Non-linear Elliptical Equations on the Sierpiński Gasket |
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Authors: | Kenneth J Falconer Jiaxin Hu |
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Institution: | a Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, United Kingdom |
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Abstract: | This paper investigates properties of certain nonlinear PDEs on fractal sets. With an appropriately defined Laplacian, we obtain a number of results on the existence of non-trivial solutions of the semilinear elliptic equation
with zero Dirichlet boundary conditions, where u is defined on the Sierpiński gasket. We use the mountain pass theorem and the saddle point theorem to study such equations for different classes of a and f. A strong Sobolev-type inequality leads to properties that contrast with those for classical domains. |
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Keywords: | Sierpiń ski gasket Laplacian operator weak solution mountain pass theorem saddle point theorem Sobolev-type inequality |
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