Homogenization of some linear and nonlinear partial differential equations
Sammanfattning: In this thesis is studied two-scale convergence for sequences of linear and nonlinear evolution problems. In particular homogenization procedures, correctors and a numerical algorithm for such problems are studied. The main results are the following: We prove a crucial compactness results of two-scale convergence type. The compactness result is used to homogenize and prove corrector results for nonlinear parabolic problems with nonperiodic coefficients which oscillate with different speeds in the time and space variables. Moreover, we derive a numerical algorithm for this kind of problems. We prove compactness results of two-scale type, useful for homogenizing the linear Maxwell equations. We homogenize Maxwell´s equations by the use of two-scale convergence and prove some new corrector results. We prove two-scale compactness results applicable to nonlinear Maxwell´s systems. We define and characterize some function spaces which are well suited for Maxwell´s equations equipped with nonlinear constitutional relations. We prove the existence of a unique solution to the Maxwell system when the electric conductivity is uniformly monotone. Moreover, we homogenize the system and prove new corrector results in the case of nonperiodic coefficients. We prove a compactness result of two-scale type useful for Maxwell´s equations in the case of nonlinear permittivity and magnetic permeability. We prove the existence of solutions and homogenize and prove new corrector results for the Maxwell equations describing electromagnetic fields in nonperiodic materials with nonlinear permeability, permittivity and conductivity. We give some new examples of applications for homogenized linear and nonlinear Maxwell´s equations, e.g., the induced polarization phenomenon which is extensively used in prospecting for impregnation ores, ceramic varistors which are used as devices to protect electrical equipment against overvoltages, nonlinear optical applications and semiconductors in general.
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