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ON THE ACCURACY OF A CARLEMAN ESTIMATE
A. G. Aslanyan
Consider the eigenvalue problem
\[ \Delta u+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}+c(x,y)u+\lambda u=0, \tag{1.1} \]
\[ u\big|_{\Gamma}=0, \]
where \(\Gamma\) is the smooth boundary of a simply connected domain \(G\); \(a(x,y)\), \(b(x,y)\), \(c(x,y)\) are functions, acting and continuous in \(G\). As is well known, the spectrum of problem (1.1) is discrete.
In [1] Carleman showed that the eigenvalues of this non-self-adjoint problem lie in the complex \(\lambda\)-plane inside a certain parabola
\[ \sigma=\alpha\tau^2+\beta, \tag{1.2} \]
where \(\lambda=\sigma+i\tau\); \(\alpha\) and \(\beta\) are real constants \((\alpha>0)\), depending on the coefficients of equation (1.1). However, the question of the accuracy of this estimate has until now remained open. Moreover, since for constant \(a(x,y)\) and \(b(x,y)\) the eigenvalues of problem (1.1) are real, one might have supposed that estimate (1.2) is not sharp and that, for example, for sufficiently smooth functions \(a(x,y)\) and \(b(x,y)\) the eigenvalues of problem (1.1) lie closer to the real axis.
The following simple example shows that Carleman’s estimate is sharp. Consider the equation
\[ \Delta u+y\frac{\partial u}{\partial x}-x\frac{\partial u}{\partial y}+\lambda u=0, \tag{1.3} \]
and let
\[ u\big|_{\Gamma}=0, \tag{1.4} \]
where \(\Gamma\) is the circle of unit radius. In polar coordinates, instead of equations (1.3) and (1.4) we have
\[ \frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \varphi^2}-\frac{\partial u}{\partial \varphi}+\lambda u=0, \tag{1.5} \]
\[ u(1,\varphi)=0. \tag{1.6} \]
We shall seek a solution of equation (1.5) in the form
\[ u(r,\varphi)=v(r)e^{in\varphi},\quad n=0,\pm1,\pm2,\ldots \tag{1.7} \]
Substituting \(u(r,\varphi)\) from (1.7) into equation (1.5), we obtain, for determining the function \(v(r)\), Bessel’s equation; solving it, we find that
\[ v(r)=J_{|n|}\bigl(\sqrt{\lambda-in}\,r\bigr). \]
From the boundary condition (1.6) we conclude that the eigenvalues of problem (1.3), (1.4) are the roots of the equation
\[ J_{|n|}\bigl(\sqrt{\lambda-in}\bigr)=0. \tag{1.8} \]
It follows that the eigenvalues of problem (1.3), (1.4) have the form
\[ \lambda_{n,j}=\chi^2_{|n|,j}+in,\quad n=0,\pm1,\pm2,\ldots,\quad j=1,2,\ldots, \tag{1.9} \]
where \(x_{|n|,j}\) \((j=1,2,\ldots)\) is the \(j\)-th zero of the Bessel function of index \(|n|\). Using the estimate
\[ x_{|n|,1}=|n|+c_0 |n|^{1/3}+O\left(|n|^{-1/3}\right) \tag{1.10} \]
(see, for example, [2]), we obtain that the eigenvalues of problem (1.3), (1.4) corresponding to \(j=1\) lie on a curve asymptotically approaching a parabola of the form (1.2). The remaining eigenvalues, corresponding to the values \(j=2,3,\ldots\), lie inside this curve.
Thus, the example constructed proves that Carleman’s estimate is sharp.
It is curious that for \(n=0;\ j=1,2,\ldots\) we obtain an infinite series of real eigenvalues of the non-self-adjoint problem (1.3), (1.4) (they correspond to the simple eigenvalues of the unperturbed problem \(\Delta u+\lambda u=0\)). The remaining eigenvalues of problem (1.3), (1.4) (\(|n|>0;\ j=1,2,\ldots\)) are obtained as a result of the splitting of double eigenvalues of the unperturbed problem and are arranged along straight lines parallel to the real axis, in symmetric pairs.
References
- Carleman T. Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen, Ber. Verh. Sächs. Akad., 88, 1936.
- Watson G. N. Theory of Bessel Functions, Part 1. IL, 1949.
Received by the editors
May 27, 1965
Moscow Institute of Physics and Technology
AT 14558. Submitted for typesetting 23/VIII 1965. Signed for printing 15/X 1965.
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