Regularity for solutions to parabolic systems and nonlocal minimization problems

The goal of this dissertation is to contribute to both the nonlocal and the local settings of regularity theory within the calculus of variations. In the nonlocal theory, we first establish the existence of minimizers for two classes of functionals. However, the main result of Chapter 2 states an analogue for higher differentiability of minimizers in the setting of nonlocal functionals, which is established through an application of the difference quotient method. This nonlocal analogue is stated in terms of the fractional order difference quotient, which corresponds to the order of the Besov space to which the solution belongs. In the third chapter, we investigate the regularity of solutions to the parabolic system [special characters omitted]In particular, we show that, under subquadratic growth and ellipticity conditions, solutions of the above system will be Hölder continuous with exponent α ∈ (0, 1) when the coefficients are continuous. In other words, it is shown that there is an open subset of full measure, when compared to the domain for the problem, on which the solution is Hölder continuous. In order to prove the result, we appeal to the A-caloric Approximation Method