Mathematical Sciences: "Wave and Heat Processes in Fractal Boundary Layers".

该提案的目标是通过分形边界研究反射、吸收和散射现象。著名物理学家M. Berry提出了关于分形可能产生的物理和数学效应的三个假设：1)散射振幅的间歇性结构；2)散射波（尾波）的较慢衰减；3)在薄边界层中存在大量状态，导致谱函数的非标准渐近行为。结果表明，分形边界域中的波和热的传播特性取决于分形的几何特性，更深入的分析首先需要分形的分类。该提案的作者挑选出三大类分形：白菜型（区域包含一个积累到外部边界的平切口系统），气泡型（区域包含一个积累到外部边界的气泡系统）和网状型（具有薄圆柱体系统）。有许多自然物体具有类似的边界结构：地球岩石圈、老化过程中高分子材料的边界层、金属的再结晶或回火层等。类似的结构也可以人工形成，以创造一种具有新特性的材料。证明了白菜型域上Dirichlet Laplacian的特征值计数函数具有两项渐近性，且第二项阶数等于边界Minkowski维数的二分之一，即修正的Weyl- Berry假设成立。他们还表明，这种假设在泡沫和网络类型的域名中是不成立的。提出了泡型和网型区域拉普拉斯Dirichlet问题的谱渐近解（特征值计数函数、theta函数、ζ函数），以及决定渐近解第二项的分形几何特征。提出了对包菜型域和泡型域的诺伊曼拉普拉斯谱m进行分类（确定谱是绝对连续、密集纯点、离散的），并找出Weyl律（谱渐近的主项存在）成立或谱渐近的第二项存在的条件。本文还提出了波浪方程在气泡和腹板型域中的外混合问题解的大时态，特别是散射波的大时态。该提案的目标是研究具有非常不规则（分形）边界的障碍物对波的反射、吸收和散射现象。结果表明，在边界非常不规则的域中，波和热的传播性质取决于边界的几何性质，而要进行更深入的分析，首先需要对边界的不规则性进行几何分类。提议者把障碍物分成了三大类：白菜型（障碍物包含一个平坦的裂缝系统，一直累积到外部边界），气泡型（障碍物包含一个气泡系统，一直累积到外部边界），以及网型（有一个薄圆柱体系统）。有许多自然物体具有类似的边界结构：地球岩石圈、老化过程中高分子材料的边界层、金属的再结晶或回火层等。类似的结构也可以人工形成，以创造一种具有新特性的材料。提出寻找障碍物的特性（白菜型、气泡型和网型障碍物的特性不同），这些特性决定了障碍物薄边界层反射、吸收或散射的波和热能，并找到波和热过程的长时间行为。＊＊＊

9623727 Molchanov/Vainberg The goal of the proposal is to study the phenomena of reflection, absorption and scattering by fractal boundaries. Well known physicist M. Berry formulated three hypotheses about possible physical and mathematical effects of fractalness: 1) intermittent structure of the scattering amplitude 2) a slower decay of scattered (coda) waves, 3) existence of a large number of states in a thin boundary layer which leads to a nonstandard asymptotic behavior of the spectral function . It turns out that properties of wave and heat propagation in domains with fractal boundaries depend on geometrical properties of fractals, and more profound analysis requires, first of all, classification of fractals. The authors of the proposal singled out three broad classes of fractals: cabbage type (domain contains a system of flat cuts accumulated to the outer boundary), bubble type (domain contains a system of bubbles accumulated to the outer boundary), and web type (with a system of thin cylinders). There are many natural objects with a similar boundary structure: the earth lithosphere, boundary layer of polymer materials in the process of aging, recrystallized or tempered layer of metals, etc. Similar structures can also be formed artificially to create a material with new properties. The authors proved that the eigenvalue counting function of the Dirichlet Laplacian in cabbage type domains has two- term asymptotics with the order of the second term equal to one half of the Minkowski dimension of the boundary, i.e. the modified Weyl- Berry hypotheses is valid. They also show that this hypotheses fails for bubble and web type domains. It is proposed to find the spectral asymptotics (eigenvalue counting function, theta-function, zeta-function) for the Dirichlet problem for the Laplacian in bubble and web type domains, and find geometrical characteristics of the fractal which determine the second term of the asymptotics. It is proposed to classify the spectru m of the Neumann Laplacian for cabbage and bubble type domains (determine when the spectrum is absolutely continuous, dense pure point, discreet) and find conditions when the Weyl law (existence of the main term of spectral asymptotics) is valid or the second term of the spectral asymptotics exists. It is also proposed to find large time behavior of the solutions to exterior mixed problems for the wave equation in bubble and web type domains, and in particular , to find large time behavior of the scattered waves. %%% The goal of the proposal is to study the phenomena of reflection, absorption and scattering of waves by obstacles with very irregular (fractal) boundaries. It turns out that properties of wave and heat propagation in domains with very irregular boundaries depend on geometrical properties of boundaries, and more profound analysis requires, first of all, geometrical classification of irregularity of the boundary. The proposers singled out three broad classes of obstacles: cabbage type (obstacle contains a system of flat cracks accumulated to the outer boundary), bubble type (obstacle contains a system of bubbles accumulated to the outer boundary), and web type (with a system of thin cylinders). There are many natural objects with a similar boundary structure: the earth lithosphere, boundary layer of polymer materials in the process of aging, recrystallized or tempered layer of metals, etc. Similar structures can also be formed artificially to create a material with new properties. It is proposed to find characteristics of the obstacles (they will be different for cabbage, bubble and web type obstacles) which determine wave and heat energy reflected, absorbed or scattered by a thin boundary layer of the obstacle and find long time behavior of the wave and heat processes. ***