Reports of the Academy of Sciences of the USSR

1. Volume 134, No. 5

MATHEMATICS

I. Ya. Bakelman

THE FIRST BOUNDARY-VALUE PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS

(Presented by Academician V. I. Smirnov on 28 V 1960)

1. Let \(\Omega\) be a domain in the plane \(x,y\), bounded by a closed convex curve \(\Gamma\). With respect to \(\Gamma\) we shall assume that it is given parametrically by functions \(x=x(s)\), \(y=y(s)\) that are three times continuously differentiable (\(s\) is the arc length on \(\Gamma\)), and that its curvature at all points is not less than \(\varkappa_0=\operatorname{const}>0\).

In the domain \(\Omega+\Gamma\) consider the first boundary-value problem for the quasilinear elliptic equation

\[ A(x,y,z,p,q)r+2B(x,y,z,p,q)s+C(x,y,z,p,q)t=D(x,y,z,p,q) \tag{1} \]

with zero boundary condition.

We shall assume that the functions \(A,B,C,D\) are continuously differentiable in all the variables \(x,y,z,p,q\), and that their first derivatives satisfy a Hölder condition with exponent \(0<\beta\leq 1\) with respect to \((x,y)\in\Omega+\Gamma\) and with respect to \(z,p,q\) for all finite values of these variables.

Let

\[ |D(x,y,0,p,q)|\leq \varphi(x,y)R\left(\sqrt{p^2+q^2}\right), \]

where \(\varphi(x,y)\geq 0\) and \(R\left(\sqrt{p^2+q^2}\right)>0\) are continuous functions of their variables, the first in the domain \(\Omega+\Gamma\), the second in the plane \(p,q\).

Using the function \(\varphi(x,y)\), we construct several auxiliary functions. Let \(M\) be an arbitrary point of the curve \(\Gamma\). Denote by \(K_M\) the circle of smallest radius that is tangent to \(\Gamma\) at the point \(M\) and contains the domain \(\Omega+\Gamma\) in its interior. Let \(O_M\) be the center of this circle and \(r_M\) its radius. On the interval \([0,r_M]\) we construct the function

\[ f_M(\rho)=\max \varphi(x,y), \]

where the greatest value of the function \(\varphi(x,y)\) is taken on the arc of the circle with center at \(O_M\) and radius \(0\leq \rho\leq r_M\), lying inside \(\Omega+\Gamma\). Let \(M\) still be an arbitrary point on \(\Gamma\), and let \(t_M\) be the tangent to \(\Gamma\) at the point \(M\). Project the domain \(\Omega+\Gamma\) onto the line \(l_M\) perpendicular to \(t_M\). Introduce in the plane \(x,y\) a new Cartesian coordinate system \(u,v\) so that the lines \(l_M,t_M\) are respectively the \(u\)- and \(v\)-axes. Let \(u_M^{(1)}\) and \(u_M^{(2)}\) \((u_M^{(1)}<u_M^{(2)})\) be the endpoints of the segment on the \(u\)-axis* which is the projection of the domain \(\Omega+\Gamma\). Put

\[ \mu_M(u)=\max_v \varphi(u,v), \]

* One of the numbers \(u_M^{(1)},u_M^{(2)}\) is equal to zero.

where the greatest value of the function \(\varphi(u,v)\) is taken over all points of the domain \(\Omega+\Gamma\) with a common projection onto the straight line \(u\). The constructed function is defined on the interval \([u_M^{(1)},u_M^{(2)}]\).

The following theorems on the solvability of the Dirichlet problem for equation (1) hold.

Theorem 1. Suppose that in the domain \(\Omega+\Gamma\) there is given the equation
\[ A(x,y,p,q)r+2B(x,y,p,q)s+C(x,y,p,q)t=D(x,y,z,p,q), \tag{2} \]
whose coefficients satisfy the conditions considered above, as well as the following requirements:

a) The inequality holds
\[ A(x,y,p,q)\xi^2+2B(x,y,p,q)\xi\eta+C(x,y,p,q)\eta^2 \ge \]
\[ \ge \alpha\!\left(\sqrt{p^2+q^2}\right)(\xi^2+\eta^2), \]
where \(\alpha\!\left(\sqrt{p^2+q^2}\right)\) is a continuous function on the \(p,q\)-plane, strictly positive for all finite values of \(p,q\).

b) For all \((x,y)\in\Omega+\Gamma\) and arbitrary \(z,p,q\), the inequality
\[ D_z(x,y,z,p,q)\ge \operatorname{const}>0 \]
is valid.

