- Research
- Open Access
Some results on the eigenvalue problem for a fractional elliptic equation
- Yujuan Tian^{1}Email author
- Received: 14 June 2018
- Accepted: 13 January 2019
- Published: 17 January 2019
Abstract
This paper deals with the eigenvalue problem for a fractional variable coefficients elliptic equation defined on a bounded domain. Compared to the previous work, we prove a quite different variational formulation of the first eigenvalue for the above problem. This allows us to give a variational proof of the fractional Faber–Krahn inequality by employing suitable rearrangement techniques.
Keywords
- Eigenvalue
- Faber–Krahn inequality
- Rearrangement
MSC
- 35P15
- 35R11
- 35J20
1 Introduction
Fractional powers of elliptic operators, whose basic case is the fractional Laplacian \((-\Delta )^{s}\), arise naturally in many applications, for instance, the obstacle problem that appears in the study of the configuration of elastic membranes, anomalous diffusion, the so-called quasi-geostrophic flow problem, and pricing of American options, as well as many modern physical problems when considering fractional kinetics and anomalous transport, strange kinetics, and Lévy processes in quantum mechanics; one can see [1–3] and the references therein. When working in the whole space domain \(\mathbb{R}^{N}\), there are several equivalent definitions of the fractional Laplacian operator \((-\Delta )^{s}\), classical references being [4–6]. However, when working on a bounded domain Ω, things get complicated because there are different options for defining \((-\Delta )^{s}\). A particular one is to define the fractional Laplacian as the Dirichlet-to-Neumann map, through an extended function defined in a cylinder \(\mathcal{C}_{\varOmega }^{+}=\varOmega \times (0,+\infty )\subset \mathbb{R}^{N+1}\) whose values are assigned to zero on the lateral boundary of \(\mathcal{C}_{\varOmega }^{+}\), as was proposed in [7, 8]. This allows reducing nonlocal problems involving \((-\Delta )^{s}\) to suitable local problems, defined in one more space dimension. A similar definition of the fractional elliptic operator \(L^{s}\) in a bounded domain is given in [9]. However, this definition seems to be contrary to the nonlocal feature of fractional operators, and thus there are some restrictions on its validity. In [10], the authors take a more usual approach to define \((-\Delta )^{s}\) in a bounded domain. It consists in keeping the definition of fractional Laplacian in \(\mathbb{R}^{N}\) through the extension method but asking the functions \(u(x)\) to vanish outside of Ω, which seems to be more natural in many applications. In this paper, we will take this approach to define the fractional elliptic operator \(L^{s}\) in bounded domains (see Sect. 2).
The study of eigenvalue problems is a classical topic and there are lots of results on local eigenvalue problems (see, for instance, [11–14] and the references therein). Recently, great attention has been focused on studying of eigenvalue problems involving fractional operators. Results for fractional linear operators were obtained in [15], where variational formulations of eigenvalues and some properties of eigenfunctions were proved. In [16–18], the eigenvalue problem associated with the fractional nonlinear operator \((-\Delta )^{s}_{p}\) was studied, and particularly some properties of the first eigenvalue and of the higher order eigenvalues were obtained. Then, Iannizzotto and Squassina [19] proved some Weyl-type estimates for the asymptotic behavior of variational eigenvalues corresponding to \((-\Delta )^{s}_{p}\).
More recently, by employing rearrangement techniques, Sire, Vázquez and Volzone [10] proved the Faber–Krahn inequality in two ways for the first eigenvalue of the fractional Laplacian operator in bounded domains. A variational proof was provided, which seems to be simpler, based on the variational characterization of the first eigenvalue and nonlocal Pólya–Szegö inequality. However, they pointed out that for the fractional variable coefficients problem \((P_{\lambda })\), it is not clear how to use the variational approach to prove such an inequality since in this case the variational formulations of the first eigenvalue given before does not seem to allow Pólya–Szegö inequality to be applied. To solve the above problem, in this paper, with the help of the extension problem of \((P_{\lambda })\) defined in one more space dimension, we will prove a quite different variational formulation of the first eigenvalue, in which the operator \(L^{s}\) is associated to a norm satisfying Pólya–Szegö inequality. Then using some properties of rearrangement, a variational proof of fractional Faber–Krahn inequality can be achieved for problem \((P_{\lambda })\).
