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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 $\partial \Omega$.

Calvin Khor
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Prakhar Gupta
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1 Answers1

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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 $$f \mapsto \int_{\partial \Omega}f \eta.$$

Hugo
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  • Observation: The determinant above is the determinant of the matrix obtained using each vector as column vector. – Hugo Aug 09 '18 at 05:52
  • I am quite unaware of Riemannian volume forms. Can you suggest me some resources to read about them? – Prakhar Gupta Aug 09 '18 at 06:08
  • 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 Aug 09 '18 at 11:30