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Civil-Comp Proceedings
ISSN 1759-3433 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. Antontsev1 and S. Shmarev2
1University of Beira Interior, Portugal
Full Bibliographic Reference for this paper
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", Civil-Comp 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 ![]() ![]() ![]() ![]() ![]()
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 Lebesgue-Orlicz spaces. The solution
of equation (37) is constructed as the limit of a sequence of
Galerkin's approximations. We claim that
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
For the uniqueness of solutions [2,3], it is shown that a.e.
bounded (small) solution of equation (37) is unique if
For equation (38), the uniqueness of bounded solutions is proved for
the assumptions that
For localization of solutions caused by the diffusion-absorption 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 diffusion-absorption the following alternative holds:
if u is a nonnegative weak solution of the exterior Dirichlet
problem for the equation
![]() ![]() ![]() ![]() ![]() ![]() Most of the results extend to solutions of equations of the type (37), (38) with the first-order terms (convection) and to systems of equations of similar structure [2,3]. References
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