1 Introduction

The estimates of Neumann eigenvalues in convex domains \(\Omega \subset \mathbb R^n\) are given in classical works [22, 23], but the estimates in non-convex domains is a long-standing complicated problem which was partially solved in [12]. The method suggested in [12] is based on the geometric theory of composition operators on Sobolev spaces [11, 30] and allows to obtain spectral estimates in non-convex domains [13].

In [9] we studied connections between quasiconformal mappings, corresponding extension operators of Sobolev spaces and Neumann eigenvalues of the Laplacian (principal frequencies of free homogeneous membranes) in non-convex domains \(\Omega \subset \mathbb R^n\). The present article is focused on applications of the theory of extension operators of Sobolev spaces to the Neumann eigenvalue problem for the two-dimensional divergence form elliptic operators

$$\begin{aligned} L_{A}(u)(z)=-\text {div} [A(z) \nabla u(z)], \quad z=(x,y)\in \Omega , \quad \left\langle A(z) \nabla u(z), n \right\rangle \big |_{\partial \Omega }=0, \end{aligned}$$

in bounded simply connected domains \(\Omega \subset \mathbb R^2\). We assume that the matrix A is defined a.e. in \(\mathbb {R}^2\) and satisfies the following regularity conditions (see, for example, [2, 5]):

  1. 1)

    The matrix A belongs to the class of all \(2 \times 2\) symmetric matrix functions \(A(w)=\left\{ a_{kl}(w)\right\} \) with measurable entries \(a_{kl}(w)\) defined in \(\mathbb {R}^2\) that satisfied to the additional condition \(\text {det} A=1\) a.e. in \(\mathbb R^2\).

  2. 2)

    The matrix A satisfies to the uniform ellipticity condition: there exists \(1\le K< \infty \) such that the inequality

    $$\begin{aligned} \frac{1}{K}|\xi |^2 \le \left\langle A(w) \xi , \xi \right\rangle \le K |\xi |^2, \,\,\, \text {a.e. in}\,\,\, \mathbb R^2, \end{aligned}$$
    (1)

    holds for every \(\xi \in \mathbb R^2\).

The uniform ellipticity condition implies the solvability of the Beltrami equation, and therefore with these conditions the matrix A induces a quasiconformal mapping \(\varphi _A:\mathbb R^2\rightarrow \mathbb R^2\) [2] which we call as an A-quasiconformal mapping [10] (see Sect. 2.2 for the exact definition). In the present work we study Neumann eigenvalues in an A-quasidisc \(\mathbb {D}_A\) that can be considered as the unit discs in the measurable Riemannian metrics, generated by the matrix A. Let us give the formal definition of A-quasidiscs:

Definition 1

Let \(\mathbb D\subset \mathbb R^2\) be the unit disc. Then a domain \(\mathbb {D}_A\) is called an A-quasidisc if there exists an A-quasiconformal mapping \(\varphi _A:\mathbb R^2\rightarrow \mathbb R^2\) such that \(\varphi _A (\mathbb {D}_A)=\mathbb D\). Since the quasiconformal mappings \(\varphi _A\) are defined by the composition factor of conformal mappings we assume that \(|\mathbb {D}_A|=|\mathbb D|=\pi \).

Remark 1

Recall that a quasidisc is the image of the unit disc \(\mathbb {D}\) under a quasiconformal mapping \(\varphi :\mathbb R^2\rightarrow \mathbb R^2\). Since any A-quasiconformal mapping \(\varphi _A:\mathbb R^2\rightarrow \mathbb R^2\) is a quasiconformal one then an A-quasidisc is a quasidisc and satisfy to Ahlfors’s 3-point condition [1] also. We use the concept of the A-quasidisc because of its relation with the matrix A.

Since the matrix A satisfies the uniform ellipticity condition, the functional

$$\begin{aligned} F_A(\nabla u,\Omega )=\left( \iint \limits _\Omega \left\langle A(z)\nabla u(z),\nabla u(z)\right\rangle \,dxdy \right) ^{\frac{1}{2}}, \end{aligned}$$

defines an equivalent seminorm in the Dirichlet space \(L^{1,2}(\Omega )\) [10]. We denote the Dirichlet space \(L^{1,2}(\Omega )\) with this new seminorm as \(L_{A}^{1,2}(\Omega )\). It is well known that quasiconformal mappings \(\varphi :\Omega \rightarrow \widetilde{\Omega }\) between domains \(\Omega ,\widetilde{\Omega }\subset \mathbb R^2\) induces an isomorphism between \(L^{1,2}(\widetilde{\Omega })\) and \(L^{1,2}(\Omega )\) according to the standard chain rule \(\varphi ^{*}(f)=f\circ \varphi \) [28]. In [10] we have refined this result for the case of the spaces \(L^{1,2}_A(\widetilde{\Omega })\) and \(L^{1,2}(\Omega )\) and on this basis we obtained estimates of semi-stability of the Neumann eigenvalues under quasiconformal mappings. Namely in [10] we proved that the quasiconformal mapping \(\varphi _A\) (induced by the matrix A) is an isometry between spaces \(L^{1,2}_A(\widetilde{\Omega })\) and \(L^{1,2}(\Omega )\) i.e it is a conformal mapping between the standard Euclidean metric and the measurable Riemannian metric induced by the matrix A.

