M. L. GERver

ON THE POSSIBLE RATE OF DECREASE OF A SOLUTION OF AN ELLIPTIC EQUATION

(Presented by Academician I. G. Petrovskii, 23 I 1964)

In this note we generalize a theorem of E. M. Landis on the possible rate of decrease of a solution of an elliptic equation.

1. This theorem of Landis was proved in the work \((^{1})\) and consists of the following. In the half-strip \(D_h\) \((x \ge 0,\ -h<y<h),\ h<1\), consider the equation

\[ A(x,y)\frac{\partial^2 u}{\partial x^2} +2B(x,y)\frac{\partial^2 u}{\partial x\,\partial y} +C(x,y)\frac{\partial^2 u}{\partial y^2} +D(x,y)\frac{\partial u}{\partial x} +E(x,y)\frac{\partial u}{\partial y} +F(x,y)u=0. \tag{1} \]

It is assumed that, for any real \(\xi\) and \(\eta\),

\[ A(x,y)\xi^2+2B(x,y)\xi\eta+C(x,y)\eta^2 >\alpha(\xi^2+\eta^2),\qquad \alpha>0. \tag{2} \]

The coefficients \(A,B\), and \(C\) are twice continuously differentiable; \(D\) and \(E\) are continuously differentiable; \(A,B\), and \(C\) themselves, their first and second partial derivatives, and \(D\) and \(E\) themselves and their first partial derivatives are bounded in modulus by unity; \(-1<F\le 0\). Under the formulated restrictions, for a solution \(u\) satisfying the requirements \(|u|<1\) in \(D_h\), \(u(0,y)>a(2^{-8}>a>0)\), the inequality

\[ M(x)>2^{-2kx/h+\log_2\log_2^{1/a}}\qquad (x>1), \]

is proved, where

\[ M(x)=\sup_{x'\ge x,\,-h<y<h}|u(x',y)|, \]

and \(K\) is a constant depending only on the constant \(\alpha\) in inequality (2).

The sharpness of the estimate is shown by the example of the harmonic function \(\operatorname{Re} e^{-e^{\pi z/4h}}\).

We shall prove that, if one considers the self-adjoint equation

\[ \frac{\partial}{\partial x}\left[A(x,y)\frac{\partial u}{\partial x} +B(x,y)\frac{\partial u}{\partial y}\right] +\frac{\partial}{\partial y}\left[B(x,y)\frac{\partial u}{\partial x} +C(x,y)\frac{\partial u}{\partial y}\right]=0, \tag{3} \]

then the theorem formulated above is valid under less restrictive assumptions: it is sufficient to assume that \(A,B\), and \(C\) are measurable, bounded in modulus by unity, and satisfy inequality (2). By a solution of equation (3) in a domain \(D\) we mean a function \(u\in W_2^1\), continuous in \(D\), satisfying the identity

\[ \iint_{D'} \left[ \left(A\frac{\partial u}{\partial x} +B\frac{\partial u}{\partial y}\right)\frac{\partial \varphi}{\partial x} +\left(B\frac{\partial u}{\partial x} +C\frac{\partial u}{\partial y}\right)\frac{\partial \varphi}{\partial y} \right]\,dx\,dy=0, \]

where \(D'\) is an arbitrary domain with smooth boundary, \(\overline{D'}\subset D\), and \(\varphi\) is an arbitrary continuous function in \(\overline{D'}\) from \(W_2^1\), \(\varphi=0\) on the boundary of \(D'\).

In what follows the following notation is used everywhere: \(P_h\) is the strip \(\{-h<y<h\}\), \(h<1\); \(\Gamma_1\) and \(\Gamma_2\) are two smooth arcs lying in \(P_h\), each of which has endpoints on both sides of \(P_h\); \(\Gamma_1\) and \(\Gamma_2\) do not intersect, and \(\Gamma_2\) is situated to the right of \(\Gamma_1\); \(G\) is the part of \(P_h\) situated between \(\Gamma_1\) and \(\Gamma_2\); \(G_i\) is the part of \(P_h\) situated to the right of \(\Gamma_i\), \(i=1,2\); \(X\) is the abscissa of the rightmost point of \(\Gamma_2\); \(G_h\) is the part of \(P_h\) situated between \(\Gamma_1\) and the line \(x=X+h\). The letter \(M\) (with indices) denotes positive constants depending only on the constant \(\alpha\) in inequality (2).

