  Let $k$ be a perfect field. Assume that the characteristic of $k$ satisfies
certain tameness assumptions \eqref{tameness}. Let $\mathcal O_{_n} :=
k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket$ and set $K_{_n} :=
\text{Fract}~\cO_{_n}$. Let $G$ be an almost-simple, simply-connected affine
Chevalley group scheme with a maximal torus $T$ and a Borel subgroup $B$. Given
a $n$-tuple ${\bf f} = (f_{_1}, \ldots, f_{_n})$ of concave functions on the
root system of $G$ as in Bruhat-Tits \cite{bruhattits1}, \cite{bruhattits}, we
define {\it {\tt n}-bounded subgroups ${\tt P}_{_{\bf f}}\subset G(K_{_n})$} as
a direct generalization of Bruhat-Tits groups for the case $n=1$. We show that
these groups are {\it schematic}, i.e. they are valued points of smooth {\em
quasi-affine} (resp. {\em affine}) group schemes with connected fibres and {\it
adapted to the divisor with normal crossing $z_1 \cdots z_n =0$} in the sense
that the restriction to the generic point of the divisor $z_i=0$ is given by
$f_i$ (resp. sums of concave functions given by points of the apartment). This
provides a higher-dimensional analogue of the Bruhat-Tits group schemes with
natural specialization properties. In \S\ref{mixedstuff}, under suitable
assumptions on $k$ \S \ref{charassum}, we extend all these results for a
$n+1$-tuple ${\bf f} = (f_{_0}, \ldots, f_{_n})$ of concave functions on the
root system of $G$ replacing $\mathcal O_{_n}$ by ${\cO} \llbracket
x_{_1},\cdots,x_{_n} \rrbracket$ where $\cO$ is a complete discrete valuation
ring with a perfect residue field $k$ of characteristic $p$. In the last part
of the paper, we give applications in char zero to constructing certain natural
group schemes on wonderful embeddings of groups and also certain families of
{\tt 2-parahoric} group schemes on minimal resolutions of surface singularities
that arose in \cite{balaproc}.
