By W. & H. Knapp Grobner

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The map → dχ (x, u(x))z is a vector-ﬁeld along the path γ : → χ (x, u(x)). We ﬁnd dχ (x, u(x))z = χ (x, u(x))i,j z j ∂xi |τ (x) . We abbreviate χ = χ (x, u(x)) and ψ = ψ (x, u(x)). By the formula for the covariant derivative along γ we have d (dχ z) d dχi ,j j i z + Γjk χj,l z l γ˙ k ∂xi |χ = d i = ξ(χ , ψ )i,j z j + Γjk χj,l z l ξ k (χ , ψ ) ∂xi |χ j i l j i α j i l k = ξ,x l (χ , ψ )χ ,j z + ξ,uα (χ , ψ )ψ ,j z + Γjk χ ,l z ξ (χ , ψ ) ∂xi |χ . Collecting terms this can be written as d (dχ z) = (Dx ξ)(χ , ψ )dχ z + (∂uα ξ)(χ , ψ ) ⊗ dψ α z d = (Dx ξ)(χ , ψ ) + (∂uα ξ)(χ , ψ ) ⊗ dψ α ◦ dχ−1 dχ z.

Similarly the partial divergence divx ξ(x, u) is deﬁned. Total derivatives Suppose u : M → Rk is a smooth Rk -valued function. For f : M × Rk → R we consider the new real-valued function f (Id ×u)(x) := f (x, u(x)). For the total gradient of f (Id ×u) : M → R we have the formula ∇(f (x, u(x))) = ∇x f (x, u(x)) + ∂uα f (x, u(x))∇uα , i written in local coordinates as f ;i ∂xi = (f ;x + f,uα uα;i )∂xi . A similar constructions leads to the total diﬀerential of f (Id ×u): d(f (x, u(x))) = dx f (x, u(x)) + ∂uα f (x, u(x))duα i k written in local coordinates as f,i dxi = (f,xi +f,uα uα ,i )dx .

10) for a large LM > 0. First, this restricts the result to all those solutions which attain values in [−M, M ]. But since M can be taken arbitrarily large the full statement is recovered. For u ∈ W01,2 (B1 (0)) let uj := B1 (0) uψj dx for j ≥ 1. Let g u := ∞ − 2 2 j=1 e (u − u0 )j ψj + P (u − u0 ) + u0 , where P : L (B1 (0)) → Lrad is the orthogonal projection. Consider the functional L[u] = B1 (0) 12 |∇u|2 −F (x, u) dx on W01,2 (B1 (0)) with F (x, s) = s 0 f (x, t) dt. 5 Classical uniqueness results d L[g u] d ∞ =0 =− ∞ µj (u − u0 )j uj + j=1 By (a) we know µj u0,j = d L[g u] d =0 d L[g u] d =0 =− µj − ∞ j=1 ∞ ∞ (f (x, u)−f (x, u0 ))(u−u0 )j ψj dx.