| The thesis consists of two problems. One is analytical and concerns partial regularity of minimizing p-harmonic maps with free and partially constrained boundary conditions. The other is numerical and deals with the Monotonicity Formula for liquid crystals.; A minimizing p-harmonic map is a map u: M {dollar}to{dollar} N, where M, and N are Riemannian manifolds, that minimizes{dollar}{dollar}intsb{lcub}M{rcub}vertnabla uvertsp{lcub}p{rcub}dM{dollar}{dollar}among {dollar}Wsp{lcub}1,p{rcub}{dollar} functions. The first problem is based on two papers by F. H. Lin and R. Hardt ( (HL1), (HL2)) and deals with the regularity of a minimizer in the neighbourhood of the free or partially constrained boundary. It involves using Morrey's lemma ( (M)) to prove Holder continuity of the minimizer outside a singular set. A monotonicity formula is then used to improve the estimate on the size of the singular set.; We show that the singular subset of the free or partially constrained boundary is empty for m {dollar}le{dollar} (p), isolated for m = (p) + 1 and of Hausdorff dimension at most m {dollar}-{dollar} (p) {dollar}-{dollar} 1 for m {dollar}>{dollar} m {dollar}-{dollar} (p) {dollar}-{dollar} 1.; It is not known if the monotonicity formula is true for the general Oseen-Frank liquid crystal functional and this is the subject of the second problem. The liquid crystal phase is a phase of matter that is intermediate between the liquid and the solid phases. One model for nematic liquid crystals is the Oseen-Frank model. Here the bulk energy of the liquid crystal at a fixed temperature is given by{dollar}{dollar}E(n) = intsbOmega W(nabla n,n)dx,{dollar}{dollar}where {dollar}n(x){dollar} is a unit vector representing the average position of the optical axis and {dollar}W(nabla n,n){dollar} is the Oseen-Frank energy, which is{dollar}{dollar}eqalign{lcub}2W(nabla n,n) &= ksb1({lcub}rm div{rcub} n)sp2 + ksb2(ncdot{lcub}rm curl{rcub} n + tau)sp2cr &quad+ksb3vert ntimes{lcub}rm curl{rcub} nvertsp2 + (ksb2 + ksb4)({lcub}rm tr{rcub}(nabla n)sp2 - {lcub}rm div{rcub} n)sp2),cr{rcub}{dollar}{dollar}where {dollar}ksb1, ksb2, ksb3{dollar} are material constants. An equilibrium configuration is a minimum of {dollar}E(n){dollar}.; The Monotonicity Formula roughly states that normalized energy is monotonic.; We use a Conjugate Gradients method to find equilibrium configurations and then calculate the normalized energy on balls of different sizes to give numerical evidence that the Monotonicity Formula does hold for the Oseen-Frank energy. |