By \(\mu_2 G\) we denote the area of the domain \(G\), and by \(\mu_1\gamma\) the length of the curve \(\gamma\).

2. Let us analyze the author’s proof of Landis’ theorem. The theorem is obtained as a simple consequence of Lemma 1.

Lemma 1. Let \(z>2^{10}h\) be an arbitrary number; let \(u\) be a solution of equation (1) in \(G_1\), having the following properties: \(u>a(2^{-8}>a>0)\) on \(\Gamma_1\), \(|u|<2^{-2z/h+\log_2\log_2 1/a}\) in \(G_2\), \(|u|<1\) in \(G_1\). Then \(\mu_2G>M_1zh\).

In turn, Lemma 1 is derived from Lemmas 2 and 3.

Lemma 2. Let the domain \(D\) lie inside the circle \(K_R\) of radius \(R<1\), contain the center \(O\) of the circle, and have boundary points on the boundary of the circle; let \(\Gamma\) be that part of the boundary of \(D\) which is situated strictly inside \(K_R\). Then for certain \(M_2\) and \(M_2^*\), if \(\mu_2D<M_2^*R^2\), then for any solution \(u\) of equation (1), continuous in \(\overline D\) and satisfying the conditions \(u=0\) on \(\Gamma\), \(u(0)>0\), the inequality
\[ \sup_D u>u(0)e^{M_2R^2/\mu_2D} \]
holds.

Lemma 3. Let \(u\) be a solution of equation (1) in some domain \(D\supset \overline G_h\), satisfying the conditions \(u=a>0\) on \(\Gamma_1\), \(u=a/2\) on \(\Gamma_2\), \(u>-a\) in \(\overline G_h\). Then \(\mu_2G>M_3h^2\).

Lemma 2 follows from Lemma \(2'\) and the maximum principle.

Lemma \(2'\). Let \(R,O,D\) and \(\Gamma\) have the same meaning as in Lemma 2. Then for some \(M_2'\), if \(\mu_2D<M_2'R^2\), then for any positive solution \(u\) of equation (1), continuous in \(\overline D\) and vanishing at \(0\) on \(\Gamma\), the inequality
\[ u(0)<\frac12\sup_D u \]
holds.

In Landis’ work \((^2)\), later than \((^1)\), Lemma \(2'\) was proved for the self-adjoint equation (3) under the minimal restrictions on the coefficients formulated above. Thus Lemma 2 also turns out to be valid for equation (3). (In this case, for the self-adjoint equation, the restriction \(R<1\) ceases to be essential—Lemmas 2 and \(2'\) for equation (3) are true for any \(R\).)

Thus, in order to generalize Landis’ theorem, it is sufficient to prove Lemma 3 for equation (3). (Similarly to Lemmas 2 and \(2'\), Lemma 3 for equation (3) is true without the assumption \(h<1\), and therefore Lemma 1 and the theorem being proved turn out to be true for any \(h>0\).)

3. We shall reduce Lemma 3 for equation (3), with minimal restrictions on the coefficients, to the following Lemma \(3'\) for equation (3), whose coefficients \(A,B\), and \(C\) are continuously differentiable, bounded in absolute value by one, and satisfy inequality (2); we shall denote such an equation by \((3')\).

Lemma \(3'\). Let \(u\) be a solution of equation \((3')\) in some domain \(D'\supset \overline G_h\), satisfying the conditions \(u>\frac78a>0\) on \(\Gamma_1\), \(u<\frac58a\) on \(\Gamma_2\), \(u>-a\) in \(\overline G_h\). Then \(\mu_2G>M_3h^2\).