Many other fractional problems were also actively studied in recent years, such as fractional Kirchhoff type problems, fractional Schrödinger problems, and also Brézis–Nirenberg problem for fractional operators. Interested readers can refer to [20–25] for details.
This paper is organized as follows: In Sect. 2, we give all the necessary functional background related to problem \((P_{\lambda })\), which is naturally connected to the very definition of the operator \(L^{s}\). In Sect. 3, the variational formulation of the first eigenvalue and some properties of eigenfunctions are obtained, while Sect. 4 is devoted to proving fractional Faber–Krahn inequality.
2 Preliminaries
In this section, we provide a self-contained description of the functional background which is necessary for the well-posedness of problem \((P_{\lambda })\). For further details, one can see [3, 9, 10, 26–28] and the references therein.
Lemma 2.1
Let \(1\leq q<2_{s}^{\sharp }=\frac{2N}{N-2s}\). Then, \(\operatorname{Tr}_{ \varOmega }(X^{s}_{\varOmega }(\mathcal{C}^{+}))\) is compactly embedded in \(L^{q}(\varOmega )\).
Proof
We know that the trace \(\operatorname{Tr}_{\varOmega }(X^{s}_{\varOmega }( \mathcal{C}^{+}))=\mathcal{H}(\varOmega )\subset H^{s}(\varOmega )\) and \(H^{s}(\varOmega )\subset \subset L^{q}\) when \(1\leq q<2_{s}^{\sharp }\), see [28]. Here ⊂⊂ denotes compact embedding. This completes the proof of the lemma. □
Then according to [9, 10], the following definition of weak solution to problem (2.6) is provided.
Definition 2.1
Remark 2.1
It is easy to see that the function \(u=\operatorname{Tr}_{\varOmega }(w)\) belongs to the space \(\mathcal{H}(\varOmega )\).
3 General results about eigenvalues
Theorem 3.1
Proof
4 Faber–Krahn inequality
Theorem 4.1
Remark 4.1
In the case \(s=1\) and \(N=2\), the above result is known as the Faber–Krahn theorem, which can be stated as follows: a membrane with the lowest principle frequency is the circular one.
Remark 4.2
- (i)Conservation of the \(L^{p}\) norms:$$ \Vert f \Vert _{L^{p}(E)}= \bigl\Vert f^{\sharp } \bigr\Vert _{L^{p}(E^{\sharp })}, \quad 1\leq p< + \infty . $$
- (ii)Hardy–Littlewood inequality:where f, g are measurable functions on E.$$ \int _{E} \bigl\vert f(x)g(x) \bigr\vert \,dx\leq \int _{E^{\sharp }}f^{\sharp }(x)g^{\sharp }(x)\,dx, $$
- (iii)Pólya–Szegö inequality:$$ \bigl\Vert \nabla f^{\sharp } \bigr\Vert _{L^{p}(E^{\sharp })}\leq \Vert \nabla f \Vert _{L^{p}(E)}, \forall f\in W_{0}^{1,p}(E), \quad 1< p< +\infty . $$
Furthermore, the following lemma holds (see [33, 34]).
Lemma 4.1
Proof of Theorem 4.1
5 Conclusions
For the fractional variable coefficients elliptic operator defined on a bounded domain, it has been pointed out that the variational formulation of the first eigenvalue given before does not allow using a variational approach to prove the fractional Faber–Krahn inequality. In this paper, we proved a different variational formulation of the first eigenvalue for \((P_{\lambda })\). Following this, a variational proof of the fractional Faber–Krahn inequality has been achieved by employing suitable rearrangement techniques. Based on this point, our work is valuable.
Declarations
Acknowledgements
The author would like to thank the referees for their useful suggestions.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
This work is supported by National Natural Science Foundation of China (Grant No. 11501333, 11571208).
Authors’ contributions
This entire work has been completed by the author, Dr. YT. The author read and approved the final manuscript.
Competing interests
The author declares that she has no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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