The stochastic p-Laplace equation on ℝd

We show well-posedness of the p-Laplace evolution equation on Rd with square integrable random initial data for arbitrary 1<p<∞ and arbitrary space dimension d∈N. The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the p-Lapla...

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Bibliographic Details
Published inStochastic analysis and applications Vol. 41; no. 5; pp. 892 - 917
Main Authors Schmitz, Kerstin, Zimmermann, Aleksandra
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis Ltd 03.09.2023
Subjects
Coercivity
Existence theorems
Laplace equation
Sobolev space
Uniqueness theorems
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Summary:We show well-posedness of the p-Laplace evolution equation on Rd with square integrable random initial data for arbitrary 1<p<∞ and arbitrary space dimension d∈N. The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the p-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of 1<p<∞ and also depends on the space dimension d. We propose a framework of functional spaces which is independent of Sobolev space embeddings and space dimension. For additive noise, we show existence using a time discretization. Then, a fixed-point argument yields the result for multiplicative noise.
ISSN:0736-2994
1532-9356
DOI:10.1080/07362994.2022.2091600