Abstract
We study the behavior of solutionsu n of the equation: −Δu n =f, satisfying the Robin boundary condition imposed on the fragmented boundaries of many tiny holes. We show that some extension\(\tilde u_n \) ofu n converges tou, if the number of holes diverges to infinity and the diameter of holes tends to zero in a special way. Hereu is the solution of the equation: −Δu+Cu=f, which is considered on a domain with no hole. HereC is determined uniquely by the coefficient of the Robin boundary condition and an individual hole’s geometric data, which are analogous to the Hausdorff dimension and measure. We also give a similar result for solutions of the Poisson equation, satisfying semilinear boundary conditons on the boundaries of holes.
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Dedicated to Professor Hiroshi Fujita on his 60th birthday
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Kaizu, S. Behavior of solutions of the Poisson equation under fragmentation of the boundary of the domain. Japan J. Appl. Math. 7, 77–102 (1990). https://doi.org/10.1007/BF03167892
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DOI: https://doi.org/10.1007/BF03167892