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1. To prove the uniqueness, if u1 u2 are two minima for J in U then J(u1 ) = J(u2 ) ≤ J(v) for all v ∈ U and, in particular, this holds for v = 12 u1 + 12 u2 which belongs to U since U is convex. On the other hand, since J is strictly convex 1 1 1 1 J( u1 + u2 ) < J(u1 ) + J(u2 ) = J(u1 ≤ J(v)) 2 2 2 2 which is impossible if we take v = 21 (u1 +u2 ). This proves the uniqueness of the minimum. We shall now pass to a sufficient condition for the existence of minima of functionals which is the exact analogue of the case of twice differentiable functions.

Here, by convexity of J1 , we have J1 (u + θ(v − u)) ≤ J1 (u) + θ(J1 (v) − J1 (u)) so that J2 (u) ≤ θ(J1 (v) − J1 (u)) + J2 (u + θ(v − u)). That is J1 (v) − J1 (u) + (J2 (u + θ(v − u)) − J2 (u))/θ ≥ 0. Taking limits as θ → 0 we get the required assertion. Conversely, since J2 is convex and is G-differentiable we have, from part (1) of Proposition 1. 1, J2 (v) − J2 (u) ≥ J2′ (u, v − u) for all u, vǫU. Now we can write, for any vǫU, J(v) − J(u) = J1 (v) − J1 (u) + J2 (v) − Ju ≥ J1 (v) − J1 (u) + J2′ (u, v − u) ≥ 0 by assumption which proves that uǫU is a minimum for J.

2. Minimisation of Functionals - Theory 34 The space V is provided with the norm v → ||v||V = ||v||H 1 (Ω) + ||v||L p (Ω) 35 for which it becomes a Banach space. If 2 ≤ p < +∞ then V is a reflexive Banach space. In order to given an interpretation of the solutions of weak formulations of the problems as solutions of certain differential equations with boundary conditions we shall need an extension of the classical Green’s formula which we recall here. 8) Green’s formula for Sobolev spaces. Let Ω be a bounded open set with sufficiently “regular” boundary Γ.

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