the separation of a certain rather broad class of quasilinear elliptic equations, including both the equations studied in the above-cited works and equation (2). We shall denote this class of equations by \(H\).

2. On the class of quasilinear elliptic equations \(H\). Let, in the cylinder \(\Omega+\Gamma \times J \times P\) with Cartesian coordinates \(x_1,x_2,\ldots,x_n;\ z;\ p_1,\ldots,p_n\), where \(J=(-\infty,+\infty)\), the functions \(a_{ik}(x,z,p)\) \((i,k=1,\ldots,n)\), \(b(x,z,p)\) satisfy the following conditions: 1) for arbitrary \(\xi_1,\xi_2,\ldots,\xi_n\) in \(\Omega+\Gamma \times J \times P\) the inequality
\[ \sum_{i,k=1}^{n} a_{ik}(x,z,p)\xi_i\xi_k>0; \]

2) for all \(x\in\Omega+\Gamma,\ z\in J,\ p\in P\) the inequalities
\[ -\varphi_-(x)/R_-(x)\leq b(x,z,p)\,[\det\|a_{ik}(x,z,p)\|]^{-1/n}\leq \varphi_+(x)/R_+(p), \tag{5} \]
hold, where \(R_\pm(p)>0\) are locally summable to degree \(n\) in \(P\), and \(\varphi_\pm(x)\geq0\) belong to \(L_n(\Omega)\).

In this paper we shall restrict ourselves to the case where the function \(b[\det\|a_{ik}\|]^{-1/n}\) satisfies inequalities (5). However, with minor changes the results of the present paper carry over to the case where the function \(b[\det\|a_{ik}\|]^{-1/n}\) increases with respect to \(z\) for fixed values of the remaining variables.

Quasilinear equations of the form
\[ \sum_{i,k=1}^{n} a_{ik}(x,z,Dz)z_{ik}=b(x,z,Dz) \tag{6} \]
will be called equations of class \(H\) if the functions \(a_{ik}\) and \(b\) satisfy conditions 1), 2).

3. Necessary conditions for solvability of the first boundary-value problem. Let \(z(x)\in C^2(\Omega)\) be a solution of the boundary-value problem
\[ \sum_{i,k=1}^{n} a_{ik}(x,z,Dz)z_{ik}=b(x,z,Dz),\qquad z\big|_{\Gamma}=h(x), \tag{7} \]
where in (7) the equation belongs to the class \(H\), and \(h(x)\in C(\Gamma)\). Denote by \(u(x)\) and \(v(x)\) the convex functions stretched respectively from below and from above over the function \(z(x)\). (By this term, according to \((^1)\), one means
\[ u(x)=\sup_{W^+}\{w(x)\}, \]
where \(W^+\) is the totality of convex functions satisfying in \(\Omega\) the inequality \(w(x)\leq z(x)\) and directed by convexity downward. The function \(v(x)\) is defined analogously.) Let \(M_u\) and \(M_v\) be, respectively, the sets of interior points of \(\Omega\) at which \(u(x)=z(x)\) and \(v(x)=z(x)\). Then the following is valid.

Theorem 1. For any solution \(z(x)\in C^2(\Omega)\) of the boundary-value problem (7), the relations
\[ \int_{\nu_z(M_u)} R_+^n(p)\,dp \leq \frac{1}{n^n}\int_{M_u}[\varphi_+(x)]^n\,dx,\qquad \int_{\nu_z(M_v)} R_-^n(p)\,dp \leq \frac{1}{n^n}\int_{M_v}[\varphi_-(x)]^n\,dx, \tag{8} \]
necessarily hold, where \(\nu_z\) is the mapping of \(\Omega\) into \(P\) given by the formulas \(p_1=\partial z/\partial x_1,\ldots,p_n=\partial z/\partial x_n\).

