Abstract
We obtain estimates of Neumann eigenvalues of the divergence form elliptic operators in Sobolev extension domains. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. The connection between Neumann eigenvalues of elliptic operators and the smallest-circle problem (initially suggested by J. J. Sylvester in 1857) is given.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
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)
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)
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
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
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
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
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
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
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
where \(\Vert E_{\Omega }\Vert \) denotes the norm of the linear continuous extension operator
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:
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
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:
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:
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:
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:
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:
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
satisfying the conditions
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
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
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
defined in a planar domain \(\Omega \) can be reduced, by means of a quasiconformal change of variables, to the canonical form
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:
where
with the complex dilatation \(\mu (z)\) is given by
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
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:
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
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.
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
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
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
where \(\Vert E_{\Omega }\Vert \) denotes the norm of the linear continuous extension operator
Proof
Since \(\Omega \) is the quasidisc and the matrix A satisfies to the uniform ellipticity condition then there exists the linear extension operator [29]
defined by the formula
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
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])
and also the continuity of the extension operator (5), i. e.,
we obtain
Combining inequalities (6) and (7) we have
where
According to the Min-Max Principle [8, 21], \(\mu _1(A,\Omega )^{-1}=B_{2,2}^2(A,\Omega )\). Thus, we have
\(\square \)
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
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
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
by the formula
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
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
with the matrix A (in polar coordinates)
in the non-convex domain singular cardioid type domain
where the coefficient \(\sqrt{\frac{128}{35}}\) is defined by the normalization condition \(|\mathbb D_{A}|=\pi \).
Consider the mapping \(\varphi :\mathbb R^2 \rightarrow \mathbb R^2\) defined by formula
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
The mapping \(\varphi \) satisfies the Beltrami equation with
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
Given that \(K=2\) and \(R_{\mathbb {D}_A}\approx 1\). Then by Theorem A we have
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
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
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
\(\square \)
Now recall that
where
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
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
Hence we obtain:
Theorem B
Let \(\mathbb D_A\) be an A-quasidisc. Then the following inequality holds
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].
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Ahlfors, L.: Lectures on Quasiconformal Mappings. D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London (1966)
Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton and Oxford (2008)
Ashbaugh, M. S.: Isoperimetric and universal inequalities for eigenvalues. In E. B. Davies and Y. Safarov (eds.), Spectral theory and geometry. Papers from the ICMS Instructional Conference held in Edinburgh, March 30-April 9, 1998. London Mathematical Society Lecture Note Series, 273. Cambridge University Press, Cambridge, 95–139 (1999)
Ashbaugh, M.S., Benguria, R.D.: Universal bounds for the low eigenvalues of Neumann Laplacians in \(n\) dimensions. SIAMJ. Math. Anal. 24, 557–570 (1993)
Bojarski, B., Gutlyanskiĭ, V., Martio, O., Ryazanov, V.: Infinitesimal Geometry of Quasiconformal and bi-Lipschitz Mappings in the Plane. EMS, Zurich (2013)
Calderón, A.P.: Lebesgue spaces of differentiable functions and distributions. Proc. Symp. Pure Math. 4, 33–49 (1961)
Courant, R.: Dirichlet’s Principle. Conformal Mapping, and Minimal Surfaces. Springer-Verlag, Berlin-Heidelberg-New York, (1977)
Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995)
Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Sobolev extension operators and Neumann eigenvalues. J. Spectr. Theory 10, 337–353 (2020)
Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Quasiconformal mappings and Neumann eigenvalues of divergent elliptic operators. Complex Var. Elliptic Equ. 67(9), 2281–2302 (2022)
Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361, 3829–3850 (2009)
Gol’dshtein, V., Ukhlov, A.: On the first eigenvalues of free vibrating membranes in conformal regular domains. Arch. Rational Mech. Anal. 221, 893–915 (2016)
Gol’dshtein, V., Ukhlov, A.: The spectral estimates for the Neumann-Laplace operator in space domains. Adv. in Math. 315, 166–193 (2017)
Graff, K.F.: Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs, Oxford Univ. Press (1993)
Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1–2), 71–88 (1981)
Koskela, P., Rajala, T., Zhang, Yi Ru-Ya: A geometric characterization of planar Sobolev extension domains, arXiv:1502.04139
Laugesen, R.S., Morpurgo, C.: Extremals of eigenvalues of laplacians under conformal mapping. J. Func. Anal. 155, 64–108 (1998)
Lehto, O., Virtanen, K. I.: Quasiconformal mappings in the plane. Die Grundlehren der mathematischen Wissenschaften, vol. 126 (Second ed.), Springer-Verlag, (1973)
Martin, G.J.: The distortion theorem for quasiconformal mappings, Shottky’s theorem and holomorphic motions. Proc. AMS 125(4), 1095–1103 (1997)
Maz’ya, V.: Sobolev Spaces: with Applications to Elliptic Partial Differential Equations. Springer, Berlin (2011)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5, 286–292 (1960)
Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N.J., (1951)
Shvartsman, P., Zobin, N.: On planar Sobolev \(L_p^m\)-extension domains. Adv. Math. 287, 237–346 (2016)
Shvartsman, P.: Whitney-type extension theorems for jets generated by Sobolev functions. Adv. Math. 313, 379–469 (2017)
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, New Jersey (1970)
Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3, 343–356 (1954)
Vodop’yanov, S.K., Gol’dstein, V.M.: Lattice isomorphisms of the spaces \(W^1_n\) and quasiconformal mappings. Siberian Math. J. 16(2), 174–189 (1975)
Vodop’yanov, S.K., Gol’dstein, V.M., Latfullin, T.G.: Criteria for extension of functions of the class \(L_2^1\) from unbounded plain domains. Siberian Math. J. 20(2), 298–301 (1979)
Vodop’yanov, S.K., Ukhlov, A.D.: Set functions and their applications in the theory of Lebesgue and Sobolev spaces. Siberian Adv. Math. 14(4), 78–125 (2004)
Acknowledgements
The second author was supported by RSF Grant No. 20-71-00037 (Sect. 3).
Funding
Open access funding provided by Ben-Gurion University.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this work.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no Conflict of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gol’dshtein, V., Pchelintsev, V. & Ukhlov, A. Neumann eigenvalues of elliptic operators in Sobolev extension domains. Anal.Math.Phys. 14, 64 (2024). https://doi.org/10.1007/s13324-024-00926-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-024-00926-x