In the present work we establish a connection between the Neumann eigenvalue problem of elliptic operators in the divergence form and the smallest-circle problem. The smallest-circle problem is a mathematical problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane. This connection will be given in the form of sharp lower estimates for the first non-trivial Neumann eigenvalues via the radii of the smallest enclosing discs (Theorem A).

Recall that according to the Min–Max Principle (see, for example, [8, 21]) the first non-trivial Neumann eigenvalue \(\mu _1(A,\Omega )\) of the divergence form elliptic operator \(L_{A}(u)(z)\) can be represented as

$$\begin{aligned} \mu _1(A,\Omega )=\min \left\{ \frac{\Vert u \mid L^{1,2}_{A}(\Omega )\Vert ^2}{\Vert u \mid L^{2}(\Omega )\Vert ^2}: u \in W^{1,2}_{A}(\Omega ) \setminus \{0\},\,\, \iint \limits _{\Omega }u\, dxdy=0 \right\} . \end{aligned}$$

Hence, the value \(\mu _1(A,\Omega )^{-\frac{1}{2}}\) is the best constant \(B_{2,2}(A,\Omega )\) in the following Poincaré type inequality

$$\begin{aligned} \inf \limits _{c \in \mathbb R} \Vert u-c \mid L^2(\Omega )\Vert \le B_{2,2}(A,\Omega ) \Vert u \mid L^{1,2}_{A}(\Omega )\Vert , \quad u \in W^{1,2}_{A}(\Omega ). \end{aligned}$$

Let us remark that by the uniform ellipticity condition \(B_{2,2}(A,\Omega )\le \sqrt{K} B_{2,2}(I,\Omega )\). Hence, by using the isometry of spaces \(L^{1,2}_A(\widetilde{\Omega })\) and \(L^{1,2}(\Omega )\) we obtain more sharp estimates than estimates that can be obtained by using uniform ellipticity condition and known estimates in domains \(\Omega \).

The methods based on applications of Poincaré type inequalities to spectral problems of elliptic operators goes back to classical works [23, 27]. Spectral estimates of Neumann eigenvalues in convex domains \(\Omega \subset \mathbb R^n\) were obtained in [22]. We also note works [3, 4, 18] devoted to the spectral theory of elliptic operators.

The main result of the article gives a connection between the first non-trivial Neumann eigenvalue of the elliptic operator \(L_{A}(u)(z)\) and the smallest-circle problem. Namely:

Theorem A

Let \(\mathbb D_A\) be an A-quasidisc. Then the following inequality holds

$$\begin{aligned} \mu _1(A,\mathbb {D}_A) \ge \frac{1}{4K} \cdot \left( \frac{j'_{1,1}}{R_{\mathbb {D}_A}}\right) ^2, \end{aligned}$$

where K is the ellipticity constant of the matrix A, \(R_{\mathbb {D}_A}=\max \limits _{\mathbb {D}(0,1)}|\varphi _A^{-1}(x)-\varphi _A^{-1}(0)|\) and \(j'_{1,1}\approx 1.84118\) denotes the first positive zero of the derivative of the Bessel function \(J_1\).

Recall that the mapping \(\varphi _A\) is the A-quasiconformal mapping \(\varphi _A:\mathbb R^2\rightarrow \mathbb R^2\), \(\varphi _A(\mathbb D_A)=\mathbb D\), \(|\mathbb D_A|=|\mathbb D|=\pi \).

In this work we give estimates of exponential type for the radius \(R_{\mathbb {D}_A}\) of the smallest disc that contains \(\mathbb {D}_A\) (see Theorem 2). Such type estimates represent a partial solution of the smallest-circle problem (initially suggested by J. J. Sylvester in 1857).

The exponential type estimates of the radius \(R_{\mathbb {D}_A}\) leads to the following estimates of the first non-trivial Neumann eigenvalue of the elliptic operator \(L_{A}(u)(z)\) in quasiconformal terms:

Theorem B

Let \(\mathbb D_A\) be an A-quasidisc. Then the following inequality holds

$$\begin{aligned} \mu _1(A,\mathbb {D}_A) \ge \frac{ 64 (j'_{1,1})^2}{K \exp \{2\pi K\}}, \end{aligned}$$

where K is the ellipticity constant of the matrix A and \(j'_{1,1}\approx 1.84118\) denotes the first positive zero of the derivative of the Bessel function \(J_1\).

Let us remind that a simply connected domain \(\Omega \subset \mathbb R^2\) is called a Sobolev \(L^{1,2}_A\)-extension domain if there exists a continuous extension operator

$$\begin{aligned} E:L^{1,2}_{A}(\Omega ) \rightarrow L_A^{1,2}(\mathbb R^2). \end{aligned}$$

The existence of this extension operator does not depend on the matrix A, but only its norm depends on the matrix A. The quasidiscs [19] represent an important subclass of Sobolev \(L^{1,2}\)-extension domains [29].