4. Two-sided estimates of the solution of the boundary-value problem (7).
Let \(f(p)>0\) be a locally summable function in the space \(P\). Put
\[ g(\rho)=\int_{|p|\leq \rho} f(p)\,dp,\qquad A(f)=\int_P f(p)\,dp. \]
Since \(g(\rho)\) is a strictly increasing continuous function of \(\rho\) on \([0,+\infty]\), \(g(\rho)\) has an inverse function \(\rho=F(f,\tau)\), strictly increasing and continuous on the interval \([0,A(f))\), and
\[ \lim_{\tau\to A(f)} F(f,\tau)=+\infty . \]

Theorem 2. Let \(z(x)\in C^2(\Omega)\) be a solution of the boundary-value problem (7). Then, if for the functions \(\varphi_+(x)\), \(\varphi_-(x)\), \(R_+(p)\), \(R_-(p)\) the inequalities
\[ \omega_\pm=\frac{1}{n^n}\int_\Omega [\varphi_\pm(x)]^n\,dx<A(R_\pm^n), \tag{9} \]
hold, then for \(z(x)\) in \(\Omega+\Gamma\) the estimates
\[ h_2-F(R_+^n,\omega_+)d\leq z(x)\leq h_1+F(R_-^n,\omega_-)d, \tag{10} \]
are valid, where
\[ h_1=\sup_\Gamma h(x),\qquad h_2=\inf_\Gamma h(x),\qquad d \]
is the diameter of \(\Omega\).

For the problem with mean curvature (equation (2)) these conditions and estimates take the form
\[ \omega_\pm=\int_\Omega H_\pm^n(x)\,dx<\sigma_n^*, \tag{11} \]
\[ h_2-\left[\frac{\omega_+^{2/n}}{\sigma_n^{2/n}-\omega_+^{2/n}}\right]^{1/2}d \leq z(x)\leq h_1+\left[\frac{\omega_-^{2/n}}{\sigma_n^{2/n}-\omega_-^{2/n}}\right]^{1/2}d, \tag{12} \]
where \(\sigma_n\) is the volume of the \(n\)-dimensional ball of radius one. If \(\Omega\) is a ball, \(H(x)=H_0=\mathrm{const}>0\) in \(\Omega\) and \(z|_\Gamma=0\), then inequality (11) takes the form
\[ rH_0<1, \]
where \(r\) is the radius of the ball \(\Omega\). It is easy to show that inequality (11) in this special case is a necessary condition for solvability of the boundary-value problem (7) in the class of functions \(C^2(\Omega+\Gamma)\).

5. Estimates of the normal derivative on \(\Gamma\) for a solution of problem (7).
Let \(\Omega\) be a domain in \(E^n\), bounded by a continuously differentiable closed surface \(\Gamma\). Let \(S\) be an \((n-1)\)-dimensional surface in \(E^{n+1}\), constructed from the boundary condition
\[ x|_\Gamma=h(x)\in C^1(\Gamma). \]
Let \(X\) be any point of \(S\), and \(Q\) a hyperplane in \(E^{n+1}\) passing through the tangent plane \(T\) to \(S\) at the point \(X\) and leaving \(S\) above itself. Let
\[ \widetilde z=\sum_{i=1}^n a_i x_i+b \]
be the equation of \(Q\). The number
\[ M_+(S)=\sup_S\left\{\inf \sum_{i=1}^n a_i^2\right\}, \]
where the exact lower bound is taken over all planes \(Q\) passing through \(T\) and leaving \(S\) above themselves, will be called the lower bending of the surface \(S\). Similarly one introduces the upper bending \(M_-(S)\). If \(S\) and \(\Gamma=\partial\Omega\) belong to \(C^2\), and at each point \(x\in\Gamma\) all normal curvatures of this surface are bounded below by one and the same number \(\chi_0>0\), then the numbers \(M_\pm(S)\) are finite. Here and below we shall assume that the functions \(R_\pm(p)\), occurring in inequalities (5), depend on the length of the vector \(p\) and are continuous on \(P\). Then the functions
\[ N_\pm\bigl(|p|,R_\pm^n(|p|)\bigr)=\inf R_\pm^n(|p|), \]
where the exact lower bounds are taken respectively in balls of radii \(\sqrt{M_+}\) and \(\sqrt{M_-}\) with center at the point \(p\), are also continuous functions of \(|p|\).

\[ \text{* By } H_\pm(x)\geq 0 \text{ are denoted, respectively, the positive and negative parts of the function } H(x). \]

