Could someone explain me (roughly) the interest of Bounded Variation (BV) Spaces for PDEs ? Is there any numerical application of those space to real problems or is it just a theoretic way to formulate specific problems ?
Thank you very much.
$BV$ spaces are exactly what you want in many inverse problem. The point is that in many inverse problems you try to determine a function $q(x)$ that describes the internal properties of a body -- say, the water content (MRI), the density (X-ray) or elastic coefficients (ultrasound). A good approximation is that this function $q(x)$ is constant (or, at least, varying little) within each part of the body (e.g. body vs. soft tissue, one organ vs. the next) but that it is discontinuous between different parts. In other words, $q(x)$ is discontinuous but the area of the surfaces where it can be discontinuous is bounded. Functions that satisfy such properties are exactly in the space $BV$ but not in any of the usual Sobolev spaces.
A separate application for $BV$ is in plasticity with weakening -- think of materials like rocks that, when they break due to excessive strain, become dramatically weaker and keep deforming in the very same places. Tectonic fault lines are examples of this. In such cases, the displacement is discontinuous along these lines but continuous in the rest of the domain. Again, $BV$ is the right space to seek solutions in in such cases.