<div>This thesis mainly focuses on the PDE theories that arise from the study of hydrodynamics of nematic liquid crystals. </div><div><br></div><div>In Chapter 1, we give a brief introduction of the Ericksen--Leslie director theory and Beris--Edwards <i>Q</i>-tensor theory to the PDE modeling of dynamic continuum description of nematic liquid crystals. In the isothermal case, we derive the simplified Ericksen--Leslie equations with general targets via the energy variation approach. Following this, we introduce a simplified, non-isothermal Ericksen--Leslie system and justify its thermodynamic consistency. </div><div><br></div><div>In Chapter 2, we study the weak compactness property of solutions to the Ginzburg--Landau approximation of the simplified Ericksen--Leslie system. In 2-D, we apply the Pohozaev type argument to show a kind of concentration cancellation occurs in the weak sequence of Ginzburg--Landau system. Furthermore, we establish the same compactness for non-isothermal equations with approximated director fields staying on the upper semi-sphere in 3-D. These compactness results imply the global existence of weak solutions to the limit equations as the small parameter tends to zero. </div><div><br></div><div>In Chapter 3, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards system for both the Landau–De Gennes and Ball–Majumdar bulk potentials in 3-D, and then study its partial regularity by proving that the 1-D parabolic Hausdorff measure of the singular set is 0.</div><div><br></div><div>In Chapter 4, motivated by the study of un-corotational Beris--Edwards system, we construct a suitable weak solution to the full Ericksen--Leslie system with Ginzburg--Landau potential in 3-D, and we show it enjoys a (slightly weaker) partial regularity, which asserts that it is smooth away from a closed set of parabolic Hausdorff dimension at most 15/7.</div>

In this dissertation we study free boundary problems that model the evolution of interfaces in the presence of elasticity, such as thin film profiles and material void boundaries. These problems are characterized by the competition between the elastic bulk energy and the anisotropic surface energy. First, we consider the evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate. The film is strained due to the mismatch between the crystalline lattices of the two materials and anisotropy is taken into account. We present the results contained in [62] where the author establishes short time existence, uniqueness and regularity of the solution using De Giorgi’s minimizing movements to exploit the L2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity. Second, we consider the relaxed energy introduced in [20] that depends on admissible pairs (E, u) of sets E and functions u defined only outside of E. For dimension three this energy appears in the study of the material voids in solids, where the pairs (E, u) are interpreted as the admissible configurations that consist of void regions E in the space and of displacements u of the atoms of the crystal. We provide the precise mathematical framework that guarantees the existence of minimal energy pairs (E, u). Then, we establish that for every minimal configuration (E, u), the function u is C 1,γ loc -regular outside an essentially closed subset of E. No hypothesis of starshapedness is assumed on the voids and all the results that are contained in [18] hold true for every dimension d ≥ 2.

[pt] Primeiro examinamos soluções de viscosidade Lp para equações elípticas totalmente não lineares com ingredientes de fronteira mensuráveis. Ao considerar p0 < p < d, focamos nas estimativas da regularidade dos gradientes derivadas de potenciais não lineares. Encontramos condições para Lipschitz-continuidade local das soluções e continuidade do gradiente. Examinamos avanços recentes na teoria da regularidade decorrentes de estimativas potenciais (não lineares). Nossas descobertas decorrem de – e são inspiradas por – fatos fundamentais na teoria de soluções de Lp-viscosidade, e resultados do trabalho de Panagiota Daskalopoulos, Tuomo Kuusi e Giuseppe Mingione (DKM2014). Na segunda parte provamos a regularidade parcial de mapas harmônicos com peso fracamente estacionários com dados de fronteira livre em um cone. Como ponto de partida, damos uma olhada na teoria da regularidade parcial interior para mapas harmônicos fracionários de minimização de energia intrínseca do espaço euclidiano em variedades Riemannianas compactas e suaves para potências fracionárias estritamente entre zero e um. Mapas harmônicos fracionários intrínsecos podem ser estendidos para mapas harmônicos com peso, então provamos regularidade parcial para mapas harmônicos minimizantes locais com dados de fronteira (parcialmente) livres em meios-espaços, mapas harmônicos fracionários então herdam essa regularidade. / [en] We first examine Lp-viscosity solutions to fully nonlinear elliptic equations with bounded measurable ingredients. By considering p0 < p < d, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from – and are inspired by – fundamental facts in the theory of Lp-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione (DKM2014). In the second part we prove partial regularity of weakly stationary weighted harmonic maps with free boundary data on a cone. As a starting point we take a look at the interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps can be extended to weighted harmonic maps, so we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces, fractional harmonic maps then inherit this regularity.