Thus, suppose that Lemma \(3'\) for equation \((3')\) has been proved, and let \(u\) be a solution of equation (3), satisfying the conditions of Lemma 3.

Take such a domain \(D'\) that
\[ D\supset \overline{D'}\supset D'\supset \overline G_h . \]
Construct a sequence of equations defined in \(D'\):
\[ L^{(m)}u^{(m)}\equiv \frac{\partial}{\partial x}\left[ A^{(m)}(x,y)\frac{\partial u^{(m)}}{\partial x} + B^{(m)}(x,y)\frac{\partial u^{(m)}}{\partial y} \right]+ \]
\[ +\frac{\partial}{\partial y}\left[ B^{(m)}(x,y)\frac{\partial u^{(m)}}{\partial x} + C^{(m)}(x,y)\frac{\partial u^{(m)}}{\partial x} \right]=0, \qquad m=1,2,\ldots, \]
such that their coefficients are continuously differentiable and satisfy the conditions
\[ |A^{(m)}(x,y)|<1,\qquad |B^{(m)}(x,y)|<1,\qquad |C^{(m)}(x,y)|<1. \]
\[ A^{(m)}\xi^2+2B^{(m)}\xi\eta+C^{(m)}\eta^2>\alpha(\xi^2+\eta^2), \]
\[ \iint_{D'} \left[(A-A^{(m)})^2+(B-B^{(m)})^2+(C-C^{(m)})^2\right]\,dx\,dy \underset{m\to\infty}{\longrightarrow}0. \]

Let \(u^{(m)}\) be a solution of the equation \(L^{(m)}u^{(m)}=0\) in \(D'\), coinciding with \(u\) on the boundary \(D'\). The family \(\{u^{(m)}\}\), by the maximum principle, is uniformly bounded and, by the Giorgi theorem \((^3)\), is equicontinuous in \(G_h\). Therefore there exists a sequence \(u^{(m_k)}\), uniformly convergent in \(\overline{G}_h\) to some function \(u^*\). Since \(u^{(m)}\) converge in the mean to \(u\), we have \(u^*=u\). Choose the index \(k\) so large that the function \(u^{(m_k)}\) satisfies the conditions \(u^{(m_k)}> \frac78 a\) on \(\Gamma_1\), \(u^{(m_k)}< \frac58 a\) on \(\Gamma_2\), and \(u^{(m_k)}>-a\) in \(G_h\). Since \(L^{(m_k)}u^{(m_k)}=0\) is an equation of the form \((3')\), according to our assumption, \(\mu_2 G > M_3 h^2\).

1. In proving Lemma \(3'\) we use the following theorem of analysis, which is an analogue of the mean-value theorem for functions of two variables. (For functions of two variables this theorem was proved by me; it was then generalized by Landis and me to the case of \(n\) variables \((^4)\).)

Let, in the ring \(K\) in the \(x,y\)-plane, situated between two concentric squares with ratio of sides \(\varkappa\), there be defined a twice continuously differentiable function \(u(x,y)\) and a quadratic form
\[ A(x,y)\xi^2+2B(x,y)\xi\eta+C(x,y)\eta^2, \]
satisfying the inequality
\[ A(x,y)\xi^2+2B(x,y)\xi\eta+C(x,y)\eta^2>\alpha(\xi^2+\eta^2),\quad \alpha>0, \tag{2'} \]
the coefficients \(A,B\), and \(C\) being continuously differentiable in \(K\) and bounded in modulus by unity. Then there is a closed piecewise-smooth curve \(l\), separating the inner boundary of \(K\) from the outer one, such that
\[ \oint_l \left|\frac{\partial u}{\partial \nu}\right|\,ds < M\frac{\varkappa}{\varkappa-1}\operatorname{osc} u, \]
where
\[ \frac{\partial u}{\partial \nu} =(A\gamma_1+B\gamma_2)\frac{\partial u}{\partial x} +(B\gamma_1+C\gamma_2)\frac{\partial u}{\partial y} \]
(\(\gamma_1,\gamma_2\) are the direction cosines of the normal to \(l\)); \(\operatorname{osc}u=\max_{(x,y)\in K}u(x,y)-\min_{(x,y)\in K}u(x,y)\) is the oscillation of \(u\) in \(K\); \(M\) is a constant depending only on the constant \(\alpha\) in inequality \((2')\).

