  We develop a variant of Coleman and Perrin Riou's methods giving, for a de
Rham $p$-adic Galois representation, a construction of $p$-adic $L$ functions
from a compatible system of global elements. As a result, we construct analytic
functions on an open set of the $p$-adic weight space containing all locally
algebraic characters of large enough conductor. Applied to Kato's Euler system,
this gives $p$-adic $L$-functions for elliptic curves with additive bad
reduction and, more generally, for modular forms which are supercuspidal at
$p$. In the case of dimension $2$, we prove a functional equation for our
$p$-adic $L$-functions.
