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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! , for Riemannian surfaces), formula (3) implies that\[\sharp\{\text{critical points of }u\text{ in }M\}=\sum_{j=1}^{L}\ell_{j}-\chi(M), \tag{4}\]where \(L\) is the number of connected components of \(tial M\) where \(u\) changes sign exactly \(\ell_{j}\) times on the \(j\)-th component. We remark that only the connected components of the boundary where \(u\) changes sign influence the number of critical points of \(u\).Finally, we prove that for surfaces, and for the second eigenfunction \(u_{2}\), formula (4) holds for any metric on a given manifold (not just generically). This is a consequence of the fact that the second eigenfunction is always a Morse function, along with other properties which we prove in the paper.We remark that for simply connected surfaces, formula (4) implies that there are no interior critical points. This fact is known for simply connected planar domains by [10], however the proof of [10] readily extends to the case of simply connected surfaces.We point out that the same results can be proved, using essentially the same arguments, for the _weighted_ Steklov problem\[\begin{cases}\Delta u=0&\text{in }M tial_{\nu}u=\sigma\rho u&\text{on }tial M,\end{cases}\]where \(\rho=\rho(x)>0\) is a strictly positive smooth function.It would be interesting to investigate the same issues in the case of the _sloshing_ problem, namely when \(\rho\) can equal zero in some portions of \(tial M\) (see for instance [10]).The present paper is organized as follows: in Section 2 we prove and identity which relates the sum of the indexes of an harmonic function \(u\) on \(M\), the sum of the indexes of its restriction on \(tial M\), and \(\chi(M)\), which holds under suitable assumptions on \(u\) (see .1). In section 3 we apply .1 to Steklov eigenfunctions. In fact, for generic metrics, the hypotheses of .1 are satisfied by Steklov eigenfunction. This allows to prove (2) (see .1). In Section 3 we also obtain (3) and (4) as consequences of (2). In Section 4 we observe that (4) holds for _any_ Riemannian surface when we consider the second Steklov eigenfunction. To prove the result, a crucial observation is that the second Steklov eigenfunction is always a Morse function. Finally, in Appendix A we prove that for a generic metric on \(M\), all Steklov eigenfunctions are Morse functions图片描述

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