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

In this work, we show the existence/uniqueness of \(L^p\) $L^p$ -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 \(W^{2,p}_{loc}\) $W_{\mathit{loc}}^{2,p}$ regularity estimates (with \(p \in \left( \frac{n}{2}, \infty \right) \) $p\in \left(\frac{n}{2},,,\infty \right)$ ) for such solution. Our findings are newsworthy even for the simplest model case: \(\begin{aligned} \left\{ \begin{array}{rclcl} \Delta u + b(x)\cdot Du +\mu (x)\Vert Du\Vert ^m &{} = &{} f(x) &{} \text {in} &{} \{u> \varphi \}\cap \Omega \\ u(x) &{} = &{} g(x) &{} \text {on} &{} \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccccc}\hfill \mathrm{\Delta }u+b(x)·Du+{\mu (x)\Vert Du\Vert }^m& =& f(x)\hfill & \text{in}& \{u>\phi \}\cap \mathrm{\Omega }\hfill \\ \hfill u(x)& =& g(x)\hfill & \text{on}& \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(f \in L^p(\Omega )\) $f\in L^p(\mathrm{\Omega })$ , \(\varphi \in W^{2, p}(\Omega )\) $\phi \in W^{2,p}(\mathrm{\Omega })$ if \(m=1\) $m=1$ , and \(\varphi \in W^{2, 2p}(\Omega )\) $\phi \in W^{2,2p}(\mathrm{\Omega })$ if \(m \in (1, 2]\) $m\in (1,2]$ , for \(b \in L^{\varrho }(\Omega )\) $b\in L^{\varrho }(\mathrm{\Omega })$ and \(\mu \in L^q(\Omega )\) $\mu \in L^q(\mathrm{\Omega })$ with \(\varrho , q>n\) $\varrho ,q>n$ , 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 \(m=1\) $m=1$ ), we obtain \(W^{2,p}_{loc}\) $W_{\mathit{loc}}^{2,p}$ regularity estimates for the range of integrability \(p \in (p_0, n]\) $p\in (p_0,n]$ . These estimates may be of independent mathematical interest and complement the Sobolev estimates recently addressed when \(p>n\) $p>n$ ..