Prove that the electric field is uniquely determined when the charge density p is given and either V or the normal derivative ôV/ôn is specified on each boundary surface. Do not assume that the boundaries are conductors, or that V is constant over any given surface. Hint: proceed as in the proof of the second uniqueness theorem, up to a certain point. But on each surface either V3 = 0 (if V is specified on the surface) or E31 = 0 (if aV/ən = -E_is specified).

Prove that the electric field is uniquely determined when the charge density p is given and either V or the normal derivative ôV/ôn is specified on each boundary surface. Do not assume that the boundaries are conductors, or that V is constant over any given surface. Hint: proceed as in the proof of the second uniqueness theorem, up to a certain point. But on each surface either V3 = 0 (if V is specified on the surface) or E31 = 0 (if aV/ən = -E_is specified).