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) = u_{0}, (0.1) where A : H → 2^{H} and B : V → 2^{V ∗} 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 u_{0} 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 , A^{ε} denotes the εYosida approximation of A for every ε > 0, and A^{1} : 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 (H4^{0}) in Sec. 2 of Ref. 1, namely, H4^{0} 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 (H4^{0}). Lemma 0.1. (Replacing Lemma 3.5 in Ref. 1) Let y ∈ H, x := A^{1}(y) ∈ H, and for any λ > 0 set x_{λ} := Ã^{−}_{λ}^{1}(y) ∈ V, with Ã_{λ}(w):= λRw + A^{λ}(w) for any w ∈ V . Then, as λ&0, it holds that x_{λ} * x in H and A^{λ}(x_{λ}) → y in H. Moreover, if x ∈ V it also holds that x_{λ} → x in V and D((Ã_{λ})^{1})(y) * D(A^{1})(y) in L_{w}(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 y_{1}, y_{2} ∈ H, setting x^{i}_{λ} := Ã^{−}_{λ}^{1}(y_{i}), for i = 1, 2, one has (Formula Present), so that testing by x^{1}_{λ} − x^{2}_{λ} and exploiting the uniform strong monotonicity of A^{λ} (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((Ã_{λ})^{1})(y) * L(y) in L_{w}(H, H). In order to complete the proof, we need to show that L(y)h = D(A^{1})(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 k_{λ} := D((Ã_{λ})^{1})(y)h and h_{λ} := D((A^{λ})^{1})(A^{λ}(x_{λ}))h, by Lemma 3.2 of Ref. 1 we have k_{λ} + λ^{2}Rk_{λ} + λD(A^{1})(A^{λ}(x_{λ}))Rk_{λ} = h_{λ}, (0.2) where h_{λ} = λh + D(A^{1})(A^{λ}(x_{λ}))h. Since (h_{λ})_{λ} is clearly bounded in H, by testing (0.2) by λ^{2}Rk_{λ} 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 (x_{λ})_{λ} is bounded in V , it follows that (R^{η}h_{λ})_{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 (x_{λ})_{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 h_{λ} * D(A^{1})(y)h in H, one obtains L(y)h = D(A^{1})(y)h, and concludes.
Original language  English 

Pages (fromto)  27592761 
Number of pages  3 
Journal  Mathematical Models and Methods in Applied Sciences 
Volume  32 
Issue number  13 
DOIs 

Publication status  Published  15 Dec 2022 
Austrian Fields of Science 2012
 101002 Analysis