= Compactness of normalized weak elliptic solutions
For a fixed bounded measurable <uniformly elliptic operator> in divergence form, solutions bounded by one have uniform local <Caccioppoli inequalities> and <Hölder continuity> estimates. A <diagonal subsequence argument> yields a local weak $W^{1,2}$ limit, while the <Arzelà-Ascoli theorem> gives <uniform convergence> on a fixed smaller closed <Euclidean ball>. The weak equation passes to the limit because the coefficients are fixed and bounded. A global supremum normalization does not force the interior limit to be nonzero.
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