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We remark also that the case when Y is independent of u is equivalent to the optimal transportation case.
In the optimal transportation case, we obtain a different and more direct proof of the improved global regularity result in [15].
The next conditions are expressed in terms of the matrix function A and extend condition A3w in the optimal transportation case [20].
We then have the following improvement and extension of the fundamental lemma, Lemma 2.1 in [4] in the optimal transportation case.
In the optimal transportation case, Theorem 1.2 improves Theorem 1.1 in [9] and Theorem 2.1 in [4] by removing the barrier and subsolution hypotheses assumed there.
We note that in [4], U = R n × R × R n so that these conditions are automatically satisfied and in the optimal transportation case J x = R. Similar conditions are also needed for the convexity theory in [16].
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