c) For fixed \(x,y,z\), the ratios \(|D_x/D_z|\), \(|D_y/D_z|\), \(|C_x/C|\), \(|A_y/A|\), \(|D/D_z|\) are uniformly bounded as functions of \(z\).

d) One of the two relations is fulfilled
\[ \sup_{M\in\Gamma}\left\{\pi\int_0^{r_M} f_M(r)\,r\,dr\right\} < 2\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \frac{\alpha^2\!\left(\sqrt{p^2+q^2}\right)} {R^2\!\left(\sqrt{p^2+q^2}\right)} \,dp\,dq \tag{3} \]
or
\[ \sup_{M\in\Gamma}\int_{u_M^{(1)}}^{u_M^{(2)}} \mu_M(u)\,du < \min\left\{ \int_{-\infty}^{0}\frac{\alpha(\tau)}{R(\tau)}\,d\tau,\, \int_{0}^{+\infty}\frac{\alpha(\tau)}{R(\tau)}\,d\tau \right\}. \tag{4} \]

Then the first boundary value problem for equation (2) with zero boundary condition is uniquely solvable in the class of functions whose third derivatives satisfy in the domain \(\Omega+\Gamma\) a Hölder condition with exponent \(0<\beta'<\beta\).

Theorem 2. Suppose that all conditions of Theorem 1 are fulfilled, except condition b), which we replace by the following:

b′) Suppose that for all \(x,y\in\Omega+\Gamma\) and arbitrary \(z,p,q\), the inequality
\[ D_z(x,y,z,p,q)\ge 0 \]
is valid, the functions \(A,B,C,D\), for fixed \(x,y\) and \(z\), behave as polynomials in the variables \(p,q\) as \(p^2+q^2\to+\infty\), and the function \(D(x,y,z,p,q)\) is represented in the form
\[ D(x,y,z,p,q)=D_1(x,y,z,p,q)+D_2(x,y,z,p,q), \]
where
\[ D_1(x,y,z,p,q)\ge 0 \]
for all \((x,y)\in\Omega+\Gamma,\ z,p,q\);
\[ \frac{D_2}{Ap^2+2Bpq+Cq^2}\le k=\operatorname{const}<+\infty \]
for \((x,y)\in\Omega+\Gamma,\ p^2+q^2>1\), and arbitrary \(z\).

Then the conclusion of Theorem 1 holds.

Theorem 3. Suppose that in the domain \(\Omega+\Gamma\) the equation

\[ A(x,y,z,p,q)r+2B(x,y,z,p,q)s+C(x,y,z,p,q)t=D(x,y,z,p,q), \tag{5} \]

is given, whose coefficients satisfy the following conditions:

a) The functions \(A, B, C, D\) are continuously differentiable with respect to all variables, and their first derivatives satisfy the Hölder condition with exponent \(0<\beta\leqslant 1\) in \((x,y)\in\Omega+\Gamma\) and for all finite \(z,p,q\).

Further, for all admissible values of \(x,y,z,p,q\), the quadratic form

\[ A_z(x,y,z,p,q)\xi^2+2B_z(x,y,z,p,q)\xi\eta+C_z(x,y,z,p,q)\eta^2 \leqslant -\lambda_0(\xi^2+\eta^2) \]

\[ (\lambda_0=\mathrm{const}>0), \]

and, finally,

\[ A(x,y,0,p,q)\xi^2+2B(x,y,0,p,q)\xi\eta+C(x,y,0,p,q)\eta^2\geqslant \]

\[ \geqslant \alpha\!\left(\sqrt{p^2+q^2}\right)(\xi^2+\eta^2), \]

where \(\alpha\!\left(\sqrt{p^2+q^2}\right)\) is a continuous function on the \(p,q\)-plane, strictly positive for all finite values of \(p,q\).

b) For all admissible values of \(x,y,z,p,q\) we have

\[ D_z(x,y,z,p,q)\geqslant 0,\qquad D(x,y,z,p,q)\geqslant \mathrm{const}>0. \]

c) Condition c) of Theorem 1 is fulfilled and, moreover, \(A_y, A_z, A;\ C_x, C_z, C;\ D_x, D_y, D_z, D\) have the same orders of growth in \(p,q\).

d) Condition d) of Theorem 1 is fulfilled.