The proof of Theorem A is based on the following general result:

Theorem C

Let \(\Omega \) and \(\widetilde{\Omega }\) be quasidiscs in \(\mathbb R^2\). If \(\widetilde{\Omega }\supset \Omega \) then

$$\begin{aligned} \mu _1(A,\Omega ) \ge \frac{\mu _1(A,\widetilde{\Omega })}{\Vert E_{\Omega }\Vert ^2}, \end{aligned}$$

where \(\Vert E_{\Omega }\Vert \) denotes the norm of the linear continuous extension operator

$$\begin{aligned} E_{\Omega }: L^{1,2}_A(\Omega )\rightarrow L_A^{1,2}(\widetilde{\Omega }). \end{aligned}$$

Theorem C gives the quasi-monotonicity property of Neumann eigenvalues in quasidiscs. Remarks, that quasidiscs can have fractals structure and the famous Van Koch snowflakes are example of quasidiscs.

The suggested method is based on estimating of the norm of extension operators on Sobolev spaces with the divergence norm:

$$\begin{aligned} \Vert Eu \mid L_A^{1,2}(\mathbb R^2)\Vert \le 2 \cdot \Vert u \mid L^{1,2}_{A}(\mathbb D_A)\Vert , \,\,\,u\in L^{1,2}_{A}(\mathbb D_A). \end{aligned}$$

Remark 2

The extension problem of quasiconformal planar mappings was studied in details by L. Ahlfors [1]. He obtained a geometric description of such domains (quasidiscs or Ahlfors domains in another terminology). Namely, a domain \(\Omega \) is called the Ahlfors domain, if its boundary \(\partial \Omega \) satisfies the so-called three point condition: there exists a constant \(C\ge 1\) having the property that, given any pair of points \(z_1,z_2\in \partial \Omega \) and a point \(z_3\) on the arc of smaller diameter between \(z_1\) and \(z_2\), such that

$$\begin{aligned} |z_1-z_3|+|z_2-z_3|\le C |z_1-z_2|. \end{aligned}$$

2 Extension operators and A-quasiconformal mappings

2.1 Sobolev spaces

Let \(\Omega \) be a simply connected domain in \(\mathbb {R}^2\). For any \(1\le p<\infty \) we consider the Lebesgue space \(L^p(\Omega )\) of measurable functions \(u: \Omega \rightarrow \mathbb {R}\) equipped with the following norm:

$$\begin{aligned} \Vert u\mid L^p(\Omega )\Vert =\left( \iint \limits _{\Omega }|u(z)|^{p}\, dxdy\right) ^{\frac{1}{p}}<\infty . \end{aligned}$$

The Sobolev space \(W^{1,p}(\Omega )\), \(1\le p<\infty \), is defined as a Banach space of locally integrable weakly differentiable functions \(u:\Omega \rightarrow \mathbb {R}\) equipped with the following norm:

$$\begin{aligned} \Vert u\mid W^{1,p}(\Omega )\Vert =\Vert u\,|\,L^{p}(\Omega )\Vert +\Vert \nabla u\mid L^{p}(\Omega )\Vert . \end{aligned}$$

The Sobolev space \(W^{1,p}_{{{\,\textrm{loc}\,}}}(\Omega )\) is defined as follows: \(f\in W^{1,p}_{{{\,\textrm{loc}\,}}}(\Omega )\) if and only if \(f\in W^{1,p}(U)\) for every open and bounded set \(U\subset \Omega \) such that \(\overline{U} \subset \Omega \), where \(\overline{U} \) stands for the closure of the set U.

The seminormed Sobolev space \(L^{1,p}(\Omega )\), \(1\le p< \infty \), is the space of all locally integrable weakly differentiable functions \(u:\Omega \rightarrow \mathbb {R}\) equipped with the following seminorm:

$$\begin{aligned} \Vert u\mid L^{1,p}(\Omega )\Vert =\Vert \nabla u\mid L^p(\Omega )\Vert , \,\, 1\le p<\infty . \end{aligned}$$

We also need a weighted seminormed Sobolev space \(L_{A}^{1,2}(\Omega )\) (associated with the matrix A), defined as the space of all locally integrable weakly differentiable functions \(u:\Omega \rightarrow \mathbb {R}\) equipped with the following seminorm:

$$\begin{aligned} \Vert u\mid L_{A}^{1,2}(\Omega )\Vert =\left( \iint \limits _\Omega \left\langle A(z)\nabla u(z),\nabla u(z)\right\rangle \,dxdy \right) ^{\frac{1}{2}}. \end{aligned}$$

The corresponding Sobolev space \(W^{1,2}_{A}(\Omega )\) is defined as the normed space of all locally integrable weakly differentiable functions \(u:\Omega \rightarrow \mathbb {R}\) equipped with the following norm:

$$\begin{aligned} \Vert u\mid W^{1,2}_{A}(\Omega )\Vert =\Vert u\,|\,L^{2}(\Omega )\Vert +\Vert u\mid L^{1,2}_{A}(\Omega )\Vert . \end{aligned}$$

In this work we consider elements of Sobolev spaces as equivalence classes up to a set of p-capacity zero [15, 21].

