Related papers: Approximating singularities by a cuspidal edge on …
The aim of this fisrt part is to introduce, for a rather large class of hypersurface singularities with 1 dimensionnal locus, the analog of the Brieskorn lattice at the origin (the singular point of the singular locus). The main results are…
We show global uniqueness of the solution to a class of constrained variational problems, using scaling properties. This is used to establish the essential uniqueness of solutions of a large deviations problem in multiple dimensions. The…
We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…
Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field $\nu$ is well-defined as a smooth vector field of the surface. A cuspidal edge…
The Bj\"orling problem amounts to the construction of a minimal surface from a real-analytic curve with a given real-analytic normal vector field. We approximate that solution locally by discrete minimal surfaces as special discrete…
We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the…
We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schr\"odinger equation in three spatial dimensions. The proof is computer-assisted and relies on a fixed point argument that shows…
On a compact surface, we prove existence and uniqueness of the conformal metric whose curvature is prescribed by a negative function away from finitely many points where the metric has prescribed angles presenting cusps or conical…
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic $n$-type edge, which is invariant under a helicoidal motion in Euclidean $3$-space admits non-trivial isometric…
We study a global theory of affine maximal surfaces with singularities, which are called affine maximal maps and defined by Aledo--Mart\' inez--Mil\' an. In this paper, we define a special subclass of such surfaces other than improper…
The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this…
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 study the geometry of cuspidal $S_k$ singularities in $\mathbb R^3$ obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We study…
We study existence and uniqueness of bounded solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the…
Characterizing face-number-related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regard, one of the well-known invariants is $g_2$. Let $K$ be a normal $3$-pseudomanifold…
We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.
We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit…
In this paper we are concerned with existence of positive solutions for a Schr\"odinger-Maxwell system with singular or strongly-singular terms. We overcome the difficulty given by the singular terms through an approximation scheme and…
An arbitrary outward cuspidal domain is shown to be bi-Lipschitz equivalent to a Lipschitz outward cuspidal domain via a global transformation. This allows us to extend earlier Sobolev extension results on Lipschitz outward cuspidal domains…