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We study Neumann functions for divergence form, second order elliptic systemswith bounded measurable coefficients in a bounded Lipschitz domain or aLipschitz graph domain. We establish existence, uniqueness, and variousestimates for the Neumann functions under the assumption that weak solutions ofthe system enjoy interior H\"older continuity. Also, we establish globalpointwise bounds for the Neumann functions under the assumption that weaksolutions of the system satisfy a certain natural local boundedness estimate.Moreover, we prove that such a local boundedness estimate for weak solutions ofthe system is in fact equivalent to the global pointwise bound for the Neumannfunction. We present a unified approach valid for both the scalar and thevectorial cases.

Neumann functions for second order elliptic systems with measurable coefficients