Lusin-type theorem for functions with prescribed gradient

In the first part of this thesis we discuss and prove a theorem by Giovanni Alberti whose statement shares similarities to that of Lusin's Theorem, hence the "Lusin-type theorem" definition. The theorem states that given a Borel vector field f on a finite measure set and some epsilon greater than zero, it is always possible to find a set of measure less than epsilon such that there exists a function whose gradient is exactly f outside this set and with p-norm bounded by the norm of f; in other words, the theorem states that for every vector field there exists a function which is a potential for it except on a small set. The proof of the theorem uses some auxuliary lemmas and an iterative construction to describe the function as a series, and common tools of Real Analysis and Measure Theory to prove the p-norm inequalities. In the second section of this work, we refine Alberti's result requesting higher regularity for the function whose gradient is f, namely Lipschitz continuity and Holder continuity of the gradient, and we obtain a result similar to Alberti's theorem; we do this with the same tools used previously and some notions on fractal sets and Hausdorff dimensions. Lastly, we show that for a specific vector field, imposing even higher regularities strongly limits the Hausdorff dimension of the set where the gradient agrees with the vector field, up to the point where the Hausdorff dimension is halved; this is proved using more specific results of Geometric Measure Theory on porosity and Hausdorff dimension.