In this erratum we correct a mistake in the proof of Lemma 3.5 of Ref. 1. This requires a slight refinement of the assumptions leading to the existence result of Ref. 1. In our paper,1 existence of martingale solutions is proved for doubly nonlinear stochastic evolution equations of the form dA(u) + B(u)dt 3 F(u)dt + G(u)dW, u(0) = u0, (0.1) where A : H → 2H and B : V → 2V ∗ are maximal monotone operators on some separable real Hilbert spaces H and V , with V,→ H compactly and densely, F : [0, T] × H → H and G : [0, T] × H → L 2(U, H) are Lipschitzcontinuous in the second variables uniformly in time, U is a separable Hilbert space, W is a cylindrical Wiener process on U, and u0 is a given initial datum. For precise assumptions on the mathematical setting, notation, and the precise statements of the results, we refer the reader to Sec. 2 of Ref. 1. In particular, we recall that R : V → V ∗ is the Riesz isomorphism of V , Aε denotes the εYosida approximation of A for every ε > 0, and A1 : H → H is Gâteauxdifferentiable. The proof in Ref. 1 relies on a preliminary technical lemma (Lemma 3.5), whose proof however appears to be incomplete. We record here an alternative argument, hinging on a slight technical refinement of assumption (H40) in Sec. 2 of Ref. 1, namely, H40 There exists a separable Hilbert space Z ⊂ V , densely embedded in H, a constant η ∈ (1/3, 1/2), and an increasing function f : [0, +∞) → [0, +∞) such that, for every x ∈ V it holds that (Formula Present). Note that it is possible to show that the relevant example of multivalued operator A in graph form treated in Sec. 7.1 of Ref. 1 satisfies also the structural hypothesis (H40). Lemma 0.1. (Replacing Lemma 3.5 in Ref. 1) Let y ∈ H, x := A1(y) ∈ H, and for any λ > 0 set xλ := Ã−λ1(y) ∈ V, with Ãλ(w):= λRw + Aλ(w) for any w ∈ V . Then, as λ&0, it holds that xλ * x in H and Aλ(xλ) → y in H. Moreover, if x ∈ V it also holds that xλ → x in V and D((Ãλ)1)(y) * D(A1)(y) in Lw(H, H). Proof. The first three statements follow exactly as in Lemma 3.5 of Ref. 1. As for the fourth one, we first note that for every y1, y2 ∈ H, setting xiλ := Ã−λ1(yi), for i = 1, 2, one has (Formula Present), so that testing by x1λ − x2λ and exploiting the uniform strong monotonicity of Aλ (see Lemma 3.1 of Ref. 1), one deduces that there exists C > 0 independent of λ such that (Formula Present). It follows that (Formula Present) for every λ > 0, hence that there exists an operator L(y) ∈ L(H, H) such that D((Ãλ)1)(y) * L(y) in Lw(H, H). In order to complete the proof, we need to show that L(y)h = D(A1)(y)h for all h ∈ H: by density of Z in H it is enough to check this equality for h ∈ Z. We follow a similar argument as in Lemma 3.3 of Ref. 1 (where we take ε = λ). For h ∈ Z fixed, setting kλ := D((Ãλ)1)(y)h and hλ := D((Aλ)1)(Aλ(xλ))h, by Lemma 3.2 of Ref. 1 we have kλ + λ2Rkλ + λD(A1)(Aλ(xλ))Rkλ = hλ, (0.2) where hλ = λh + D(A1)(Aλ(xλ))h. Since (hλ)λ is clearly bounded in H, by testing (0.2) by λ2Rkλ one readily gets (Formula Present). (0.3) Now, by interpolation we have (Formula Present), so that (0.3) yields also (Formula Present) Moreover, by (H4') and the fact that (xλ)λ is bounded in V , it follows that (Rηhλ)λ is bounded in H: hence, testing (0.2) by (Formula Present). Taking (0.4) into account we infer that . (0.5) Hence, again by interpolation we find that (Formula Present). (0.6) Now, given z ∈ Z arbitrary, one has that (Formula Present) and the fact that (xλ)λ is bounded in V , so that (0.6) yields (Formula Present). By recalling that η ∈ (1/3, 1/2) one finds (Formula Present), while (0.3) yields directly (Formula Present). Passing to the weak limit in Z∗ as λ&0 in Eq. (0.2), and noting that hλ * D(A1)(y)h in H, one obtains L(y)h = D(A1)(y)h, and concludes.
