My Ph. D. Thesis is in the area of regularity of local minimisers of quasiconvex variational integrals in the calculus of variations. The title of my PhD thesis is "Partial Regularity of Local Minimisers in the Calculus of Variations" and the abstract of the thesis is provided below.

PH. D Thesis Abstract.

The theory of local minimisers of the general variational integral

$$\int_{\Omega} F(Du(x))\, dx$$

is discussed, where latexit-0 is an open bounded domain and latexit-0. The focus is on the partial regularity of such minimisers. Certain partial regularity results are proved for a new class of local minimisers. As background to the result a number of topics important for the result are discussed. The first of these is quasiconvexity of the integrand latexit-0, important for existence and partial regularity of minimisers of the variational integral, above. This is followed by an introduction and discussion of Morrey, Campanato and latexit-0 spaces. Finally the regularity of latexit-0 Harmonic functions and elliptic systems of partial differential equations with continuous coefficients is established before the results of the manuscript are presented. The results are as follows: An a priori Campanato type regularity condition is established for a class of latexit-0 local minimisers latexit-0 of the general variational integral above, where latexit-0 is an open bounded domain, latexit-0 is of class latexit-0, latexit-0 is strongly quasi-convex and satisfies the growth condition

latexit-0

for a latexit-0 and where the corresponding Banach spaces latexit-0 are the Morrey-Campanato space latexit-0, latexit-0, Campanato space latexit-0 and the space of bounded mean oscillation latexit-0. The admissible maps latexit-0 are of Sobolev class latexit-0, satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for latexit-0 local minimisers is extended from Lipschitz maps to this admissible class.

Acknowledgements


Thanks to my supervisors Jan Kristensen and Bryan Rynne for there invaluable help and
advice during my Ph. D. Financial support was provided by the departmental doctoral training account.

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