Then the first boundary-value problem with zero boundary condition for equation (5) has at least one solution in the class of functions whose third derivatives satisfy the Hölder condition in the domain \(\Omega+\Gamma\) with exponent \(\beta'<\beta\).

Let us note that if the function \(\mu_M(u)\) is constructed so that its graph is symmetric with respect to the ordinate passing through the midpoint of the interval \([u_M^{(1)},u_M^{(2)}]\) (for any point \(M\in\Gamma\)), then condition (4) may be replaced by the following:

\[ \sup_{M\in\Gamma}\int_{u_M^{(1)}}^{u_M^{(2)}} \mu_M(u)\,du < \int_{-\infty}^{+\infty}\frac{\alpha(\tau)}{R(\tau)}\,d\tau. \]

2. Theorems 1 and 2 represent a further development of S. N. Bernstein’s results on quasilinear elliptic equations of the form (2) (see \((^1)\)). S. N. Bernstein considers equations for which the coefficients \(A,B,C,D\) have coordinated orders of growth in \(p,q\) as \(p^2+q^2\to+\infty\). This coordination is expressed by the condition that for \(p^2+q^2>1\)

\[ \frac{|D|}{Ap^2+2Bpq+Cq^2}\leqslant k=\mathrm{const}<+\infty. \tag{6} \]

The first boundary-value problem for the equation

\[ (1+q^2)r-2pqs+(1+p^2)t=(1+p^2+q^2)^{3/2}, \]

which does not obey condition (6), is solvable, generally speaking, only in sufficiently small domains.

The integral conditions d) in Theorems 1 and 2 give sufficient conditions for the solvability of the first boundary-value problem for quasilinear elliptic equations, if the orders of growth of the coefficients with respect to the first derivatives do not obey S. N. Bernstein’s coordination condition. Let us note that in a number of cases these conditions are very close to necessary ones; for example, if

the domain \(\Omega\) is a circle of radius 1, and equation (2) has the form

\[ (1+q^2)r-2pqs+(1+p^2)t=(1+p^2+q^2)^{3/2}. \]

Geometrically, this concerns the construction of a surface with mean curvature equal to one. The solution of the problem under consideration will be a hemisphere of radius one.

Theorem 3 is an analogue of the results of J. Schauder \(\left({}^{2}\right)\), L. Nirenberg \(\left({}^{3}\right)\), and O. A. Ladyzhenskaya for equations

\[ A(x,y,z,p,q)r+2B(x,y,z,p,q)s+C(x,y,z,p,q)t=0 \]

in the general case of nonhomogeneous quasilinear elliptic equations.

1. Let us briefly outline the proof of the theorems stated. From the results of S. N. Bernstein \(\left({}^{1}\right)\), J. Schauder \(\left({}^{2}\right)\), and L. Nirenberg \(\left({}^{3}\right)\), it follows that, for this purpose, it suffices to prove the possibility of obtaining a priori estimates for the proposed solution in the space \(C^1\). We first obtain an a priori estimate for the modulus of the solution in \(\Omega+\Gamma\) and for its first derivatives on the boundary. To this end, a convex shell is stretched over the solution of the equation; it consists of two convex caps, with their convexities turned in opposite directions and joined to each other along the curve \(\Gamma\). Conditions b) and c) in Theorems 1 and 2, and conditions a), b), and c) in Theorem 3, make it possible to give an a priori estimate for the slopes of the supporting planes on the boundary of both caps; from this the a priori estimates required for us for the modulus and the first derivatives on the boundary follow easily. A priori estimates for the moduli of the first derivatives of the solution in the interior are obtained by means of S. N. Bernstein’s method of auxiliary functions.

After this, the completion of the proof of Theorems 1, 2, and 3 is obtained by reduction to Schauder’s topological principle, using the results of Nirenberg and Schauder on linear elliptic equations (this scheme is described in detail in \(\left({}^{3}\right)\)).

In conclusion we note that the results obtained above are naturally generalized to the case of a sufficiently smooth nonhomogeneous boundary condition; for lack of space we shall not dwell on them.

Leningrad Pedagogical Institute
named after A. I. Herzen

Received
17 V 1960

REFERENCES

\({}^{1}\) S. N. Bernstein, UMN, 8, 75 (1941).
\({}^{2}\) J. Schauder, Math. Zs., 34, No. 4 (1933).
\({}^{3}\) L. Nirenberg, Comm. Pure and Appl. Math., 6, No. 3 (1953).
\({}^{4}\) O. A. Ladyzhenskaya, DAN, 120, No. 5 (1958).