Publicação: Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation
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World Scientific Publ Co Pte Ltd
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We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type partial derivative u/partial derivative n + g( x, u) = 0 when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by partial derivative u/partial derivative n+gamma(x) g(x, u) = 0, where gamma(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary,. is a bounded function and we show the convergence of the solutions in H-1 and C-alpha norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.
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varying boundary, oscillations, nonlinear boundary conditions, elliptic equations
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Inglês
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Mathematical Models & Methods In Applied Sciences. Singapore: World Scientific Publ Co Pte Ltd, v. 17, n. 10, p. 1555-1585, 2007.
