Integration by parts and the function of bounded variation 1000-1M15CPC

Here is the well-known integration by parts formula: the integral of divergence of a vector field X over region U equals the integral of the
normal component of field X over the boundary of U. This formula require that we know what is:
1) the boundary of region U and the surface a measure on it;
2) the normal component of field X.
The situation is well-known when the boundary of U is a smooth manifold and field X is also smooth.

In many interesting situation these objects are not regular. They naturally appear in the calculus of variation, e.g. in the least gradient problem, or shape
optimization. We will consider two particular cases:
a) when the characteristic function of set U has bounded total variation. This approach requires introducing the theory of BV spaces
or
b) vector field X and its divergence are from L^p, but then we have to explain what is the normal component of a vector field.

We will present the necessary elements of a theory of BV spaces and measure theory.

The results we will present will be applied to natural examples arising in the calculus of variations, e.g. those mentioned above and theory of
PDEs.