2023/03/18 by Sabri Bensid, Bensid, Sabri
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2303.10412
openalex publication_date 2023/03/18 · openalex created_date 2023/03/22 · openalex updated_date 2026/07/28
In this paper, we consider the following free boundary problem (P)\Δu = λϕ(x)\Sumi=1n H(u-μi ) · amp; in Ω=Ω2∖ Ω1,
u =0 · amp; on ∂ Ω2,
u =M · amp; on ∂ Ω1. . The domain Ω is a convex ring where Ω1 and Ω2 are bounded convex domain in ℝN, N≥ 2, H is the Heaviside step function, λ, M,μi (i=1,..,n) are a given positive real parameters and ϕ is a given function . We show that under suitable conditions, there exists a solution to problem (P) with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function ϕ which show that there are more solutions.