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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! 6 (and of the more general [9, Theorem3]) is much longer and technical with respect to the proof of .1 using the double manifold. The general version of .6, [9, ], is a generalization of the Poincare-Hopf theorem for manifolds with boundary when the vector field is not outward pointing everywhere on the boundary._We note that .6 applies to \(u_{2}\) in case we have no singular zeros on \(tial M\), since interior critical points are Morse by 4.1. However, it applies also in the case when there are boundary singular points, since by .2 they are isolated and the behavior of \(u_{2}\) in a neighborhood of such points is explicit (see also [5, Appendix A]). In fact, let us consider for simplicity the Euclidean case. Let \(x_{0}\) be a singular boundary zero. We can assume \(x_{0}=(0,0)\) (in Euclidean coordinates \((x,y)\)), and also that the tangent to the boundary at \(x_{0}\) is given by \(y=0\). Hence, in a neighborhood of \(x_{0}\) we have that \(u_{2}(x,y)=c(x^{2}-y^{2})+O((x^{2}+y^{2})^{1+\epsilon})\). We can perform a perturbation of \(\nabla u_{2}\) which is supported in a small neighborhood of \(x_{0}\) in such a way that the new vector field has no singular points at the boundary, and no new interior critical points have been produced. Hence one can apply .6 to \(M\). Doing so we are counting the sum of the indexes of the _interior_ critical points of \(u_{2}\). Note that for the perturbed vector field the point \(x_{0}\) is an interior point of a connected component of \(\mathscr{P}\) or of its complement. We have just proved the following non-generic result.**Corollary**.: _Let \(M\) be a smooth compact surface with boundary and let \(u_{2}\) be the second Steklov eigenfunction on \(M\). Then_\[\sharp\{\text{interior critical points of }u_{2}\}=\sum_{j=1}^{L}\ell_{j}- \chi(M), \tag{18}\]_where \(L\) is the number of connected components of \(tial M\) where \(u_{2}\) changes sign exactly \(\ell_{j}\) times (in the case of singular boundary zeros, they must not be counted as sign changing)._We can verify the validity of Corollary 4.8 in various situations. For example, let \(A=\{x\in\mathbb{R}^{2}:r0\) such that, for \(0T^{*}\) however the second eigenvalue is simple, and \(u_{2}(\theta,z)=z\). It has constant sign on each connected component of the boundary and, therefore, from (18) we deduce that it has no interior singular points (which is an immediate check). We observe then that only the connected components of the boundary where the eigenfunction changes sign influence the number of interior critical points图片描述

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