Evolution equations for shapes and geometries
The authors consider the evolution problem \(dx(t)/dt=V(t,x(t))\), where \(V\) is a given velocity field in \(L^{\infty }(]0,\tau [ ,C_{0}^{0,1}( \overline{D},\mathbb{R}^{N}))\), at least, where \(D\) is an open subset of \( \mathbb{R}^{N}\) with Lipschitz boundary. The solution \(x\) starts from a given data \(x_{0}\) at \(t=0\). The authors define the transformation mapping \( T_{t}\) from \(D\) to \(D\) through \(T_{t}(x_{0})=x(t)\). For any measurable subset \(\Omega \) of \(D\), they finally define the domain \(\Omega _{t}=T_{t}(\Omega )\). The first main result of the paper proves some continuity and differentiability properties of the functions defined through \(t\mapsto \int_{\Omega _{t}}\varphi\,dx\), for every \(\varphi \in L^{1}(D)\). Then the authors prove that the characteristic function \(\chi \) of \(\Omega _{t}\) is the solution of the first-order differential equation \(d\chi (t)/dt+\nabla \chi (t)\cdot V=0\), \(\chi \) starting from the characteristic function of \(\Omega \) at \(t=0\). The next part of the paper proves quite similar similar results but for the characteristic function of the boundary \( \Gamma _{t}\) of \(\Omega _{t}\). A weak formulation of this equation is introduced. In the last part of the paper, the authors introduce the oriented distance function \(b_{\Omega }=d_{\Omega }-d_{\complement \Omega }\) . The last main result of the paper proves that \(b_{\Omega _{t}}\) is the solution of a quite similar first-order evolution equation, which involves the projection on \(\Gamma _{t}\). Here \(\Omega \) is supposed to have a thin boundary.
scientific article; zbMATH DE number 5118022