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An algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume, a problem often arising in cell biology. It applies to pixilated surfaces obtained from experimental images and performs at low computational cost. In the method, the Laplace-Beltrami operator is approximated locally by the Laplacian on the tangential plane and then a finite volume discretization scheme based on a Voronoi decomposition is applied. Convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2. The method is applied to a cellbiological problem where a signaling molecule, G-protein Rac, cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior. Simulations on realistic cell geometry are performed to validate, and determine the accuracy of, a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching. The method is implemented within the Virtual Cell computational framework freely accessible at www.vcell.org.

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In this paper, we study global positive C 4 solutions of the geometrically interesting equation: 2 u + u −q = 0 with q > 0 in R 3 . We will establish several existence and non-existence theorems, including the classification result for q = 7 with exactly linear growth condition.

Abstract.We consider the problemUyy with a > 0 , ¿> > 0 , on a smooth convex bounded region in R2 with Dirichlet boundary conditions. We show that if the positive function p is uniformly bounded away from zero, then the problem has a classical solution.

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