Reference for transformation of integrals over Lipschitz boundaries

Question

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function $f:\partial D\to \mathbb{R}$ on the boundary $\partial D$ of $D$ by \begin{align} \int_{\partial D} f(x) ~dS(x)=\sum_{r=1}^m \int_{\Delta_r}f(x_r',0)\varphi_r(x_r',0)\left[1+\sum_{i=1}^{d-1} \left(\frac{\partial a_r(x_r')}{\partial x_{ri}}\right)^2\right]^{1/2} dx_r', \end{align} where $(\varphi_r)_{r=1}^m$ is a partition of one (w.r.t. some open sets covering the boundary), $x_r'\in \Delta_r\subseteq\mathbb{R}^{d-1}$ are local coordinates (upon rotating and translating the domain), and the boundary is locally given as the graph of the functions $a_r$. (I hope it is clear what I mean, but I don't want to write down the whole construction)

I am wondering whether someone knows a reference of what happens when $\Psi:D'\to D$ is a bi-Lipschitz map (i.e. bijective, Lipschitz and the inverse is Lipschitz as well) for some Lipschitz domain $D'$: how can one then transfer \begin{align} \int_{\partial D} f~dS \end{align} to $D'$? For example, I know that if $D,D'$ are $C^1$ manifolds and $T$ is a $C^1$ diffeo, then \begin{align} \int_{\partial D} f~dS=\sum_j \int_{\partial D'\cap O_j} f\circ\Psi \cdot \sqrt{\frac{\det D(\Psi\circ \chi_j)^TD(\Psi\circ\chi_j)}{\det D\chi_j^TD\chi_j}}(\chi_j^{-1}(x))~ dS'(x), \end{align} where $\chi_j$ denote the charts and $O_j\subseteq \mathrm{ran}\chi_j$ is a partition of $\partial D'$. Does the same hold if the transformation $\Psi\in W^{1,\infty}$ is only bi-Lipschitz?

Any help is very much appreciated.

Answer 1

Yes, the change of variables holds for bi-Lipschitz maps. Let's clarify the picture first. It's not really relevant that $S=\partial D$ and $S'=\partial D'$ are boundaries of some domains; the could be any two Lipschitz surfaces. Also, we don't need $\Psi'$ to be defined on $D'$, we need it as a bi-Lipschitz map $\Psi:S'\to S$. (Of course you can extend a bi-Lipschitz map to the boundary, and the extension will be bi-Lipschitz).

The Change of Variable formula for Lipschitz maps is stated and proved in Evans-Gariepy, p.99.

Let $f:\mathbb R^n\to\mathbb R^m$ be Lipschitz, $n\le m$. Then for each integrable function $g:\mathbb R^n\to\mathbb R$, $$ \int_{\mathbb R^n} g(x)Jf(x)\,dx = \int_{\mathbb R^m}\bigg(\sum_{x\in f^{-1}(y)} g(x)\bigg) \,d\mathcal H^n(y) $$

Here $Jf(x) = \sqrt{\det (Df(x)^* Df(x))}$ where $Df$ is the matrix of first order derivatives of $f$. If $f$ is injective, then on the right we just have $g(f^{-1}(y))$.

Using this formula together with a partition of unity, you can express an integral over a Lipschitz surface in terms of integrals over $\mathbb R^n$ involving the Jacobian of parametrization. When the parametrization is composed with a bi-Lipschitz map $\Psi$, the chain rule applies (almost everywhere, since Lipschitz maps are differentiable a.e.) And $J(\Phi\circ f) = (J(\Phi)\circ f)\, Jf$ by the definition of $J$.