A two phase boundary obstacle-type problem for the bi-Laplacian. (January 2022)

• Record Type: Journal Article
• Authors: Danielli, Donatella; Haj Ali, Alaa
• Is Part Of: Nonlinear analysis. Volume 214(2022)
• Publication Date: 2022-01
• Subjects: 35B65 -- 35R35 -- 35J35; Free boundary problems -- Variational methods -- Biharmonic operator -- Monotonicity formulas; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112583
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian ( − Δ ) 3 / 2, as first observed in Yang (2013). We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type.