arXiv subjects
Publications and source records attributed to Olivaine S. de Queiroz.
Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev exponent on the boundary.
We study local behavior of positive solutions to the fractional Yamabe equation with a singular set of fractional capacity zero.
We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_γ\to $ min, ruled by nonlinear, $p$-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl-Batchelor type; singular degenerate elliptic equations; and obstacle type systems. The Euler-Lagrange equation associated to $\mathcal{J}_γ$ becomes singular along the free interface $\{u= 0\}$. The degree of singularity is, in turn, dimed by the parameter $γ\in [0,1]$. For $0< γ< 1$ we show local minima is locally of class $C^{1,α}$ for a sharp $α$ that depends on dimension, $p$ and $γ$. For $γ= 0$ we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.
This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.