线性代数网课代修|ENGO361代写|ENGO361英文辅导|ENGO361

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! 4. The case of the second eigenfunctionIn this section we consider the second Steklov eigenfunction \(u_{2}\) on _any_ smooth connected Riemannian surface \(M\), and we prove that (16) holds (i.e., the result is not generic). To do so we prove a few properties of \(u_{2}\) when \(n=2\).**Theorem**.: _The function \(u_{2}\) is a Morse function._Proof.: The proof follows the same lines as that of [10, Lemma 3]. Let \(u_{2}\) be a second eigenfunction and let \(p\in M\) be a critical point of \(u_{2}\). Assume it is degenerate. This implies that the Hessian at \(p\) vanishes. Consider the function \(w=u_{2}-u_{2}(p)\). Clearly \(\Delta w=0\), and \(p\) is a zero of \(w\) such that \(\nabla w(p)=0\), \(D^{2}w(p)=0\). Hence the function \(w\) has at least _three_ nodal curves intersecting at \(p\) and forming equal angles. Also, these curves cannot form a closed loop, being \(w\) harmonic. Therefore these curves meet the boundary at \(2k\) distinct points, \(2k\geq 6\). In particular, on each connected component \(tial M_{i}\) of \(tial M\) we have \(2k_{i}\) zeros, and \(\sum_{i}2k_{i}=2k\). We deduce that there are at least \(4\) interior nodal domains for \(w\).Let then \(\Omega_{i}\), \(i=1,…,m\), \(m\geq 4\), the interior nodal domains of \(w\), and let \(w_{i}=w|_{\Omega_{i}}\) (extended by \(0\) outside \(\Omega_{i}\)). Let\[\phi:=\sum_{i=1}^{m}a_{i}w_{i}\]where \(a_{i}\in\mathbb{R}\) are not all zero and are chosen such that\[\int_{tial M}\phi=\sum_{i=1}^{m}a_{i}\int_{tial\Omega_{i}}w_{i}=0\]and\[\sum_{i=1}^{m}a_{i}^{2}\int_{tial\Omega_{i}}w_{i}=0.\]Here \(tial\Omega_{i}=tial M\cap\overline{\Omega}_{i}\). By construction, \(\phi\in H^{1}(M)\). It is always possible to find such \(a_{i}\) since \(m\geq 4\). In fact, let us set \(\gamma_{i}:=\int_{tial\Omega_{i}}w_{i}\). We need to solve the system\[\begin{cases}\sum_{i=1}^{m}\gamma_{i}a_{i}=0 \sum_{i=1}^{m}\gamma_{1}a_{i}^{2}=0.\end{cases}\]Without loss of generality, we may assume that \(\gamma_{i}>0\) for \(i\) odd and \(\gamma_{i}0\}\), and \(\nu\) is the outer unit normal to \(tial M\)._**Remark**.: _We note that .1 in the case \(n=2\) can be also deduced from .6. However the proof of 图片描述

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