Related papers: Distance functions with dense singular sets

In this note, which is the second part of a three-part series, we focus on uniqueness sets specifically in the case of spaces of entire functions of exponential type. As in the first part, we consider sets with angular density; however, now…

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also…

This paper studies singularity structures of the linear inviscid damping of two-dimensional Euler equations in a finite periodic channel. We introduce a recursive definition of singularity structures which characterize the singularities of…

If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\dist^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance…

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing…

In this paper, we consider the Dirichlet problem of three-dimensional inhomogeneous incompressible micropolar equations with density-dependent viscosity. Under the assumption that the coefficients are power functions of the density, we…

A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define…

The paper deals with singularities of nonconfluent hypergeometric functions in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such…

This paper studies hypersurface exceptional singularities in $\mathbb C^n$ defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional…

We study the vanishing viscosity method for the eikonal equation $|Du|=V$ in $B(0,1)$ with homogeneous Dirichlet boundary value condition. By assuming $V$ is radially symmetric and restricting attention to radially symmetric solutions, we…

Let $\xi$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=\xi$, and such that 1 is not a singularity of…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

The property that a one to one function from the natural numbers to itself preserves the density of sub-sets is shown to be equivalent to a condition on the covering of intervals in the range of the function by images of intervals in the…

We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the…

We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in…

We describe recent results obtained by G. Barles, P. Cardaliaguet, R. Monneau and the author recently. They are concerned with nonlocal Eikonal equations arising in the study of the dynamics of dislocation lines in crystals. These equations…

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as…

We prove that certain suitably renormalized value functions associated with the $d$-dimensional ($d\geq2$) $N$-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is…