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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 16
A Theory of Elliptic Equations with Variable Nonlinearity S. Antontsev^{1} and S. Shmarev^{2}
^{1}University of Beira Interior, Portugal
S. Antontsev, S. Shmarev, "A Theory of Elliptic Equations with Variable Nonlinearity", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 16, 2006. doi:10.4203/ccp.84.16
Keywords: elliptic equation, nonstandard growth conditions, variable nonlinearity, localization, anisotropy.
Summary
We study the Dirichlet problem for the elliptic equations with
variable anisotropic nonlinearity
The boundary is Lipschitzcontinuous. The coefficients , c and the exponents of nonlinearity , , are prescribed functions of their arguments. Equations of the type (37), (38) emerge from the mathematical modelling of various physical phenomena, e.g., the processes of image restoration, flows of electrorheological fluids, thermistor problem, filtration through inhomogeneous media. For the existence of solutions [1,2], we prove that under suitable restrictions on the coefficients and the nonlinearity exponents the Dirichet problem for equations (37) and (38) admit an a.e. bounded weak solutions which belong to the anisotropic analogs of the generalized LebesgueOrlicz spaces. The solution of equation (37) is constructed as the limit of a sequence of Galerkin's approximations. We claim that , , , and that the exponents are continuous with a logarithmic module of continuity. The existence of an a.e. bounded weak solution of equation (38) is proved by means of the Schauder fixed point principle. It is requested that and either in and , or and . For the uniqueness of solutions [2,3], it is shown that a.e. bounded (small) solution of equation (37) is unique if and either , and is Lipschitzcontinuous with respect to s, or if , and . For equation (38), the uniqueness of bounded solutions is proved for the assumptions that , , , . For localization of solutions caused by the diffusionabsorption balance [2], a localization (or vanishing on a set of nonzero measure) is an intrinsic property of solutions to nonlinear elliptic equations. It is known that for the solutions of equations of the type (37), (38) with constant (but possibly anisotropic) nonlinearity such an effect appears due to a suitable balance between the diffusion and absorption terms of the equation. We show that the same is true for equations with variable exponents of nonlinearity. The proof relies on the method of local energy estimates. For directional localization caused by anisotropic diffusion [1,3], it is known that for the solutions of nonlinear equations of the type diffusionabsorption the following alternative holds: if u is a nonnegative weak solution of the exterior Dirichlet problem for the equation with constant exponents p and , then
Most of the results extend to solutions of equations of the type (37), (38) with the firstorder terms (convection) and to systems of equations of similar structure [2,3]. References
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