Related papers: Uniform Lipschitz regularity of flat segregated in…

In this paper, we extend several results established for stable minimal hypersurfaces to $\delta$-stable minimal hypersurfaces. These include the regularity and compactness theorems for immersed $\delta$-stable minimal hypersurfaces in…

We prove that a complete, two-sided, stable minimal immersed hypersurface in $\mathbf{R}^{4}$ is flat.

In this paper, we consider inverse time-harmonic acoustic and electromagnetic scattering from locally perturbed rough surfaces in three dimensions. The scattering interface is supposed to be the graph of a Lipschitz continuous function with…

This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper…

We prove essentially sharp incidence estimates for a collection of $\delta$-tubes and $\delta$-balls in the plane, where the $\delta$-tubes satisfy an $\alpha$-dimensional spacing condition and the $\delta$-balls satisfy a…

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique…

Consider the Ising model at low-temperatures and positive external field $\lambda$ on an $N\times N$ box with Dobrushin boundary conditions that are plus on the north, east, and west boundaries and minus on the south boundary. If $\lambda =…

We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating $n$ noisy training data points in…

In this work, we are concerned with the inverse scattering by interfaces for the linearized and isotropic elastic model at a fixed frequency. First, we derive complex geometrical optic solutions with linear or spherical phases having a…

We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz…

Let $\Delta$ be a connected, pure $2$-dimensional simplicial complex embedded in $\mathbb{R}^2$ and let $C^{r}(\hat{\Delta})$ be the homogenized spline module of $\Delta$ with smoothness $r$. To study $C^{r}(\hat{\Delta})$, Schenck and…

We provide an account for the existence and uniqueness of solutions to rough differential equations under the framework of controlled rough paths. The case when the driving path is $\beta$-H\"older continuous, for $\beta>1/3$, is widely…

We will prove a global estimate for the gradient of the solution to the {\it Poisson differential inequality} $|\Delta u(x)|\le a|\nabla u(x)|^2+b$, $x\in B^{n}$, where $a,b<\infty$ and $u|_{S^{n-1}}\in C^{1,\alpha}(S^{n-1}, \Bbb R^m)$. If…

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the…

A 7-dimensional area-minimizing embedded hypersurface $M$ will in general have a discrete singular set. The same is true if $M$ is stable, or has bounded index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such…

We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension…

We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension $11$ in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, $n+1 \geq 12$, we prove in the…

We apply iteration schemes and perturbation methods to provide a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n -…

We consider discontinuous perturbations of smooth endomorphisms and show that if the perturbed family satisfies uniform mixing assumptions on standard pairs the physical measure is Lipschitz in the parameter defying the perturbation. We…

We deal with the problem of the linearized and isotropic elastic inverse scattering by interfaces. We prove that the scattered $P$-parts or $S$-parts of the far field pattern, corresponding to all the incident plane waves of pressure or…