This paper first proposes a new approximate scheme to construct a harmonic heat flow $u$ between a parabolic cylinder to a sphere. Y.Chen and M.Struwe have proved an existence and discussed a partial regularity of harmonic heat flows by using Ginzburg-Landau heat flow and passing to the limit of a parameter appeared in the equation. To construct a new harmonic heat flow, we propose a Ginzburg-Landau type heat flow with a time-dependent parameter. and we next establish the existence of a harmonic heat flow into spheres with (i) a global energy inequality, (ii) a monotonicity for the scaled energy, (ii) a reverse Poincare inequality. These inequalities (i), (ii) and (iii) improves the Hausdorff dimensional estimates on it's singular set contrast to the former results. I believe that inequalities (i), (ii) and (iii) allow us to analyze how it behaves around its singularities.
Payne-Weinberger证明了\textit{`among the class of membranes with given area $A$, free along the interior boundaries and fixed along the outer boundary of given length $L_0$, the annulus $Ω^\#$ has the highest fundamental frequency,'}，其中$Ω^\#$是与$Ω$具有相同面积且与$L_0$具有相同外边界长度的同心环。我们将这一结果推广到高维域和$p$ -拉普拉斯算子，并在外边界为球面的假设下，用$p\in (1,\infty),$推广。作为应用，我们证明了球和同心环上$p$ -Laplacian（具有混合边界条件）第二特征函数的节点集不可能是同心球.
PPW and Chitti type inequalities for the Robin Laplacian operator
本文证明了Robin本征值问题的两个结果。一个是前两个特征值之比的上界，它可以用来恢复M. S. Ashbaugh和R. D. Benguria证明的PPW猜想;另一个是第一特征函数的反向Holder不等式，它是Dirichlet Laplacian第一特征函数的Chiti反向Holder不等式的自然推广.
Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Institute of Mathematics, Ufa Federal Research Centre, RAS（俄罗斯科学院乌法联邦研究中心数学研究所）；Ufa University of Science and Technology（乌法科学技术大学）；Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia（西波希米亚大学应用科学学院数学与NTIS系）
