  We study the minimization problem for the Yang-Mills energy under fixed
boundary connection in supercritical dimension $n\geq 5$. We define the natural
function space A_{G} in which to formulate this problem in analogy to the space
of integral currents used for the classical Plateau problem. The space A_{G}
can be also interpreted as a space of weak connections on a "real measure
theoretic version" of reflexive sheaves from complex geometry. We prove the
weak closure result which ensures the existence of energy-minimizing weak
connections in A_{G}. We then prove that any weak connection from A_{G} can be
obtained as a L^2-limit of classical connections over bundles with defects.
This approximation result is then extended to a Morrey analogue. We prove the
optimal regularity result for Yang-Mills local minimizers. On the way to prove
this result we establish a Coulomb gauge extraction theorem for weak curvatures
with small Yang-Mills density. This generalizes to the general framework of
weak $L^2$ curvatures previous works of Meyer-Rivi\`ere and Tao-Tian in which
respectively a strong approximability property and an admissibility property
were assumed in addition.