Erratum: Doubly nonlinear stochastic evolution equations (Mathematical Models and Methods in Applied Sciences (2020) 30:5 (9911031) DOI: 10.1142/S0218202520500219)
Scarpa L.;
20220101
Abstract
In this erratum we correct a mistake in the proof of Lemma 3.5 of Ref. 1. This requires a slight refinement of the assumptions leading to the existence result of Ref. 1. In our paper,1 existence of martingale solutions is proved for doubly nonlinear stochastic evolution equations of the form dA(u) + B(u)dt 3 F(u)dt + G(u)dW, u(0) = u0, (0.1) where A : H → 2H and B : V → 2V ∗ are maximal monotone operators on some separable real Hilbert spaces H and V , with V,→ H compactly and densely, F : [0, T] × H → H and G : [0, T] × H → L 2(U, H) are Lipschitzcontinuous in the second variables uniformly in time, U is a separable Hilbert space, W is a cylindrical Wiener process on U, and u0 is a given initial datum. For precise assumptions on the mathematical setting, notation, and the precise statements of the results, we refer the reader to Sec. 2 of Ref. 1. In particular, we recall that R : V → V ∗ is the Riesz isomorphism of V , Aε denotes the εYosida approximation of A for every ε > 0, and A1 : H → H is Gâteauxdifferentiable. The proof in Ref. 1 relies on a preliminary technical lemma (Lemma 3.5), whose proof however appears to be incomplete. We record here an alternative argument, hinging on a slight technical refinement of assumption (H40) in Sec. 2 of Ref. 1, namely, H40 There exists a separable Hilbert space Z ⊂ V , densely embedded in H, a constant η ∈ (1/3, 1/2), and an increasing function f : [0, +∞) → [0, +∞) such that, for every x ∈ V it holds that (Formula Present). Note that it is possible to show that the relevant example of multivalued operator A in graph form treated in Sec. 7.1 of Ref. 1 satisfies also the structural hypothesis (H40). Lemma 0.1. (Replacing Lemma 3.5 in Ref. 1) Let y ∈ H, x := A1(y) ∈ H, and for any λ > 0 set xλ := Ã−λ1(y) ∈ V, with Ãλ(w):= λRw + Aλ(w) for any w ∈ V . Then, as λ&0, it holds that xλ * x in H and Aλ(xλ) → y in H. Moreover, if x ∈ V it also holds that xλ → x in V and D((Ãλ)1)(y) * D(A1)(y) in Lw(H, H). Proof. The first three statements follow exactly as in Lemma 3.5 of Ref. 1. As for the fourth one, we first note that for every y1, y2 ∈ H, setting xiλ := Ã−λ1(yi), for i = 1, 2, one has (Formula Present), so that testing by x1λ − x2λ and exploiting the uniform strong monotonicity of Aλ (see Lemma 3.1 of Ref. 1), one deduces that there exists C > 0 independent of λ such that (Formula Present). It follows that (Formula Present) for every λ > 0, hence that there exists an operator L(y) ∈ L(H, H) such that D((Ãλ)1)(y) * L(y) in Lw(H, H). In order to complete the proof, we need to show that L(y)h = D(A1)(y)h for all h ∈ H: by density of Z in H it is enough to check this equality for h ∈ Z. We follow a similar argument as in Lemma 3.3 of Ref. 1 (where we take ε = λ). For h ∈ Z fixed, setting kλ := D((Ãλ)1)(y)h and hλ := D((Aλ)1)(Aλ(xλ))h, by Lemma 3.2 of Ref. 1 we have kλ + λ2Rkλ + λD(A1)(Aλ(xλ))Rkλ = hλ, (0.2) where hλ = λh + D(A1)(Aλ(xλ))h. Since (hλ)λ is clearly bounded in H, by testing (0.2) by λ2Rkλ one readily gets (Formula Present). (0.3) Now, by interpolation we have (Formula Present), so that (0.3) yields also (Formula Present) Moreover, by (H4') and the fact that (xλ)λ is bounded in V , it follows that (Rηhλ)λ is bounded in H: hence, testing (0.2) by (Formula Present). Taking (0.4) into account we infer that . (0.5) Hence, again by interpolation we find that (Formula Present). (0.6) Now, given z ∈ Z arbitrary, one has that (Formula Present) and the fact that (xλ)λ is bounded in V , so that (0.6) yields (Formula Present). By recalling that η ∈ (1/3, 1/2) one finds (Formula Present), while (0.3) yields directly (Formula Present). Passing to the weak limit in Z∗ as λ&0 in Eq. (0.2), and noting that hλ * D(A1)(y)h in H, one obtains L(y)h = D(A1)(y)h, and concludes.File  Dimensione  Formato  

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