Existence of minimizers in magnetoelasticity and a ferronematic model with a stray field energy

This thesis investigates the existence of solutions and examines the underlying physical phenomenon in mathematical models that describe the physical states of complex materials, by using the framework of non-linear partial differential equations. The model problems pertain to magnetoelastic and ferronematic materials. In the context of the magnetoelastic model, we have proven the existence of an energy minimizer using the direct method in the calculus of variations. The key contribution in this part is the introduction of a new admissible space that extends beyond those considered in previous studies. The proposed admissible space for deformations consists of mappings of finite distortion. Notably, we have proved the compactness result under a weaker condition in the integrability of the outer distortion coefficient. The establishment of our compactness result is closely related to the Iwanie-ˇSver´ ak conjecture. Employing the Ciarlet-Neˇ cas condition, we provide a positive instance of the Iwaniec-ˇSver´ak conjecture. As a result, the admissible space for deformations is characterized by homeomorphic mappings. Furthermore, the existence result has been obtained for compressible magnetoelastic solids. In the context of the ferronematic model, we have proposed a ferronematic energy for a ferronematic material in a bounded two-dimensional domain. The proposed energy comprises the Landau-de Gennes energy for Q-tensor order parameter, the micromagnetics energy for magnetization M, and a coupling energy due to the interaction between Q and M. In the energy functional, we explicitly incorporate the contribution of the stray field effect by including the stray field energy and the energy due to the coupling between the nematic order and the stray field. In particular, we have incorporated the stray field energy in the two-dimensional model by using the limiting micromagnetic energy for very thin films, derived by Gioia and James GJ97. We then study the key aspects related to the proposed ferronematic energy: we prove the existence of an energy minimizer using the direct method in the calculus of variations, we derive local L^∞-bounds for weak solutions of the derived Euler-Lagrange equations using a maximum principle argument, and we prove the uniqueness of the minimizer using derived L^∞-bounds. Moreover, we construct the numerical solutions of the corresponding gradient flow system by employing the Crank-Nicolson finite difference scheme. The numerical results reveal the influence of the stray field on nematic and magnetic configurations: the stray field has an impact on the localization of the nematic defects and magnetic vortices, and it contributes to the change in shape of the core of localized defects.