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Does the LegendreHadamard condition imply a generalized Gårding inequality?
Terry, thanks again for your help. As Denis' answer shows, the tangential Fourier transform does indeed provide the answer. 
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Does the LegendreHadamard condition imply a generalized Gårding inequality?
Many thanks! I haven't had a chance to study your paper carefully, but it looks good. I've learned that elliptic systems of PDE's can have much more subtle behavior than scalar elliptic PDE's. 
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accepted  Does the LegendreHadamard condition imply a generalized Gårding inequality? 
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Jan 23 
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Does the LegendreHadamard condition imply a generalized Gårding inequality?
Terry, perhaps I'm trying to do the reflection too naively but I can't get it to work. If I reflect the function about the hyperplane $x^n = 0$, then the argument seems to work only if $A^{in}_{ab} = 0$ for $1 \le i \le n1$. Or if $u=0$ along the hyperplane. 
Jan 22 
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Does the LegendreHadamard condition imply a generalized Gårding inequality?
Terry, thanks! Obvious after you said it. I'll try that. 
Jan 21 
revised 
Does the LegendreHadamard condition imply a generalized Gårding inequality?
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Jan 21 
asked  Does the LegendreHadamard condition imply a generalized Gårding inequality? 
Jan 20 
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The Minkowski sum of two curves
In the generic situation it should be possible to approximate by smooth curves and take a limit. But there will definitely be singularities, where the sum is not open. I don't know how to identify this when the curves are not differentiable. 
Jan 20 
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The Minkowski sum of two curves
It's instructive to look at some specific examples. Setting $\gamma $ to $y = 0$ and $y = 1$ are instructive. More generally, if you parameterize both $\gamma$ and $\gamma'$ with two different parameters, say, $s$ and $t$, then you get a map from $\mathbb{R}^2$ to the Minkowski sum, and you can compute the rank of this map. 
Jan 18 
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Gauß Bonnet operator
I would say that the classical definition has won and $\Delta$ usually denotes the negative operator. But you always need to check. 
Jan 17 
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Gauß Bonnet operator
Well, is it a positive or negative operator? 
Jan 17 
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Gauß Bonnet operator
This really should be asked on math.stackexchange.com, since it's not a research level question. Nevertheless, here's the answer: There is an ambiguity in how $\Delta$ is defined. Overall, people in PDE theory define it to be the negative operator, but some (but not all) people in differential geometry prefer to define it to be the positive operator. So you always have to watch out for this when learning or doing anything involving a Laplacian. 
Jan 16 
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Immersability and applications of a particular Riemannian metric
The calculation of curvature is probably easier using moving frames. You simply start with a moving frame $e_1, \dots, e_n$ of the flat metric with dual frame $\omega^1, \cdots \omega^n$, where $e_n = N$. There are corresponding connection $1$forms $\omega^i_j$, and the MaurerCartan equations give the curvature. Now define a new set of forms $\eta^1 = \omega^1, \dots, \eta_{n1} = \omega^{n1}, \eta^n = (1+\alpha)\omega^n$. This is an orthonormal frame for $G$. Now solve for the connection $1$forms for $G$ and compute its curvature. 
Jan 14 
revised 
Cauchy problem for an overdetermined system of PDE
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Jan 14 
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Cauchy problem for an overdetermined system of PDE
Sorry but if $E_1$ and $E_2$ are linearly independent, you need to solve both equations. In particular, if $p + q = 1$, then $E_1 + pE_3$ and $E_1 + qE_3$ are linearly dependent and therefore do not imply solutions to $E_1$ and $E_2$. So you have to assume that $p + q \ne 1$ in order to get two linearly independent equations. 
Jan 13 
revised 
Cauchy problem for an overdetermined system of PDE
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Jan 13 
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Cauchy problem for an overdetermined system of PDE
Yes, this case is handled in section 1.7 of "Involutive Hyperbolic Differential Systems" (Memoirs of the AMS, #370). In the generic case, the socalled reduced Cartan characters are, I believe, $3, 2, 2$. Theorem (1.26) gives an invariant way to identify whether the procedure yields a hyperbolic system of PDE's or not. Alas, as proud as I am of this paper, it's rather difficult to read. My advice is to play around with the calculations described above. I've appended some additional information. 
Jan 13 
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Cauchy problem for an overdetermined system of PDE
I may have misspoken. I'm not sure an invariant way has been worked out. If it has, it would be in the reference cited by Ben McKay above (yes, I wrote it but I don't have a copy handy and I don't remember whether I dealt with this case). I'll look at it when I get a chance. 