## The diameter of the first nodal line of a convex domain.(English)Zbl 0831.35115

This paper gives estimations for the location of the nodal line of the second eigenfunction $$u_2$$ of a membrane (with fixed boundary) on a convex plane domain $$\Omega$$, as well as upper and lower bounds for the eigenvalues $$\lambda_1$$ and $$\lambda_2$$ of $$\Omega$$.
As a characteristic dimension for $$\Omega$$, the author defines the length $$L$$ of the rectangle contained in $$\Omega$$ with the smallest first eigenvalue. The eigenfunctions $$u_1$$ and $$u_2$$ are essentially supported near this “optimal” rectangle.
A most appealing auxiliary one-dimensional eigenvalue problem of Schrödinger’s type (corresponding to $$\Omega )$$ is defined in the following way. Let $$\Omega$$ be contained in the strip $$a < x < b$$ and let $$h(x_0)$$ be the length of the segment $$x = x_0$$ lying in $$\Omega$$. The auxiliary eigenvalue problem is $$\varphi''(x) + \{\mu - [\pi/h (x)]^2\} \varphi (x) = 0$$ with $$\varphi (a) = \varphi (b) = 0$$. Thus, the first eigenvalue $$[\pi/h(x)]^2$$ of the vertical homogeneous string plays the role of the potential in Schrödinger’s equation. The first eigenvalue $$\mu_1$$ of the auxiliary problem is a lower bound for $$\lambda_1$$. Of course, this bound depends on the orientation of the axes $$x$$ and $$y$$ with respect to the domain $$\Omega$$. Note that $$\mu_1$$ is also a lower bound for the first eigenvalue of the domain obtained by Steiner symmetrization of $$\Omega$$ with respect to the $$x$$-axis. We have $$\mu_1 = \lambda_1$$ if $$\Omega$$ is any rectangle with sides parallel to the axes.
This method is closely related (though different) to that of “one- dimensional auxiliary problems” initiated by L. E. Payne and H. F. Weinberger [J. Soc. Ind. Appl. Math. 5, 171-182 (1957; Zbl 0083.194)], who constructed lower bounds for $$\lambda_1$$ by applying Rayleigh’s principle to horizontal and vertical homogeneous vibrating strings as auxiliary problems. For the use of nonhomogeneous auxiliary strings, see the reviewer [Z. Angew. Math. Phys. 11, 387-413 (1960; Zbl 0104.414); J. Math. Phys. 43, 15-26 (1964; Zbl 0126.313)] and R. Klötzler’s book [Mehrdimensionale Variationsrechnung, (Birkhäuser Verlag and VEB, 1970; Zbl 0199.429), Section 27].
In the present paper upper bounds for $$\lambda_1$$ and $$\lambda_2$$ are constructed by using test functions of the form $$\varphi (x) \sin [\alpha (x)y + \beta (x)]$$. It is shown that, for an “optimal” orientation of the axes $$x$$ and $$y$$, the nodal line of $$u_2$$ lies near the line $$x = \overline x$$, where $$\overline x$$ is the unique zero of $$\varphi_2 (x)$$.
While the main ideas are very intuitive, most of the proofs are “technical”.
Reviewer: J.Hersch (Zürich)

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs 31A35 Connections of harmonic functions with differential equations in two dimensions

### Citations:

Zbl 0083.194; Zbl 0104.414; Zbl 0126.313; Zbl 0199.429
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