Convergence of solutions to the p-Laplace evolution equation as p goes to 1

2011/03/01 by Jonas M. Tölle, Tölle, Jonas M.
Computer Science · Mathematics · #35K67 #49J45 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1103.0229

openalex publication_date 2011/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the set of solutions to the parabolic singular p-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain Ω for all space dimensions is continuous in the parameter p∈ [1,+∞) and the initial data. The highly singular limit case p=1 is included. In particular, we show that the solutions u_p converge strongly in L²(Ω), uniformly in time, to the solution u₁ of the parabolic 1-Laplace equation as p→ 1.

Related

• Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term
• Gradient estimates for singular parabolic p-Laplace type equations with measure data
• The Dirichlet-to-Neumann operator associated with the 1-Laplace operator and evolution problems
• The Time Derivative in a Singular Parabolic Equation
• Weak solutions to the parabolic p-Laplace equation in a moving domain under a Neumann type boundary condition