Recall that a continuous operator

$$\begin{aligned} E:L^{1,2}_{A}(\Omega ) \rightarrow L_A^{1,2}(\mathbb R^2) \end{aligned}$$

satisfying the conditions

$$\begin{aligned} Eu|_{\Omega }=u \quad \text {and} \quad \Vert E\Vert :=\sup \limits _{u \in L^{1,2}_{A}(\Omega )} \frac{\Vert Eu\Vert _{L^{1,2}_A(\mathbb R^2)}}{\Vert u\Vert _{L^{1,2}_{A}(\Omega )}}<\infty \end{aligned}$$

is called a continuous extension operator on Sobolev spaces \(L^{1,2}_A\).

We say that \(\Omega \subset \mathbb R^2\) is a Sobolev \(L^{1,2}_{A}\)-extension domain if there exists a continuous extension operator

$$\begin{aligned} E:L^{1,2}_{A}(\Omega ) \rightarrow L_A^{1,2}(\mathbb R^2). \end{aligned}$$

Remark 3

The spaces \(L^{1,2}(\Omega )\) and \(L^{1,2}_{A}(\Omega )\), \(\Omega \subset \mathbb R^2\), both are spaces of weak differentiable functions with square integrable first derivatives equipped with equivalent seminorms. Hence Sobolev \(L^{1,2}\)-extension domains and Sobolev \(L^{1,2}_{A}\)-extension domains coincide, and are quasidiscs [29].

It is well known that existence of an extension operator from \(L^{k,p}(\Omega )\) to \(L^{k,p}(\mathbb R^n)\), \(k\in \mathbb N\), \(n\ge 2\), depends on the geometry of the domain \(\Omega \). In the case of Lipschitz domains, Calderón [6] constructed an extension operator on \(L^{k,p}(\Omega )\) for \(1< p < \infty \), \(k\in \mathbb N\). Stein [26] extended Calderón’s result to the endpoints \(p=1, \infty \). Jones [16] introduced an extension operator for locally uniform domains. This class is much broader then previous classes and includes examples of domains with highly non-rectifiable boundaries.

In [29] necessary and sufficient conditions were obtained for \(L^{1,2}\)-extension operators from planar simply connected domains in terms of the quasiconformal geometry of domains. On this base in [9] were obtained sharp estimates of norms of the \(L^{1,2}\)-extension operators from quasidiscs. Necessary and sufficient conditions for \(L^{1,p}\)-extension operators from planar simply connected domains were obtained in [24, 25] for \(p>2\) and in the case \(1<p<2\) in [17]. Note, that in the case \(p\ne 2\) estimates of extension operators norms are unknown.

2.2 Quasiconformal mappings associated with the matrix A

In this subsection we use the Beltrami equation and so we work on the complex plane \(\mathbb C\). In subsequent sections we don’t use complex numbers and we identify the complex plane \(\mathbb C\) and two-dimensional Euclidean space \(\mathbb R^2\) by the standard setting \(\mathbb C\ni z=x+iy=(x,y)=z\in \mathbb R^2\).

Recall that a homeomorphism \(\varphi : \Omega \rightarrow \widetilde{\Omega }\), where \(\Omega ,\, \widetilde{\Omega } \subset \mathbb C\) are domains is called a K-quasiconformal mapping if \(\varphi \in W^{1,2}_{{{\,\textrm{loc}\,}}}({\Omega })\) and there exists a constant \(1\le K<\infty \) such that

$$\begin{aligned} |D\varphi (z)|^2\le K |J(z,\varphi )|\,\,\text {for almost all}\,\,z \in \Omega . \end{aligned}$$

The quasiconformal mappings arose as a natural generalization of conformal mappings [1] and have the significant applications to elliptic equations, see, for example, [2].

Let us give a construction of A-quasiconformal mappings connected with the A-divergent form elliptic operators [2]. We suppose that the matrix A belongs to the class of all \(2 \times 2\) symmetric matrix functions \(A(w)=\left\{ a_{kl}(w)\right\} \) with measurable entries defined in \(\mathbb {R}^2\), such that \(\text {det} A=1\) a.e. in \(\mathbb R^2\), and satisfy the uniform ellipticity condition (1).

The basic idea is that every positive quadratic form

$$\begin{aligned} ds^2=a_{11}(x,y)dx^2+2a_{12}(x,y)dxdy+a_{22}(x,y)dy^2 \end{aligned}$$

defined in a planar domain \(\Omega \) can be reduced, by means of a quasiconformal change of variables, to the canonical form

$$\begin{aligned} ds^2=\Lambda (du^2+dv^2),\,\, \Lambda \ne 0,\,\, \text {a.e. in}\,\, \widetilde{\Omega }, \end{aligned}$$

such that \(a_{11}a_{22}-a^2_{12} \ge \kappa _0>0\), \(a_{11}>0\), almost everywhere in \(\Omega \) [1, 5]. We will use the complex variable \(z=x+iy\) that is more convenient for this study of the operator \(\text {div} [A(z) \nabla f(z)]\).

