  The notion of $p$-ellipticity has recently played a significant role in
improving our understanding of issues of solvability of boundary value problems
for scalar complex valued elliptic PDEs. In particular, the presence of
$p$-ellipticity ensures higher regularity of solutions of such equations.
  In this work we extend the notion of $p$-ellipticity to second order elliptic
systems. Recall that for systems, there is no single notion of ellipticity,
rather a more complicated picture emerges with ellipticity conditions of
varying strength such as the Legendre, Legendre-Hadamard and integral
conditions. A similar picture emerges when $p$-ellipticity is considered. In
this paper, we define three new notions of $p$-ellipticity, establish
relationships between them and show that each of them does play an important
role in solving boundary value problems.
  These important roles are demonstrated by establishing extrapolation results
for solvability of the $L^p$ Dirichlet problem for elliptic systems, followed
by applications of this result in two different scenarios: one for the Lam\'e
system of linear elasticity and another in the theory of homogenization.