Theorem 3. If \(z(x)\in C^2(\Omega+\Gamma)\) is a solution of the boundary-value problem (7) and the surfaces \(S\) and \(\Gamma\in C^{2,\beta}\), and if the normal curvatures at each point of \(\Gamma\) are bounded below by one and the same number \(\chi_0\), then, provided the inequalities

\[ \psi_{\pm}\equiv {1\over n^n\chi_0^n}\sup_{\Omega}[\varphi_{\pm}(x)]^n < \int_{\dot P} N_{\pm}(|p|,R_{\pm}^n(|p|))\,dp \tag{13} \]

hold, the following estimates hold on \(\Gamma\):

\[ \sqrt{M_+(S)}+F(N_+,\psi_+)\ge {\partial z\over \partial n}\bigg|_{\Gamma} \ge -\sqrt{M_-(S)}-F(N_-,\psi_-). \tag{14} \]

6. The problem of constructing a hypersurface with prescribed mean curvature. For lack of space, in this article we shall confine ourselves only to formulations of existence theorems for the solution of the first boundary-value problem for equation (2), i.e. the case of constructing a hypersurface with prescribed mean curvature.

Theorem 4. Let the domain \(\Omega\) and the function \(h(x)\) prescribed on its boundary belong to \(C^{m,\beta}\) \((m\ge 3,\ 0<\beta<1)\), and suppose, moreover, that the normal curvatures at the boundary points of \(\Omega\) are bounded below by a constant \(\chi_0>0\). Suppose further that \(H(x)\in C^{m-2,\beta}\) and that the conditions

\[ H_{\pm}<\chi_0 T_n\bigl(\sqrt{M_{\pm}(S)}\bigr), \tag{15} \]

are satisfied, where

\[ H_+=\max\{\sup_{\Omega}H(x),0\},\qquad H_-=\max\{-\inf_{\Omega}H(x),0\}; \]

\[ T_n(Q)=\left[n\int_0^{+\infty}p^{\,n-1}\bigl[1+(p+Q)^2\bigr]^{-(n+2)/2}\,dp\right]^{1/n}. \]

Then the boundary-value problem of constructing a surface with prescribed mean curvature equal to \(H(x)\) in \(\Omega\) has in \(C^2(\Omega)\) a unique solution which, moreover, belongs to the space \(C^{m,\delta}(\Omega+\Gamma)\) \((0<\delta<\beta)\).

Condition (15) is a necessary solvability condition for the problem under consideration if \(\Omega\) is a ball, \(H(x)=\mathrm{const}\), and \(z|_{\partial\Omega}=\mathrm{const}\) (in this case \(M_{\pm}(S)=0\)). If one imposes on the boundary of the domain \(\Omega\) restrictions that are, in a certain sense, stronger, then one can obtain conditions under which the problem under consideration is solvable for arbitrary regular boundary conditions.

Theorem 5. Let the domain \(\Omega\) and the function \(h(x)\) prescribed on its boundary belong to \(C^{m,\beta}\) \((m\ge 3,\ 0<\beta<1)\). Suppose further that \(H(x)\in C^{m-2,\beta}\) and that at all points of \(\Gamma=\partial\Omega\) the inequality

\[ L\ge {n\over n-1}\sup_{\Omega+\Gamma}|H(x)| \]

holds, where \(L\) is the mean curvature of \(\Gamma\) in the direction of the inward normal to \(\Gamma\). Then the boundary-value problem of constructing a surface whose mean curvature coincides with \(H(x)\) and whose boundary is \(z|_{\partial\Omega}=h\) has in \(C^2\) a unique solution which, moreover, belongs to \(C^{m,\delta}(\Omega+\Gamma)\) \((0<\delta<\beta)\).

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
5 IV 1967

REFERENCES

1. I. Ya. Bakel′man, Geometric Methods for Solving Elliptic Equations, “Nauka,” 1965. 2. N. Bernstein, Collected Works, vol. 3, 1960. 3. O. A. Ladyzhenskaya, N. N. Ural′tseva, Linear and Quasilinear Elliptic Equations, “Nauka,” 1964.