For simplicity, we restrict to constant coefficients. Let $A^{ij}\_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition:
$$
A^{ij}\_{ab}\xi\_i\xi\_jv^av^b \ge \lambda |\xi|^2|v|^2
$$
for some $\lambda > 0$ and any $\xi \in \mathbb{R}^n$ and $v \in \mathbb{R}^m$. Let $B$ be the unit ball in $\mathbb{R}^n$.


It is straightforward to use the Fourier transform to prove that there exists $c > 0$ such that given any $u \in C^\infty\_0(B,\mathbb{R}^m)$,
$$
\int\_{B} A^{ij}\_{ab}\partial\_iu^a\partial\_ju^b \ge c\int\_B |\partial u|^2.
$$
If the (stronger) Legendre condition
$$
A^{ij}\_{ab}p\_i^ap\_j^b \ge \lambda\sum\_{i,a}(p\_i^a)^2
$$
for any $p\_i^a \in \mathbb{R}$ holds, it is easy to use an extension operator to extend the inequality to any function $u \in C^\infty(\overline{B},\mathbb{R}^m)$.


Question: Does the Legendre-Hadamard condition imply a Gårding inequality of the form
$$
\int\_{B} A^{ij}\_{ab}\partial\_iu^a\partial\_ju^b \ge \int\_B c|\partial u|^2 - c'|u|^2,
$$
for any $u \in C^\infty(\overline{B},\mathbb{R}^m)$? If not, what is a counterexample?



Does the Legendre-Hadamard condition imply a generalized Gårding inequality?
The answer is **No**, and this is an interesting question.


As Terry Tao commented, the test case is when $B$ is replaced by a half-space $H$, so let us consider the latter case for a moment and fields $u$ with compact support in $\bar B$ (i.e. not vanishing at the boundary). 


Because $H$ is dilation invariant, a Garding inequality implies actually an inequality of the form
$$
\int\_H A^{ij}\_{ab}\partial\_iu^a\partial\_ju^b \ge c\int\_H |\partial u|^2.
$$
So we are led to the question whether such an inequality holds true or not.
I analysed this problem in my paper "Second-order initial-boundary value problems of variational type". *J. of Functional Analysis*, **236** (2006), pp 409-446.
The existence of such an inequality is equivalent to a Lopatinskii-type property, which can be expressed in terms of a Lopatinskii determinant associated with the symbol $A$. The technique uses a Fourier transform in the $n-1$ coordinates along the boundary. You will find in Section 6 examples and counter-examples. 


I do not resist to give the following counter-example: take the linear isotropic elasticity, satisfying the Legendre-Hadamard condition. It has two wave velocities, $c\_S$ for shear (transverse) waves and $c\_P$ for pressure (longitudinal) waves. In real media, we have $c\_S<c\_P$. However, L.-H. does not prevent from the possibility of $c\_P<c\_S$, in which case the inequality above does not hold. Actually, the lack of inequality reveals the ill-posedness of the Initial-boundary-value problem in the half-space when the boundary condition is Neumann-type (here free displacement).


Now, when the inequality above holds true in every half-space, I proved that the Garding inequality holds true in bounded domains. It is not contained in my article (it was in the submission) because it is essentially an exercise.


[Link to my JFA paper](http://www.sciencedirect.com/science/article/pii/S0022123606000978). The main result is Theorem 3.5. Isotropic elasticity is treated in Paragraph 6.3, see Theorem 6.2.


