Related papers: Boundary regularity via Uhlenbeck-Rivi\`ere decomp…

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving…

We prove existence and regularity of solutions to degenerate and singular elliptic free boundary problems, where the volume of the positivity set of the solution is prescribed.

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal…

This paper is addressed to a stabilization problem of a system coupled by a wave and a Euler-Bernoulli plate equation. Only one equation is supposed to be damped. Under some assumption about the damping and the coupling terms, it is shown…

Let $u$ be a weak solution of the free boundary problem $$\mathcal L u=\lambda_0 \mathcal H^1\lfloor\partial\{u>0\}, u\ge 0,$$ where $\mathcal L u={\text{div}}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and $g(\xi)$ is a given…

We prove the regularity of solutions to the strain tensor equation on degenerated hyperbolic surfaces $S$ where the Gauss curvature is zero on a part of boundary. Furthermore, we obtain the density property that smooth infinitesimal…

Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show,…

We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.

We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all…

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on `zooming in' on a presumed singularity to the local setting near a curved portion $\Gamma \subset \partial\Omega$ of the boundary. Suppose that $u$ is a…

We develop a mathematically and physically sound definition of the spectrally-hyperviscous Navier-Stokes equations (SHNSE) on general bounded domains \Omega with zero (no-slip) boundary conditions prescribed on \varGamma=\partial\varOmega.…

We prove optimal H\"older boundary regularity for a non-local operator with a singular, symmetric kernel that depends on the distance to the boundary of the underlying domain. Additionally, we prove higher boundary regularity of solutions.

We study second order hyperbolic equations with initial conditions, a nonhomogeneous Dirichlet boundary condition and a source term. We prove the solution possesses $H^1$ regularity on any piecewise $C^1$-smooth non-timelike hypersurfaces.…

We establish the existence of solutions to common noise McKean-Vlasov martingale problems for coefficients with low regularity. Our approach is able to handle the key challenge posed by drift coefficients that are discontinuous with respect…

We study a class of degenerate hyperbolic equations in a bounded domain whose degeneracy occurs at a boundary point. We first develop the weighted functional framework, prove well-posedness of the degenerate problem, and establish…

The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to H\"{o}rmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We…

We prove the H\"older regularity of continuous isentropic solutions to multi-dimensional scalar balance laws when the source term is bounded and the flux satisfies general assumptions of nonlinearity. The results are achieved by exploiting…

This paper is devoted to give a simplified proof of the trace theorem for functions of bounded deformation defined on bounded Lipschitz domains of $\mathbb{R}^n$. As a consequence, the existence of one-sided Lebesgue limits on countably…

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.