Let \(\xi (z)={{\,\textrm{Re}\,}}\varphi (z)\) be a real part of a quasiconformal mapping \(\varphi (z)=\xi (z)+i \eta (z)\), which satisfies to the Beltrami equation:

$$\begin{aligned} \varphi _{\overline{z}}(z)=\mu (z) \varphi _{z}(z),\,\,\, \text {a.e. in}\,\,\, \Omega , \end{aligned}$$
(2)

where

$$\begin{aligned} \varphi _{z}=\frac{1}{2}\left( \frac{\partial \varphi }{\partial x}-i\frac{\partial \varphi }{\partial y}\right) \quad \text {and} \quad \varphi _{\overline{z}}=\frac{1}{2}\left( \frac{\partial \varphi }{\partial x}+i\frac{\partial \varphi }{\partial y}\right) , \end{aligned}$$

with the complex dilatation \(\mu (z)\) is given by

$$\begin{aligned} \mu (z)=\frac{a_{22}(z)-a_{11}(z)-2ia_{12}(z)}{\det (I+A(z))},\quad I= \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix}. \end{aligned}$$
(3)

We call this quasiconformal mapping (with the complex dilatation \(\mu \) defined by (3)) as an A-quasiconformal mapping.

Note that the uniform ellipticity condition (1) can be written as

$$\begin{aligned} |\mu (z)|\le \frac{K-1}{K+1},\,\,\, \text {a.e. in}\,\,\, \Omega , \end{aligned}$$

and therefore the Beltrami equation (2) is a non-degenerate elliptic equation.

Conversely we can obtain from the complex dilatation (3) (see, for example, [2], p. 412) the matrix A by the following way:

$$\begin{aligned} A(z)= \begin{pmatrix} \frac{|1-\mu |^2}{1-|\mu |^2} &{} \frac{-2 {{\,\textrm{Im}\,}}\mu }{1-|\mu |^2} \\ \frac{-2 {{\,\textrm{Im}\,}}\mu }{1-|\mu |^2} &{} \frac{|1+\mu |^2}{1-|\mu |^2} \end{pmatrix},\,\,\, \text {a.e. in}\,\,\, \Omega . \end{aligned}$$
(4)

Hence, the given matrix A one produced, by (3), the complex dilatation \(\mu (z)\), for which, in turn, the Beltrami equation (2) induces a quasiconformal homeomorphism \(\varphi _A:\Omega \rightarrow \widetilde{\Omega }\) as its solution, by the Riemann measurable mapping theorem (see, for example, [1, 2]). We will say that the matrix function A induces the corresponding A-quasiconformal homeomorphism \(\varphi _A\) or that A and \(\varphi _A\) agree.

So, by the given A-divergent form elliptic operator \(L_A\) defined in a domain \(\Omega \subset \mathbb C\) we can construct a so-called an A-quasiconformal mapping \(\varphi _A:\Omega \rightarrow \widetilde{\Omega }\) with a quasiconformal coefficient

$$\begin{aligned} K_A=\frac{1+\Vert \mu \mid L^{\infty }(\Omega )\Vert }{1-\Vert \mu \mid L^{\infty }(\Omega )\Vert }, \end{aligned}$$

where \(\mu \) defined by (3).

Note that the inverse mapping to an A-quasiconformal mapping \(\varphi _A: \Omega \rightarrow \widetilde{\Omega }\) is an \(A^{-1}\)-quasiconformal mapping [10].

Theorem 1

([10]) Let \(\Omega ,\widetilde{\Omega }\) be domains in \(\mathbb R^2\). Then a homeomorphism \(\psi : \Omega \rightarrow \widetilde{\Omega }\) is an A-quasiconformal mapping if and only if \(\psi \) generates by the composition rule \(\psi ^{*}(\widetilde{u})= \widetilde{u} \circ \psi \) an isometry of Sobolev spaces \(L^{1,2}_A(\Omega )\) and \(L^{1,2}(\widetilde{\Omega })\), i.e.

$$\begin{aligned} \Vert \psi ^{*}(\widetilde{u}) \mid L^{1,2}_A(\Omega )\Vert = \Vert \widetilde{u} \mid L^{1,2}(\widetilde{\Omega })\Vert \end{aligned}$$

for any \(\widetilde{u} \in L^{1,2}(\widetilde{\Omega })\).

This theorem generalizes the well known property of conformal mappings to generate the isometry of uniform Sobolev spaces \(L^{1,2}(\Omega )\) and \(L^{1,2}(\widetilde{\Omega })\) (see, for example, [7]) and refines (in the case \(n=2\)) the functional characterization of quasiconformal mappings in the terms of isomorphisms of uniform Sobolev spaces [28].

3 Spectral estimates

Let \(\Omega \) be a quasidisc in \(\mathbb R^2\). In [10] it was proved that in quasidiscs the embedding operator

$$\begin{aligned} i_{\Omega }:W^{1,2}_{A}(\Omega ) \hookrightarrow L^2(\Omega ) \end{aligned}$$

is compact. Hence in quasidiscs \(\Omega \subset \mathbb R^2\) the spectrum of the divergence form elliptic operators \(L_A(u)(z)\) with the Neumann boundary condition is discrete and can be written in the form of a nondecreasing sequence

$$\begin{aligned} 0=\mu _0(A,\Omega )< \mu _1(A,\Omega ) \le \mu _2(A,\Omega ) \le \ldots \le \mu _n(A,\Omega ) \le \ldots , \end{aligned}$$

where each eigenvalue is repeated as many times as its multiplicity [21].

The theory of extension operators on Sobolev spaces permit us to obtain lower estimates of the first non-trivial Neumann eigenvalues of the divergence form elliptic operators \(L_A(u)(z)\) in quasidiscs \(\Omega \subset \mathbb R^2\).

