TY - JOUR
T1 - Correction to
T2 - Doubly nonlinear stochastic evolution equations II (Stochastics and Partial Differential Equations: Analysis and Computations, (2023), 11, 1, (307-347), 10.1007/s40072-021-00229-3)
AU - Scarpa, Luca
AU - Stefanelli, Ulisse
N1 - Publisher Copyright:
© 2022, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2023/12
Y1 - 2023/12
N2 - In our paper [1], we show existence of martingale solutions for doubly nonlinear stochastic evolution equations of the form (Formula presented.) (Formula presented.) (Formula presented.) where (Formula presented.) and (Formula presented.) are maximal monotone operators on some separable real Hilbert spaces H and V, with (Formula presented.) compactly and densely, (Formula presented.) and (Formula presented.) satisfy suitable Lipschitz-continuity conditions in the second variable, U is a separable Hilbert space, W is a cylindrical Wiener process on U, and (Formula presented.) is a given initial datum. We refer to [1, Sec. 2] for the rigorous assumptions, notations, and statement of the main results. We limit ourselves in recalling that (Formula presented.) is assumed to be strongly monotone and Gâteaux-differentiable with (Formula presented.) , such that (Formula presented.) belongs to (Formula presented.) for every (Formula presented.) and (Formula presented.). Moreover, we recall that (Formula presented.) for every (Formula presented.) and that the symbols (Formula presented.) denote the Yosida approximation and the resolvent of B, respectively. The proof in [1] relies on a preliminary technical Lemma [1, Lem. 3.1], whose proof however appears to be incomplete. We present here an alternative argument, which is based on a slight technical refinement of assumptions B and G in [1, Sec. 2]. B3there exists (Formula presented.) such that for every (Formula presented.) , (Formula presented.) is weakly Gâteaux differentiable with (Formula presented.) , i.e. (Formula presented.) and (Formula presented.) for all (Formula presented.) and (Formula presented.). there exists (Formula presented.) such that for every (Formula presented.) , (Formula presented.) is weakly Gâteaux differentiable with (Formula presented.) , i.e. (Formula presented.) and (Formula presented.) for all (Formula presented.) and (Formula presented.). (Replacing Lemma 3.1 in [1]) For every (Formula presented.) , the resolvent (Formula presented.) uniquely extends to a (Formula presented.) -Hölder-continuous operator (Formula presented.) such that (Formula presented.) is weakly Gâteaux-differentiable with (Formula presented.). Moreover, as (Formula presented.) , for all (Formula presented.) it holds that (Formula presented.) In particular, it holds that (Formula presented.) The fact that (Formula presented.) extends uniquely to a (Formula presented.) -Hölder-continuous operator (Formula presented.) , the first two convergences, and the last estimate of the lemma follow exactly as in [1]. Step 1. Let (Formula presented.) and (Formula presented.) be fixed. Now, given (Formula presented.) and (Formula presented.) we have (Formula presented.) so that by assumption B3 we have that (Formula presented.) is weakly Gâteaux-differentiable in x along the whole H with (Formula presented.) and (Formula presented.) as well. Moreover, by the mean value theorem there exists (Formula presented.) for some (Formula presented.) such that (Formula presented.) from which it follows (Formula presented.) Letting (Formula presented.) one has (Formula presented.) and (Formula presented.) in V, so that, as (Formula presented.) and (Formula presented.) , we get (Formula presented.) Step 2. Let us focus on the last three convergences as (Formula presented.). Fix (Formula presented.). Since (Formula presented.) is 1-Lipschitz continuous for every (Formula presented.) , there exists (Formula presented.) such that along a subsequence (Formula presented.) it holds that (Formula presented.) in (Formula presented.). Moreover, by (5) and (6) it readily follows that (Formula presented.) Since (Formula presented.) , it follows by assumption G in [1] that (Formula presented.) which implies the fourth convergence of the lemma and that (Formula presented.). Hence, it holds that (Formula presented.) for all (Formula presented.) , yielding (Formula presented.). By means of assumption B3, since (Formula presented.) for every (Formula presented.) , a classical (Formula presented.) argument ensures also that (Formula presented.) in (Formula presented.) , and the third convergence holds along the entire sequence (Formula presented.). The fifth convergence is then a direct consequence. (Formula presented.) Note that assumption B3 is not specifically restrictive in case (Formula presented.). In particular, the PDE example (Formula presented.) of Subsection 6.1 in [1] can be included in the analysis for (Formula presented.). The case (Formula presented.) is more delicate and would require (Formula presented.) to be quadratically bounded, whereas (Formula presented.) can have p-growth.
