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. ***