Theorem C

Let \(\Omega \) and \(\widetilde{\Omega }\) be quasidiscs in \(\mathbb R^2\). If \(\widetilde{\Omega }\supset \Omega \) then

$$\begin{aligned} \mu _1(A,\Omega ) \ge \frac{\mu _1(A,\widetilde{\Omega })}{\Vert E_{\Omega }\Vert ^2}, \end{aligned}$$

where \(\Vert E_{\Omega }\Vert \) denotes the norm of the linear continuous extension operator

$$\begin{aligned} E_{\Omega }: L^{1,2}_A(\Omega )\rightarrow L_A^{1,2}(\widetilde{\Omega }). \end{aligned}$$

Proof

Since \(\Omega \) is the quasidisc and the matrix A satisfies to the uniform ellipticity condition then there exists the linear extension operator [29]

$$\begin{aligned} E_{\Omega }:L^{1,2}_A(\Omega ) \rightarrow L^{1,2}_A(\widetilde{\Omega }) \end{aligned}$$
(5)

defined by the formula

$$\begin{aligned} (E_{\Omega }u)(z) = {\left\{ \begin{array}{ll} u(z) &{} \text {if }z \in \Omega , \\ \widetilde{u}(z) &{} \text {if }z \in \widetilde{\Omega } \setminus \overline{\Omega }, \end{array}\right. } \end{aligned}$$

where \(\widetilde{u}:\widetilde{\Omega } \setminus \overline{\Omega } \rightarrow \mathbb R\) is an extension of the function u.

Hence for every function \(u \in W^{1,2}_A(\Omega )\) we have

$$\begin{aligned} \Vert u-u_{\Omega } \mid L^2(\Omega )\Vert= & {} \inf \limits _{c\in \mathbb R}\Vert u-c\mid L^2(\Omega )\Vert =\inf \limits _{c\in \mathbb R}\Vert E_{\Omega }u-c\mid L^2(\Omega )\Vert \nonumber \\\le & {} \Vert E_{\Omega }u-(E_{\Omega }u)_{\widetilde{\Omega }}\mid L^2(\Omega )\Vert \le \Vert E_{\Omega }u-(E_{\Omega }u)_{\widetilde{\Omega }} \mid L^2(\widetilde{\Omega })\Vert .\nonumber \\ \end{aligned}$$
(6)

Here \(u_{\Omega }\) and \((E_{\Omega }u)_{\widetilde{\Omega }}\) are mean values of corresponding functions u and \(E_{\Omega }u\).

Given the Sobolev-Poincaré inequality in quasidiscs (see, for example, [12])

$$\begin{aligned} \inf \limits _{c \in \mathbb R} \Vert E_{\Omega }u-c \mid L^2(\widetilde{\Omega })\Vert \le B_{2,2}(A,\widetilde{\Omega }) \Vert E_{\Omega }u \mid L^{1,2}_A(\widetilde{\Omega })\Vert , \quad u \in W^{1,2}_A(\widetilde{\Omega }), \end{aligned}$$

and also the continuity of the extension operator (5), i. e.,

$$\begin{aligned} \Vert E_{\Omega }u \mid L^{1,2}_A(\widetilde{\Omega })\Vert \le \Vert E_{\Omega }\Vert \cdot \Vert u \mid L^{1,2}_A(\Omega )\Vert , \quad \Vert E_{\Omega }\Vert < \infty , \end{aligned}$$

we obtain

$$\begin{aligned} \Vert E_{\Omega }u-(E_{\Omega }u)_{\widetilde{\Omega }} \mid L^2(\widetilde{\Omega })\Vert\le & {} B_{2,2}(A,\widetilde{\Omega }) \cdot \Vert E_{\Omega }u \mid L^{1,2}_A(\widetilde{\Omega })\Vert \nonumber \\\le & {} B_{2,2}(A,\widetilde{\Omega }) \cdot \Vert E_{\Omega }\Vert \cdot \Vert u \mid L^{1,2}_A(\Omega )\Vert . \end{aligned}$$
(7)

Combining inequalities (6) and (7) we have

$$\begin{aligned} \Vert u-u_{\Omega } \mid L^2(\Omega )\Vert \le B_{2,2}(A,\Omega ) \cdot \Vert u \mid L^{1,2}_A(\Omega )\Vert , \end{aligned}$$

where

$$\begin{aligned} B_{2,2}(A,\Omega ) \le B_{2,2}(A,\widetilde{\Omega }) \cdot \Vert E_{\Omega }\Vert . \end{aligned}$$

According to the Min-Max Principle [8, 21], \(\mu _1(A,\Omega )^{-1}=B_{2,2}^2(A,\Omega )\). Thus, we have

$$\begin{aligned} \mu _1(A,\Omega ) \ge \frac{\mu _1(A,\widetilde{\Omega })}{\Vert E_{\Omega }\Vert ^2}. \end{aligned}$$

\(\square \)

From this theorem, using inequalities (1) about the uniform ellipticity of the matrix A we obtain