AB - In our paper [1], we show existence of martingale solutions for doubly nonlinear stochastic evolution equations of the form (Formula presented.) (Formula presented.) (Formula presented.) where (Formula presented.) and (Formula presented.) are maximal monotone operators on some separable real Hilbert spaces H and V, with (Formula presented.) compactly and densely, (Formula presented.) and (Formula presented.) satisfy suitable Lipschitz-continuity conditions in the second variable, U is a separable Hilbert space, W is a cylindrical Wiener process on U, and (Formula presented.) is a given initial datum. We refer to [1, Sec. 2] for the rigorous assumptions, notations, and statement of the main results. We limit ourselves in recalling that (Formula presented.) is assumed to be strongly monotone and Gâteaux-differentiable with (Formula presented.) , such that (Formula presented.) belongs to (Formula presented.) for every (Formula presented.) and (Formula presented.). Moreover, we recall that (Formula presented.) for every (Formula presented.) and that the symbols (Formula presented.) denote the Yosida approximation and the resolvent of B, respectively. The proof in [1] relies on a preliminary technical Lemma [1, Lem. 3.1], whose proof however appears to be incomplete. We present here an alternative argument, which is based on a slight technical refinement of assumptions B and G in [1, Sec. 2]. B3there exists (Formula presented.) such that for every (Formula presented.) , (Formula presented.) is weakly Gâteaux differentiable with (Formula presented.) , i.e. (Formula presented.) and (Formula presented.) for all (Formula presented.) and (Formula presented.). there exists (Formula presented.) such that for every (Formula presented.) , (Formula presented.) is weakly Gâteaux differentiable with (Formula presented.) , i.e. (Formula presented.) and (Formula presented.) for all (Formula presented.) and (Formula presented.). (Replacing Lemma 3.1 in [1]) For every (Formula presented.) , the resolvent (Formula presented.) uniquely extends to a (Formula presented.) -Hölder-continuous operator (Formula presented.) such that (Formula presented.) is weakly Gâteaux-differentiable with (Formula presented.). Moreover, as (Formula presented.) , for all (Formula presented.) it holds that (Formula presented.) In particular, it holds that (Formula presented.) The fact that (Formula presented.) extends uniquely to a (Formula presented.) -Hölder-continuous operator (Formula presented.) , the first two convergences, and the last estimate of the lemma follow exactly as in [1]. Step 1. Let (Formula presented.) and (Formula presented.) be fixed. Now, given (Formula presented.) and (Formula presented.) we have (Formula presented.) so that by assumption B3 we have that (Formula presented.) is weakly Gâteaux-differentiable in x along the whole H with (Formula presented.) and (Formula presented.) as well. Moreover, by the mean value theorem there exists (Formula presented.) for some (Formula presented.) such that (Formula presented.) from which it follows (Formula presented.) Letting (Formula presented.) one has (Formula presented.) and (Formula presented.) in V, so that, as (Formula presented.) and (Formula presented.) , we get (Formula presented.) Step 2. Let us focus on the last three convergences as (Formula presented.). Fix (Formula presented.). Since (Formula presented.) is 1-Lipschitz continuous for every (Formula presented.) , there exists (Formula presented.) such that along a subsequence (Formula presented.) it holds that (Formula presented.) in (Formula presented.). Moreover, by (5) and (6) it readily follows that (Formula presented.) Since (Formula presented.) , it follows by assumption G in [1] that (Formula presented.) which implies the fourth convergence of the lemma and that (Formula presented.). Hence, it holds that (Formula presented.) for all (Formula presented.) , yielding (Formula presented.). By means of assumption B3, since (Formula presented.) for every (Formula presented.) , a classical (Formula presented.) argument ensures also that (Formula presented.) in (Formula presented.) , and the third convergence holds along the entire sequence (Formula presented.). The fifth convergence is then a direct consequence. (Formula presented.) Note that assumption B3 is not specifically restrictive in case (Formula presented.). In particular, the PDE example (Formula presented.) of Subsection 6.1 in [1] can be included in the analysis for (Formula presented.). The case (Formula presented.) is more delicate and would require (Formula presented.) to be quadratically bounded, whereas (Formula presented.) can have p-growth.
UR - http://www.scopus.com/inward/record.url?scp=85138973959&partnerID=8YFLogxK
U2 - 10.1007/s40072-022-00275-5
DO - 10.1007/s40072-022-00275-5
M3 - Annotation
AN - SCOPUS:85138973959
VL - 11
SP - 1740
EP - 1743
JO - Stochastics and Partial Differential Equations: Analysis and Computations
JF - Stochastics and Partial Differential Equations: Analysis and Computations
SN - 2194-0401
IS - 4
ER -