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\( W^{2, p} \)-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. (English) Zbl 1486.35103

Authors’ abstract: We prove a global \( W^{2,p}\)-estimate for the viscosity solution to fully nonlinear elliptic equations \(F(x,u,Du,D^2u)=f(x)\) with oblique boundary condition in a bounded \(C^{2, \alpha}\)-domain for every \(\alpha\in (0,1).\) Here, the nonlinearities \(F\) is assumed to be asymptotically \(\delta\)-regular to an operator \(G\) that is \((\delta,R)\)-vanishing with respect to \(x.\) We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global \(W^{2,p}\)-estimate for the viscosity solution to fully nonlinear parabolic equations \(F(x,t,u,Du,D^2u)-u_t=f(x,t)\) with oblique boundary condition in a bounded \(C^3\)-domain.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35D40 Viscosity solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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