  We show that every bounded domain $D$ in $\mathbb R^n$ with smooth $p$-convex
boundary for $2\le p < n$ admits a smooth defining function $\rho$ which is
$p$-plurisubharmonic on $\overline D$; if in addition $bD$ has no $p$-flat
points then $\rho$ can be chosen strongly $p$-plurisubharmonic on $D$. If $bD$
is $2$-convex then for any open connected conformal surface $M$ and conformal
harmonic map $f:M\to \overline D$, either $f(M)\subset D$ or $f(M)\subset bD$.
In particular, every conformal harmonic map $\mathbb D^*\to D$ from the
punctured disc extends to a conformal harmonic map $\mathbb D\to D$.
