scholarly journals The variable coefficient thin obstacle problem: Carleman inequalities

2016 ◽  
Vol 301 ◽  
pp. 820-866 ◽  
Author(s):  
Herbert Koch ◽  
Angkana Rüland ◽  
Wenhui Shi
Keyword(s):  
Obstacle Problem ◽  
Variable Coefficient ◽  
Carleman Inequalities ◽  
Thin Obstacle

scholarly journals Regularity for the fully nonlinear parabolic thin obstacle problem

2021 ◽  
pp. 2150011
Author(s):  
Georgiana Chatzigeorgiou
Keyword(s):  
Parabolic Equation ◽  
Viscosity Solution ◽  
Nonlinear Parabolic Equation ◽  
Obstacle Problem ◽  
Elliptic Case ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Nonlinear Parabolic ◽  
Parabolic Case ◽  
Thin Obstacle

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

scholarly journals Regularity results of the thin obstacle problem for the p(x)-Laplacian

2019 ◽  
Vol 276 (2) ◽  
pp. 496-519 ◽  
Author(s):  
Sun-Sig Byun ◽  
Ki-Ahm Lee ◽  
Jehan Oh ◽  
Jinwan Park
Keyword(s):  
Obstacle Problem ◽  
Regularity Results ◽  
Thin Obstacle

scholarly journals Convergence rates for the classical, thin and fractional elliptic obstacle problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140449 ◽  
Author(s):  
Ricardo H. Nochetto ◽  
Enrique Otárola ◽  
Abner J. Salgado
Keyword(s):  
Obstacle Problem ◽  
Fractional Laplacian ◽  
Convergence Rates ◽  
Finite Element Approximation ◽  
Obstacle Problems ◽  
Element Approximation ◽  
Optimal Error ◽  
Optimal Convergence Rates ◽  
Regularity Results ◽  
Thin Obstacle

We review the finite-element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results, we present an optimal error analysis for the thin obstacle problem. Finally, we discuss the localization of the obstacle problem for the fractional Laplacian and prove quasi-optimal convergence rates.

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