如果你也在线性代数linearalgebra这个学科遇到相关的难题，请随时添加vx号联系我们的代写客服。我们会为你提供专业的服务。 linearalgebra™长期致力于留学生网课服务，涵盖各个网络学科课程：金融学Finance，经济学Economics，数学Mathematics，会计Accounting，文学Literature，艺术Arts等等。除了网课全程托管外，linearalgebra™也可接受单独网课任务。无论遇到了什么网课困难，都能帮你完美解决！ # On the critical points of Steklov eigenfunctionsLuca BattagliaLuca Battaglia, Department of Mathematics, University of Michigan, Ann Arbor, MI 48524, USA battaglia@math.msu.eduAngela PistoiaAngela Pistoia, Department of Mathematics, University of Michigan, Ann Arbor, MI 48524, USA angela.pistoia@math.msu.eduLuigi ProvenzanoLuigi Provenzano, Department of Mathematics, University of Michigan, Ann Arbor, MI 48524, USA lri.provenzano@math.msu.edu Abstract.We consider the critical points of Steklov eigenfunctions on a compact, smooth \(n\)-dimensional Riemannian manifold \(M\) with boundary \(tial M\). For generic metrics on \(M\) we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to \(tial M\), and the Euler characteristic of \(M\). In dimension \(2\) this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of \(M\) and of the number of sign changes of \(u\) on \(tial M\). In the case of the second Steklov eigenfunction on a surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on \(M\) Steklov eigenfunctions are Morse functions in \(M\).Key words and phrases:Steklov eigenfunctions; critical point theory; Morse function; generic properties 2020 Mathematics Subject Classification: 58C40, 58J05, 58J20, 58J50 The first author is partially supported by MUR-PRIN-2022AKNSE4 “Variational and Analytical aspects of Geometric PDEs”. The second author is partially supported by the MUR-PRIN-20227HK33Z “Pattern formation in nonlinear phenomena”. The first and the second authors are also partially supported by INDAM-GNAMPA project “Problemi di doppia curvatura su varieta a bordo e legami con le EDP di tipo ellittico”. The third author acknowledges support of the INDAM-GNSAGA project “Analisi Geometrica: Equazioni alle Derivate Parziali e Teoria delle Sottovarieta” and of the the project “Perturbation problems and asymptotics for elliptic differential equations: variational and potential theoretic methods” funded by the MUR Progetti di Ricerca di Rilevante Interesse Nazionale (PRIN) Bando 2022 grant 2022SENJZ3 be interpreted as the eigenvalues of the Dirichlet-to-Neumann map \(\mathscr{D}:H^{1/2}(tial M)\to H^{-1/2}(tial M)\), which acts as follows: \(\mathscr{D}f=tial_{\nu}(Hf)\), where \(Hf\) is the harmonic extension of \(f\) in \(M\). It turns out that the eigenfunctions of \(\mathscr{D}\) are the traces of the Steklov eigenfunctions \(u_{k}\). Interested readers may find quite complete information on the Steklov problem in the survey [4], and in the more recent [3], along with a large number of open questions.In this paper we are interested in the geometry of the Steklov eigenfunctions, and in particular in the counting of the interior critical points. The literature on the geometry of the Steklov eigenfunctions has been mainly devoted to questions like the number and measure of nodal domains. This is not a surprise, since the study of nodal domains is perhaps one of the oldest topics in spectral geometry. In particular, the Courant’s nodal domain theorem holds for Steklov eigenfunctions: \(u_{k}\) has at most \(k\) nodal domains (see e.g., [7]). However, the proof of Courant’s nodal domain theorem cannot be generalized to the Dirichlet-to-Neumann eigenfunctions, i.e., the traces of the Steklov eigenfunctions on \(tial M\), and in fact it is an open problem to find an upper bound for the nodal count of Dirichlet-to-Neumann eigenfunctions. Bounds are available only in two dimensions as consequence of the interior nodal count and elementary topological arguments. For simply connected domains, see e.g., [2]. In dimension two, bounds on the number of nodal domains can be translated into bounds on the multiplicities of \(\sigma_{k}\). The study of multiplicity bounds is another topic which has been extensively investigated in the last decades (see e.g., [6, 7], see also [4, SS6] and [3, SS10] for a more up-to-date account on multiplicity bounds and related open problems). Concerning critical points of eigenfunctions, much less is known. Most of the results are for simply connected domains in \(\mathbb{R}^{2}\). The main reference is [2], where the authors prove a number of bounds concerning the number of interior and boundary nodal domains, the eigenvalue multiplicity, and number of interior critical points of a Steklov eigenfunction. We also refer to [1] where the authors consider more in general identities or inequalities relating the number and type of critical points of solutions to elliptic PDEs on domains of the plane, the topology of the domain, and the boundary data. We also mention [10] where the authors consider an overdetermined Steklov problem and prove, as a technical lemma, that for a simply connected planar domain, the second Steklov eigenfunction has no critical points. The proof can be adapted to any simply connected surface. Up to our knowledge, these are the only results on the counting of critical points of Steklov eigenfunctions.In our paper, we compute the number of interior critical points of \(u_{k}\) in terms of the number of critical points of \(u_{k}\) on \(tial M\) (or, to be more precise, the number of critical points of the corresponding Dirichlet-to-Neumann eigenfunction on \(tial M\)). One of the main results is valid for _generic_ metrics on a given smooth manifold \(M\), and can be summarized as follows: for a generic metric, any Steklov eigenfunction \(u\) satisfies\[\sum_{\{x\in M:\nabla u(x)=0\}}\mathrm{ind}_{x}\nabla u=\chi(M)-\sum_{\{\xi\in tial M:\bar{\nabla}u(\xi)=0,u(\xi)>0\}}\mathrm{ind}_{\xi}\bar{\nabla}u, \tag{2}\]where \(\chi(M)\) is the Euler characteristic of \(M\). Here \(\bar{\nabla}\) denotes the gradient on \(tial M\) with respect to the induced metric. By an abuse of notation, in the right-hand side of (2) we have still denoted by \(u\) the trace of \(u\) (which in the smooth case is just the restriction of \(u\) to \(tial M\)). With the term _generic_ we intend that the result holds for a dense subset of Riemannian metrics on the given manifold \(M\). As a consequence, we show that (2) implies\[\sum_{\{x\in M:\nabla u(x)=0\}}\operatorname{ind}_{x}\nabla u=\chi(M)-\sum_{i=1} ^{\ell}\chi(\mathscr{P}_{i}). \tag{3}\]where \(\mathscr{P}_{i}\) are the connected components of \(\mathscr{P}=\{\xi\intial M:u(\xi)>0\}\).In order to prove (2) we exploit the fact that, generically, a Steklov eigenfunction is Morse in \(M\), its trace is Morse on \(tial M\) (i.e., Dirichlet-to-Neumann eigenfunctions are Morse functions), and its restriction on the boundary has no singular zeros (i.e., Dirichlet-to-Neumann eigenfunctions have no singular zeros). The last two generic statements have been recently proved in [13]. In fact, in [13] the author proves generic properties of Dirichlet-to-Neumann eigenfunctions. We prove in this paper that Steklov eigenfunctions are generically Morse in \(M\) and, for completeness, we show also that, generically, there are no singular zeros in \(M\).When \(n=2\) (i.e

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