By Baldi P., Toland J.F.

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Remark 11. In Lemma 8 we have proved that L is an isomorphism of H01,ρ . However, h ∈ H02,ρ does not imply that L(h) ∈ H02,ρ , because w0′ has only regularity W 1,ρ . This means that, in the present problem, L(h) cannot be taken as test function, as was done in [14]. So instead, here we will take L(h′ ) ∈ H01,ρ as test function “of lower order”. 9) that involves only L(h′ ) as test function. First, we note that, for every h ∈ H02,ρ , 2π 0 2π ∇I0 h dτ = 0 (∇I0 − λ0 ) h dτ, where λ0 := [∇I0 ]. Integrating by parts yields 2π 0 2π (∇I0 − λ0 ) h dτ = − 0 0 2π =− τ 0 33 (∇I0 − λ0 ) h′ (τ ) dτ m0 L(h′ ) dτ, where τ m0 := (L−1 )∗ 0 (∇I0 − λ0 ) and (L−1 )∗ is the adjoint operator of L−1 in the usual L22π sense, (L−1 )∗ (f ) = w0′ f + C (1 + Cw0′ )f Now, for any h ∈ H02,ρ , let ∀f.

4] S. S. Chern, Curves and surfaces in Euclidian space. In: Studies in Global Geometry and Analysis, Studies in Mathematics, Volume 4 (editor S. S. Chern). Mathematical Association of America, 1967. [5] P. L. Duren, Theory of H p -Spaces. Dover, Mineola, 2000. [6] A. I. Dyachenko, E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov, Analytic description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Physics Letters A 221 (1996), 73–79. [7] I. Ekeland and R.

The maps 1,ρ W2π → R, 2π w→ wCw′ dτ 0 30 2π and w → 0 w2 Cw′ dτ 1,ρ are Fr´echet differentiable at w0 . 6) ∇I0 := ∇I0 (w0 ) = (c2 − 2ga0 )Cw0′ − g w0 (1 + Cw0′ ) + C(w0 w0′ ) and a0 := −[w0 Cw0′ ]. 7) In the following, we will denote k,ρ H0k,ρ := {w ∈ W2π : [w] = 0}, k = 1, 2. To investigate the differentiability (at least in the Gateaux sense) of the elastic energy term E(w, χ) with respect to w, we recall the following fact. 2,ρ Lemma 7. w0 is an interior point of A0 in the topology of W2π , that is there exists ε0 = ε0 (w0 ) > 0 such that W0 := {w ∈ H02,ρ : w − w0 2,ρ W2π < ε0 } ⊂ A0 , and there exist constants C, C ′ such that 0 < C ≤ Ω(w) ≤ C ′ ∀w ∈ W0 .

### Steady periodic water waves under nonlinear elastic membranes by Baldi P., Toland J.F.

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