Definition of the weak derivative involving the mean curvature
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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 $$int_Gamma f D_iphi ,dA =- int_Gammaphi v_i ,dA+ int_Gamma fphi Hv_id,A$$
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.
ordinary-differential-equations differential-geometry curvature weak-derivatives
asked Jan 1 at 13:08
TeslaTesla
890426
$endgroup$
add a comment |
$begingroup$
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 $$int_Gamma f D_iphi ,dA =- int_Gammaphi v_i ,dA+ int_Gamma fphi Hv_id,A$$
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.
ordinary-differential-equations differential-geometry curvature weak-derivatives
asked Jan 1 at 13:08
TeslaTesla
890426
$endgroup$
$begingroup$
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
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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
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@Dap So what is the reason that it differs from the conventional definition for a weak derivative?
$endgroup$
– Tesla
Feb 15 at 14:54
$begingroup$
@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
add a comment |
$begingroup$
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 $$int_Gamma f D_iphi ,dA =- int_Gammaphi v_i ,dA+ int_Gamma fphi Hv_id,A$$
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.
ordinary-differential-equations differential-geometry curvature weak-derivatives
asked Jan 1 at 13:08
TeslaTesla
890426
$endgroup$
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 $$int_Gamma f D_iphi ,dA =- int_Gammaphi v_i ,dA+ int_Gamma fphi Hv_id,A$$
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.
ordinary-differential-equations differential-geometry curvature weak-derivatives
ordinary-differential-equations differential-geometry curvature weak-derivatives
asked Jan 1 at 13:08
TeslaTesla
890426
asked Jan 1 at 13:08
TeslaTesla
890426
asked Jan 1 at 13:08
TeslaTesla
890426
asked Jan 1 at 13:08
TeslaTesla
890426
asked Jan 1 at 13:08
TeslaTesla
890426
890426
$begingroup$
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
$begingroup$
@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
add a comment |
$begingroup$
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
$begingroup$
@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
$begingroup$
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
$begingroup$
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
$begingroup$
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
$begingroup$
@Dap So what is the reason that it differs from the conventional definition for a weak derivative?
$endgroup$
– Tesla
Feb 15 at 14:54
$begingroup$
@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
$begingroup$
@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
add a comment |
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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
$begingroup$
@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