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Source: https://homepages.warwick.ac.uk/staff/C.M.Elliott/DziEll13a.pdf

Definition 2.11 of the weak derivative:

A function $f \in L^1(\Gamma)$ has the weak derivative $v_i=D_if \in L^1(\Gamma)$, $i \in \{1,...,n+1\}$, if for every function $\phi \in C^1_0(\Gamma)$ we have the relation $$\int_\Gamma f D_i\phi \,dA =- \int_\Gamma\phi v_i \,dA+ \int_\Gamma f\phi Hv_id\,A$$ where $\Gamma$ is a hypersurface in $\mathbb R^{n+1}$ and $H$ is its mean curvature.

Question:

I am familiar with the "standard" definition of the weak derivative (https://en.wikipedia.org/wiki/Weak_derivative). How are these two different defintions to be reconciled and what role does the mean curvature get into the equation? Some explanation/interpretation would be much appreciated.

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  • $\begingroup$ it seems like in the pdf you link the last surface integral doesn't have a $v_i$ but a $\nu_i$, where $\nu$ seems to be the surface normal. $\endgroup$ – 0x539 Jan 14 at 19:04
  • $\begingroup$ Are you happy with Theorem 2.10? The paper says "The formula for integration by parts on Γ leads to the notion of a weak derivative" i.e. Theorem 2.10 justifies Definition 2.11 - the derivative of a $C^1$ function is the weak derivative. A weak derivative is just something that makes integration by parts work. $\endgroup$ – Dap Jan 14 at 21:22
  • $\begingroup$ @Dap So what is the reason that it differs from the conventional definition for a weak derivative? $\endgroup$ – Tesla Feb 15 at 14:54
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    $\begingroup$ @Tesla: because it's using partial derivatives in the ambient space, which is a different setting to just using derivatives in $\mathbb R^n$ to define weak derivatives on functions on $\mathbb R^n.$ So for the same reason that Theorem 2.10 is different from conventional Stokes theorem/integration by parts. $\endgroup$ – Dap Feb 15 at 16:54
  • $\begingroup$ @Dap It's been a while, but I picked it up again. So how does one exactly get the partial derivatives in the ambient space? $\endgroup$ – Tesla May 14 at 13:42

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