# Meaning of surface measure

While studying PDE, I came across this trace operator which talks about the class $$L^{p}(\partial \Omega)$$ for open sets $$\Omega \subset \mathbb{R}^{n}$$ with $$C^{1}$$ boundary.

I don't understand what measure we give to $$\partial \Omega$$.

Notice $\partial \Omega$ is a Riemannian hypersurface of $\mathbb R^n$. The measure on $\partial \Omega$ is the one given by the Riemannian metric.

Let me define the measure more explicitly.

Since $\Omega$ is open (and therefore orientable), $\partial \Omega$ is orientable. Knowing $\partial \Omega$ is an orientable hypersurface we have a global normal vector field $n$. We can suppose $|n(p)|=1$ for all $p\in \partial \Omega$.

For each $p$ define $$\eta(v_1,\ldots,v_{n-1}) = \det(n(p),v_1,\ldots,v_{n-1})$$ where $v_1,\ldots,v_{n-1} \in T_p \partial \Omega$. This $\eta$ is the Riemannian volume form of $\partial \Omega$.

The measure you want is the one provided by this volume form. More precisely you can obtain the measure using the Riesz–Markov–Kakutani representation theorem on the functional $C(\partial \Omega) \to \mathbb R$ given by $$f \mapsto \int_{\partial \Omega}f \eta.$$

• Observation: The determinant above is the determinant of the matrix obtained using each vector as column vector.
– Hugo
Commented Aug 9, 2018 at 5:52
• I am quite unaware of Riemannian volume forms. Can you suggest me some resources to read about them? Commented Aug 9, 2018 at 6:08
• In the case of hypersurfaces on Euclidean spaces you may want to take a look at "Calculus on Manifold" by Michael Spivak and "Mathematical Analysis 2" by Vladimir A. Zorich. Both books talk about the volume formula using determinant. I personally enjoy more Zorich's book. For the general case, on general Riemannian manifolds, take a look at John M. Lee wonderful book "Introduction to Smooth Manifolds", second edition.
– Hugo
Commented Aug 9, 2018 at 11:30