Calderón–Zygmund estimates for the fully nonlinear obstacle problem with super-linear Hamiltonian terms and unbounded ingredients

• Published in: Mathematische Zeitschrift 2024-03, Vol.306 (3), Article 40
• Main Authors: da Silva, João Vitor, Frias, Romário Tomilhero
• Format: Article
• Language: English
• Subjects: Barriers; Estimates; Ingredients; Integral calculus; Mathematics; Mathematics and Statistics; Regularity
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-024-03444-5

In this work, we show the existence/uniqueness of L p -viscosity solutions for a fully non-linear obstacle problem with super-linear gradient growth, unbounded ingredients and irregular obstacles. In our results, we obtain Calderón–Zygmund estimates, namely W loc 2 , p regularity estimates (with p ∈ n 2 , ∞ ) for such solution. Our findings are newsworthy even for the simplest model case: Δ u + b ( x ) · D u + μ ( x ) ‖ D u ‖ m = f ( x ) in { u > φ } ∩ Ω u ( x ) = g ( x ) on ∂ Ω , where f ∈ L p ( Ω ) , φ ∈ W 2 , p ( Ω ) if m = 1 , and φ ∈ W 2 , 2 p ( Ω ) if m ∈ ( 1 , 2 ] , for b ∈ L ϱ ( Ω ) and μ ∈ L q ( Ω ) with ϱ , q > n , thereby extending recent Calderón–Zygmund estimates for the fully nonlinear obstacle problem with unbounded drift terms and irregular obstacles. Finally, in the unconstrained linear setting (i.e., without restriction on obstacle and m = 1 ), we obtain W loc 2 , p regularity estimates for the range of integrability p ∈ ( p 0 , n ] . These estimates may be of independent mathematical interest and complement the Sobolev estimates recently addressed when p > n ..