1. We proceed to the proof of Lemma \(3'\) for a solution \(u\) of equation \((3')\). Choose \(u_1\ge \frac78 a\) and \(u_2\le \frac58 a\) so that on the level sets \(u=u_1\) and \(u=u_2\) one has everywhere \(\operatorname{grad} u\ne0\). The level set \(u=u_i\) \((i=1,2)\) consists of a finite or countable number of smooth arcs. It separates \(\Gamma_1\) from \(\Gamma_2\) in \(P_h\). Therefore one can find a smooth arc \(\Gamma_i'\), on which \(u=u_i\) \((i=1,2)\), also separating \(\Gamma_1\) from \(\Gamma_2\) in \(P_h\). Denote by \(G'\) the part of \(P_h\) situated between \(\Gamma_1'\) and \(\Gamma_2'\). It can be shown that there exists a smooth arc \(\Gamma\), having endpoints on both sides of \(P_h\) and situated between \(\Gamma_1'\) and \(\Gamma_2'\), such that \(u\) is constant on \(\Gamma\), with \(u_1>u_\Gamma>u_2\), and for any arc \(\gamma\subseteq\Gamma\)
   \[ \int_\gamma \frac{\partial u}{\partial \nu}\,ds > \alpha\,\frac{(u_1-u_2)(\mu_1\Upsilon)^2}{\mu_2G'}. \]

Consider the point \(\mathscr P\) of intersection of \(\Gamma\) with the straight line \(y=0\), and construct two concentric squares \(K_h\) and \(K_{2h}\) with center at \(\mathscr P\) and with sides \(h\) and \(2h\). Let \(K\) be the closed ring enclosed between these squares. Let \(K_1\) be the component, containing the point \(\mathscr P\), of the subset of \(K_{2h}\) on which \(u\) is defined and satisfies the inequalities \(u_1>u>-a\). Every boundary point of \(K_1\) belongs either to the boundary of \(K_{2h}\), or to the level set \(u=u_1\); moreover, those boundary points of \(K_1\) at which \(u=u_1\) (and, consequently, \(\operatorname{grad} u\ne0\)) lie on no more than a countable set of smooth arcs \(\gamma^{(i)}\) \((i=1,2,\ldots)\). From the set \(K_1\) extend \(u\) in a twice continuously differentiable manner to the ring \(K\), preserving the inequality \(\operatorname{osc}u<2a\). Also extend to the ring \(K\) the coefficients \(A,B\), and \(C\) of equation \((3')\) in a continuously differentiable manner, preserving the inequalities \(|A|<1\), \(|B|<1\), \(|C|<1\) and
\[ A\xi^2+2B\xi\eta+C\eta^2>\alpha(\xi^2+\eta^2) \]
(the extensions of the functions \(u,A,B\), and \(C\) are denoted by the same letters \(u,A,B\), and \(C\)).

According to item 4, there exists a closed piecewise-smooth curve \(l\), separating the inner boundary \(K\) from the outer one and such that \(\displaystyle \oint_l \left|\frac{\partial u}{\partial \nu}\right|\,ds<4Ma\).

The curve \(l\) is intersected, obviously, by only a finite number of arcs \(\gamma^{(i)}\). The rightmost point of \(l\) lies to the left of the line \(x=X+h\). Therefore on \(l\) one can find a finite number of arcs \(l^{(i)}\), which, together with the arcs \(\gamma^{(i)}\), bound a domain \(B\) containing the point \(P^0\) and lying, together with its boundary, inside \(D'\).

