To Pei Chen, whose love and support lift me daily Abstract. In this paper we give simpler proofs of the classical continuity and lower semicontinuity theorems of Reshetnyak. In 1968, Reshetnyak  proved two important results concerning the continuity and lower semicontinuity of functionals with respect to weak-star convergence of measures. These theorems… (More)
This work presents some new characterizations of Sobolev spaces and the space of functions of Bounded Variation. Additionally it gives new proofs of continuity and lower semicontinuity theorems due to Reshetnyak. In the recent paper , Bourgain, Brezis, and Mironescu studied the limiting behavior of the semi-norm |f | W s,p (Ω) := Ω Ω |f (x) − f (y)| p… (More)
In this paper, we prove L p estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are V M O. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.
for their help during my stay here at Pittsburgh. Finally, I would like to thank the Department of Mathematical Sciences at Carnegie Mellon University for creating a comfortable environment of study. In particular, I want to thank Professor William J. Hrusa for being such an understanding and generous graduate advisor. Special thanks go to my family for… (More)
In this paper, we study nonlocal gradients and their relationship to classical gradients. As the nonlocality vanishes we demonstrate the convergence of nonlocal gradients to their local analogue for Sobolev and BV functions. As a consequence of these localizations we give new characterizations of the Sobolev and BV spaces that are in the same spirit of… (More)
The principle of material frame-indifference is a fundamental concept in classical physics. In 1972 Walter Noll showed that one can formulate constitutive laws without using any external frames of reference. Constitutive laws formulated in this way vacuously satisfy the principle of material frame-indifference. This thesis describes the basic concepts… (More)