Related papers: Reflection of Willmore surfaces with free boundari…

We study the umbilic points of Willmore surfaces in codimension 1 from the viewpoint of the conformal Gauss map. We first study the local behaviour of the conformal Gauss map near umbilic curves and prove that they are geodesics up to a…

For a smooth closed embedded planar curve $\Gamma$, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus $\mathfrak{g}\geq1$ having the curve $\Gamma$ as boundary, without any prescription on…

Conventional mirrors obey Snell's reflection law: a plane wave is reflected as a plane wave, at the same angle. To engineer spatial distributions of fields reflected from a mirror, one can either shape the reflector (for example, creating a…

We experimentally implement a virtual geometric periodicity in an elastic metamaterial. First, unwanted boundary reflections at the domain ends are cancelled through the iterative injection of the polarity reversed, reflected wavefield. The…

We study the formation of images in a reflective sphere in three configurations using caustics of the field of light rays. The optical wavefront emerging from a source point reaching a subject following passage through the optical system…

Reflection, propagation and energy analysis are crucially important in designing structures, especially plates. A thick plate is considered based on first order shear deformation theory. Wave Propagation Method (WPM) is employed to exactly…

This paper investigates the Knudsen layer equation in half-space, arising from the hydrodynamic limit of the Boltzmann equation to fluid dynamics. We consider the Maxwell reflection boundary condition with accommodation coefficient…

In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type…

We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.

We consider a convex solid cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ with vertex at the origin and boundary $\partial\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $\Sigma$ immersed in…

In this article, we establish a relationship between geometric quantities of a hypersurface restricted to its boundary, and the geometric quantities of its boundary as a hypersurface of the boundary of the ball. As a first application, we…

This is the third article in a series of three dealing with the exploitation of speckle for imaging purposes. In complex media, a fundamental limit is the multiple scattering phenomenon that completely blurs the imaging process in depth.…

We prove existence and uniqueness for semimartingale reflecting diffusions in 2-dimensional piecewise smooth domains with varying, oblique directions of reflection on each "side", under geometric, easily verifiable conditions. Our…

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave.…

We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove the solutions are unique, and prove that the…

We establish a rigidity result for the critical points, with boundary, of a four dimensional Willmore energy. These critical points satisfy a 4-Willmore equation which is a sixth order nonlinear elliptic partial differential equation. We…

It is well-known that the Willmore flow of closed spherical immersions exists globally in time and converges if the initial datum has Willmore energy below $8\pi$ - exactly the Li-Yau energy threshold below which all closed immersions are…

In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological…

We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of…

Given a subspace $U\subset\mathbb{C}[x_1,\dots,x_n]_d$ we consider the closure of the image of the rational map $\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{\dim U-1}$ given by $U$. Its coordinate ring is isomorphic to $\bigoplus_{i\ge 0}…