Denote by \(\gamma\) the smallest arc belonging to the arc \(\Gamma\) and having its endpoints on the curve \(l\). The arc \(\gamma\) divides the domain \(B\) into two domains. Denote by \(B_0\) that one of them for which the outward normal to the boundary \(\beta_0\) coincides on \(\gamma\), in direction, with \(\operatorname{grad} u\). Denote \(\beta_0\cap l\) by \(l_0\), and \(\beta_0\cap\left(\bigcup_i \gamma^{(i)}\right)\) by \(\gamma_0\). Since on \(\gamma_0\) the outward normal coincides in direction with \(\operatorname{grad} u\), we have \(\displaystyle \int_{\gamma_0}\frac{\partial u}{\partial \nu}\,ds>0\); since \(l_0\) is part of \(l\), we have \(\displaystyle \int_{l_0}\left|\frac{\partial u}{\partial \nu}\right|\,ds<4Ma\); finally, since \(\mu_1\gamma\ge h\), \(u_1-u_2\ge a/4\), and \(\mu_2G'<\mu_2G\), we have \(\displaystyle \int_{\gamma}\frac{\partial u}{\partial \nu}\,ds>\frac{\alpha ah^2}{4\mu_2G}\).

Since \(B_0\subset D'\), the equality holds

\[ \iint_{B_0}\left\{ \frac{\partial}{\partial x}\left[A(x,y)\frac{\partial u}{\partial x} +B(x,y)\frac{\partial u}{\partial y}\right]+ \frac{\partial}{\partial y}\left[B(x,y)\frac{\partial u}{\partial x} +C(x,y)\frac{\partial u}{\partial y}\right]\right\}\,dx\,dy=0. \]

Applying Green’s formula to its left-hand side, we obtain \(\displaystyle \int_{\beta_0}\frac{\partial u}{\partial \nu}\,ds=0\).

Thus,

\[ \int_{\gamma\cup\gamma_0}\frac{\partial u}{\partial \nu}\,ds = \left|\int_{l_0}\frac{\partial u}{\partial \nu}\,ds\right| \le \int_{l_0}\left|\frac{\partial u}{\partial \nu}\right|\,ds, \]

or \(\alpha h^2/4\mu_2G<4M\). Denoting \(\alpha/16M\) by \(M_3\), we obtain \(\mu_2G>M_3h^2\), as was required to prove.

1. Up to now we have considered the homogeneous self-adjoint equation (3). The theorem is also valid for the equation

\[ \frac{\partial}{\partial x}\left[A(x,y)\frac{\partial u}{\partial x} +B(x,y)\frac{\partial u}{\partial y}\right] +\frac{\partial}{\partial y}\left[B(x,y)\frac{\partial u}{\partial x} +C(x,y)\frac{\partial u}{\partial y}\right] + D(x,y)\frac{\partial u}{\partial x} + E(x,y)\frac{\partial u}{\partial y} + F(x,y)u=0, \]

the coefficients being measurable and, in modulus, less than one. If, moreover, it is assumed that \(F\le 0\), then Lemmas 2 and 3 remain true and their proof changes only insignificantly. If nothing is assumed about the sign of \(F\), then the theorem is true for sufficiently small \(h\).

In conclusion I express my deep gratitude to E. M. Landis for posing the problem and for help in the work.

Moscow State University
named after M. V. Lomonosov

Received
21 I 1964

References

1. E. M. Landis, Uspekhi Mat. Nauk, 15, issue 1 (85), 21 (1959).
2. E. M. Landis, Uspekhi Mat. Nauk, 18, issue 1 (109), 3 (1963).
3. E. D. Giorgi, Mem. Acc. Sci. Torino, Ser. \(3^a\), \(3^0\), 25 (1957).
4. M. L. Gerver, E. M. Landis, Dokl. Akad. Nauk SSSR, 146, No. 4, 761 (1962).