Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities (Q1683355)

The authors focus on variational equations defined by an elliptic operator which becomes degenerate along a portion of the domain boundary, and where no boundary condition is prescribed. They prove local supremum bounds, a Harnack inequality, Hölder continuity up to the boundary and a strong maximum principle for the solutions. Let \(\mathcal{O}\subset\mathbb{R}^n\) be an open set and consider an elliptic partial differential operator \(Au=-\overline{a}^{\mu\nu}u_{z_{\mu}z_{\nu}}-b^{\mu}u_{z_{\mu}}+cu\), where \(A\) is boundary degenerate, i.e., \((\overline{a}^{\mu\nu})\) fails to be locally strictly elliptic along a non-empty open subset of the boundary, \(\Gamma_0\subset \partial\mathcal{O}\). Let \(\Gamma_1=\partial\mathcal{O}\setminus\overline{\Gamma}_0\). The authors consider suitably defined weak solutions \(u\) to the elliptic boundary value problem \[ \begin{aligned} Au= f & \text{ on } \mathcal{O},\\ u=g & \text{ on } \Gamma_1, \end{aligned} \] and also the elliptic obstacle problem with partial Dirichlet boundary conditions \[ \begin{aligned} & \min\{Au-f, u-\psi\}=0 \text{ a.e. on } \mathcal{O},\\ & u=g \text{ on } \Gamma_1,\\ \end{aligned} \] where \(\psi:\mathcal{O}\rightarrow \mathbb{R}\) (the obstacle) satisfies a compatibility condition. The authors analyze variational equations corresponding to the elliptic boundary value problem above, defined by a class of boundary-degenerate operators that include the Heston operator. Under certain conditions on \(A\), the first main result of this paper is a local supremum estimate up to \(\partial\mathcal{O}\) for the subsolutions. Secondly, the authors obtain a Harnack inequality for the non-negative solutions on open subsets \(\mathcal{O}'\subset\subset\mathcal{O}\cup\Gamma_0\), when \(f=0\) on \(\mathcal{O}\). Finally, they also prove a strong maximum principle for the subsolutions. Furthermore, in the case of a solution \(u\) to a variational equation corresponding either to the elliptic boundary value problem above, or to the variational inequality corresponding to the elliptic obstacle problem with partial Dirichlet boundary conditions, the authors prove \(C^{\alpha}\) regularity up to \(\partial\mathcal{O}\), including the corner points where \(\Gamma_0\) and \(\Gamma_1\) meet, and a local \(C^{\alpha}\) estimate. The key ingredients of the proofs are interesting adaptations of the Moser iteration technique, the John-Nirenberg inequality and the Poincaré inequality to the setting of the relevant degenerate elliptic operators and weighted Sobolev spaces, along with a penalization method.