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**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”.

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)