论文标题
Dirichlet特征值的梯度流动
Gradient Flows for Dirichlet Eigenvalues
论文作者
论文摘要
我们对形状功能的梯度流有兴趣，尤其是对于第一拉普拉斯特征值。我们引入了不同的技术来证明存在并使用不同的配方进行梯度流。我们应用一个紧凑的论点来证明存在与几种常见指标相对于迪里奇和罗宾边界条件的普遍最小化运动的存在。此外，我们使用Brunn-Minkowski不等式来证明$α$ - 概念性和在凸体上的Dirichlet边界条件的收缩半组存在。最后，我们通过第二个域变化给出了罗宾边界条件的$α$ - 概念的证明，因为在这种情况下我们没有布鲁恩·米科夫斯基的不平等。
We are interested in existence of gradient flows for shape functionals especially for first Laplacian eigenvalues. We introduce different techniques to prove existence and use different formulations for gradient flows. We apply a compactness argument to prove existence of generalized minimizing movements for Dirichlet and Robin boundary conditions with respect to several common metrics. Moreover, we use Brunn-Minkowski inequalities to prove $α$-convexity and existence of contraction semi-groups for Dirichlet boundary conditions on convex bodies. Finally, we give a proof of $α$-convexity for Robin boundary conditions by the second domain variation since we do not have a Brunn-Minkowski inequality in this case.