Corollary D

Let \(\Omega \subset \mathbb R^2\) and \(\widetilde{\Omega }\subset \mathbb R^2\) be quasidiscs. Suppose \(\widetilde{\Omega } \supset \Omega \), then

$$\begin{aligned} \mu _1(A,\Omega ) \ge \frac{\mu _1(\widetilde{\Omega })}{K\Vert E\Vert ^2}, \end{aligned}$$

where \(\mu _1(\widetilde{\Omega })\) is the first non-trivial Neumann eigenvalues of Laplacian in \(\widetilde{\Omega }\), \(\Vert E_{\Omega }\Vert \) denotes the norm of the linear continuous extension operator

$$\begin{aligned} E: L^{1,2}_A(\Omega )\rightarrow L^{1,2}_A(\widetilde{\Omega }). \end{aligned}$$

Let us present the construction of the extension operator for domains \(\mathbb {D}_A\) that are analogs of the unit disc for the Riemannian metric induced by the matrix A. As in the classical case we will use quasiconformal reflections and Theorem 1.

Since there is a Möbius transformation of the unit disc \(\mathbb D \subset \mathbb R^2\) onto the upper halfplane \(H^{+}\) we can replace in the definition of domains \(\mathbb {D}_A\) the unit disc \(\mathbb D\) to the upper halfplane \(H^{+}\).

Consider the following diagram for A-quasiconformal mappings

where \(\omega \) is a symmetry with respect to the real axis, that extend any function \(u \in L^{1,2}(H^{+})\) to a function \(\widetilde{u}\) from \(L^{1,2}(\mathbb R^2)\). In this case \(\left\| \omega \right\| =2\).

In accordance to this diagram we define the extension operator on Sobolev spaces

$$\begin{aligned} E:L^{1,2}_A(\Omega ) \rightarrow L_A^{1,2}(\mathbb R^2) \end{aligned}$$

by the formula

$$\begin{aligned} (Eu)(z) = {\left\{ \begin{array}{ll} u(z) &{} \text {if }z \in \Omega , \\ \widetilde{u}(z) &{} \text {if }z \in \mathbb R^2 \setminus \overline{\Omega }, \end{array}\right. } \end{aligned}$$

where \(\widetilde{u}:\mathbb R^2 {\setminus } \overline{\Omega } \rightarrow \mathbb R\) is defined as \(\widetilde{u}=\varphi ^{*}\circ \omega \circ (\varphi ^{-1})^{*}u\).

By Theorem 1 the operators \(\varphi _A^{*}\) and \((\varphi _A^{-1})^{*}\) are isometries and \(\left\| \omega \right\| =2\) then \(\left\| E \right\| =2\).

Let D be a smallest disc that contains \(\mathbb {D}_A\). Putting \(\widetilde{\Omega }=D\) in Corollary D and taking into account the estimate of the norm of extension operators in \(\mathbb {D}_A\) and the corresponding value of the first non-trivial Neumann eigenvalue in the disc of the radius R (see, for example [8]) we obtain the following result:

Theorem A

Let \(\mathbb D_A\) be an A-quasidisc. Then the following inequality holds

$$\begin{aligned} \mu _1(A,\mathbb {D}_A) \ge \frac{1}{4K} \cdot \left( \frac{j'_{1,1}}{R_{\mathbb {D}_A}}\right) ^2, \end{aligned}$$

where K is the ellipticity constant of the matrix A, \(R_{\mathbb {D}_A}=\max \limits _{\mathbb {D}(0,1)}|\varphi _A^{-1}(x)-\varphi _A^{-1}(0)|\) and \(j'_{1,1}\approx 1.84118\) denotes the first positive zero of the derivative of the Bessel function \(J_1\).

By using Theorem A we give an example of the estimate of the first Neumann eigenvalue of the divergence form elliptic operator

$$\begin{aligned} -\text {div} [A(z) \nabla u(z)]=\mu u\,\, \text {in}\,\,\mathbb D_{A},\,\,\frac{\partial u}{\partial \nu }=0\,\,\text {on}\,\,\partial \mathbb D_{A}, \end{aligned}$$

with the matrix A (in polar coordinates)

$$\begin{aligned} A=\begin{pmatrix} 2\cos ^2\theta +1/2\sin ^2\theta &{}\quad 3/4\sin 2\theta \\ 3/4\sin 2\theta &{}\quad 1/2\cos ^2\theta +2\sin ^2\theta \end{pmatrix}, \end{aligned}$$

in the non-convex domain singular cardioid type domain

$$\begin{aligned} \mathbb D_{A}=\left\{ (\rho , \theta ) \in \mathbb R^2:\rho =\sqrt{\frac{128}{35}}\cos ^{4}\left( \frac{\theta }{2}\right) , \quad - \pi \le \theta \le \pi \right\} , \end{aligned}$$

where the coefficient \(\sqrt{\frac{128}{35}}\) is defined by the normalization condition \(|\mathbb D_{A}|=\pi \).

