汽车传感器检测图解.pdf

arXiv: [] 11 Dec 2009 LEMNISCATES DO NOT SURVIVE LAPLACIAN GROWTH D. KHAVINSON, M. MINEEV-WEINSTEIN, M. PUTINAR, AND R. TEODORESCU Many moving boundary processes in the plane, ., solidi?cation, electrodeposition, viscous ?ngering, bacterial growth, etc., can be math- ematically modeled by the so-called Laplacian growth [9, 13]. In a nut- shell, it can be described by the equation () V(z) =? ng ?(t)(z, ζ), whereVis the ponent of the velocity of the boundary??(t) of the moving domain ?(t)?R 2?C,z∈??(t),tis time, ??n denotes the normal derivative on??(t) andg ?(t)(z, ζ) is the Green function for the Laplace operator in the domain ?(t) with a unit source at the pointζ∈?(t). Equation () can be elegantly rewritten as the area- preserving di?eomorphism () ?(ˉz tz θ) = 1, where?denotes the imaginary part of plex number,??(t) := {z:=z(t, θ)}is the moving boundary parametrized byw=e iθon the unit circle and the conformal mapping from, say, the exterior of the unit diskD +:={|w|>1}onto ?(t) with the normalizationz(∞) = ζ, z ′(∞)>0. The equation (), namedLaplacian growthor thePolubarinova - Galinequation in modern literature, was ?rst derived by Polubarinova- Kochina [11] and Galin [7] in 1945, as a description of secondary oil recovery processes. This equation is known to be integrable [10], and as such possesses an in?nte number of conserved quantities. More precisely, it admits Part of this work was done during the ?rst and third authors’ visit to LANL and was supported by the LDRD project 20070483 “Minimal Description plex 2D Shapes” at LANL. Also, D. Khavinson and M. Putinar gratefully acknowledge partial support by the National Science Foundation. 1 2 KHAVINSON, MINEEV-WEINSTEIN, PUTINAR, AND TEODORESCU conserved momentsc n= R ?(t) z ndxdy, wherenruns over either all non- negative or all non-positive integers depending on whetherdomains ?(t) are ?nite or in?nite. At the same time () admits an impressive numb