  We consider the class of semi-stable solutions to semilinear equations
$-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$
convex in some results). This class includes all local minimizers, minimal, and
extremal solutions. In dimensions $n \leq 4$, we establish an priori $L^\infty$
bound which holds for every positive semi-stable solution and every
nonlinearity $f$. This estimate leads to the boundedness of all extremal
solutions when $n=4$ and $\Omega$ is convex. This result was previously known
only in dimensions $n\leq 3$ by a result of G. Nedev. In dimensions $5 \leq n
\leq 9$ the boundedness of all extremal solutions remains an open question. It
is only known to hold in the radial case $\Omega=B_R$ by a result of A. Capella
and the author.