Fig. 1
figure 1

Domain \(\mathbb D_{A}\)

Full size image

Consider the mapping \(\varphi :\mathbb R^2 \rightarrow \mathbb R^2\) defined by formula

$$\begin{aligned} \varphi (z)= \sqrt{\frac{35}{32}} \cdot \frac{z^{\frac{3}{8}}}{\overline{z}^{\frac{1}{8}}}-1,\,\, \varphi (0)=-1, \quad z=x+iy, \end{aligned}$$

which is the A-quasiconformal mapping. Under such mapping the preimage of the unit disc \(\mathbb D\) is the interior of this non-convex domain

$$\begin{aligned} \mathbb D_{A}=\left\{ (\rho , \theta ) \in \mathbb R^2:\rho =\sqrt{\frac{128}{35}}\cos ^{4}\left( \frac{\theta }{2}\right) , \quad - \pi \le \theta \le \pi \right\} . \end{aligned}$$

The mapping \(\varphi \) satisfies the Beltrami equation with

$$\begin{aligned} \mu (z)=\frac{\varphi _{\overline{z}}}{\varphi _{z}}=-\frac{1}{3}\frac{z}{\overline{z}}. \end{aligned}$$

We see that \(\mu \) induces, by formula (4), the matrix function A(z), which in the polar coordinates \(z=\rho e^{i\theta }\) has the form

$$\begin{aligned} A=\begin{pmatrix} 2\cos ^2\theta +1/2\sin ^2\theta &{}\quad 3/4\sin 2\theta \\ 3/4\sin 2\theta &{}\quad 1/2\cos ^2\theta +2\sin ^2\theta \end{pmatrix}. \end{aligned}$$

Given that \(K=2\) and \(R_{\mathbb {D}_A}\approx 1\). Then by Theorem A we have

$$\begin{aligned} \mu _1(A,\mathbb {D}_A) \ge \frac{(j'_{1,1})^2}{8}. \end{aligned}$$

Now we give estimates for the radius \(R_{\mathbb {D}_A}\) of the smallest disc that contains \(\mathbb {D}_A\). This result is a partial solution of the smallest-circle problem (initially suggested by J. J. Sylvester in 1857).

Theorem 2

Let \(\varphi _A: \mathbb R^2 \rightarrow \mathbb R^2\) be an A-quasiconformal mapping. Then the following estimate

$$\begin{aligned} R_{\mathbb {D}_A}\le \frac{1}{16} e^{\pi K_A} \end{aligned}$$

holds, where \(K_A\) is the quasiconformality coefficient of the mapping \(\varphi _A.\)

Proof

Let \(r_{\mathbb {D}_A}=\min \limits _{\mathbb {D}(0,1)}|\varphi _A^{-1}(x)-\varphi _A^{-1}(0)|\). Then by the global distortion theorem [20] we have

$$\begin{aligned} R_{\mathbb {D}_A}\le \frac{1}{16} e^{\pi K_A}r_{\mathbb {D}_A}. \end{aligned}$$

Because \(|\mathbb {D}(\varphi _A(0),r_{\mathbb {D}_A})|\le |\mathbb {D}_A|=\pi \) we obtain an upper estimate for \(r_{\mathbb {D}_A}\le 1\). Therefore

$$\begin{aligned} R_{\mathbb {D}_A}\le \frac{1}{16} e^{\pi K_A}. \end{aligned}$$

\(\square \)

Now recall that

$$\begin{aligned} K_A=\frac{1+\Vert \mu \mid L^{\infty }(\Omega )\Vert }{1-\Vert \mu \mid L^{\infty }(\Omega )\Vert }, \end{aligned}$$

where

$$\begin{aligned} \mu (z)=\frac{a_{22}(z)-a_{11}(z)-2ia_{12}(z)}{\det (I+A(z))},\quad I= \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix}. \end{aligned}$$

It means that \(K_A\) is a function of the matrix A only.

Hence, combining Theorem A and Theorem 2 we obtain a connection between the first non-trivial Neumann eigenvalue of the elliptic operator \(L_{A}(u)(z)\) and the Sylvester smallest-circle problem (1857).

Theorem 3

Let \(\mathbb D_A\) be an A-quasidisc. Then the following inequality holds

$$\begin{aligned} \mu _1(A,\mathbb {D}_A) \ge \frac{64 (j'_{1,1})^2}{K} \exp \left\{ -2\pi \frac{1+\Vert \mu \mid L^{\infty }(\Omega )\Vert }{1-\Vert \mu \mid L^{\infty }(\Omega )\Vert } \right\} , \end{aligned}$$

where K is the ellipticity constant of the matrix A and \(j'_{1,1}\approx 1.84118\) denotes the first positive zero of the derivative of the Bessel function \(J_1\).

Because \(|\mu (z)|\le \frac{K-1}{K+1}\) we have by the direct calculations

$$\begin{aligned} \frac{1+\Vert \mu \mid L^{\infty }(\Omega )\Vert }{1-\Vert \mu \mid L^{\infty }(\Omega )\Vert } \le K. \end{aligned}$$

Hence we obtain:

Theorem B

Let \(\mathbb D_A\) be an A-quasidisc. Then the following inequality holds

$$\begin{aligned} \mu _1(A,\mathbb {D}_A) \ge \frac{ 64 (j'_{1,1})^2}{K \exp \{2\pi K\}}, \end{aligned}$$

where K is the ellipticity constant of the matrix A and \(j'_{1,1}\approx 1.84118\) denotes the first positive zero of the derivative of the Bessel function \(J_1\).

This theorem gives estimates of the first non-trivial Neumann eigenvalue of the elliptic operator \(L_{A}(u)(z)\) in terms of the quasiconformal geometry of domains [2].