  We study weak solutions of the incompressible Euler equations on
$\mathbb{T}^2\times \mathbb{R}_+$; we use test functions that are divergence
free and have zero normal component, thereby obtaining a definition that does
not involve the pressure. We prove energy conservation under the assumptions
that $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$
  \lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|}
|u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0, $$ and an additional continuity
condition near the boundary: for some $\delta>0$ we require $u\in
L^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditions
are satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, with
H\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$.
