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We relate the existence of Noether global conserved currents associated with locally variational field equations to existence of global solutions for a local variational problem generating global equations. Both can be characterized as the…
Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…
We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the $C^1$-topology. The proof of this fact combines ideas from dynamical…
We study a class of $p$-Laplacian Dirichlet problems with weights that are possibly singular on the boundary of the domain, and obtain nontrivial solutions using Morse theory. In the absence of a direct sum decomposition, we use a…
We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} \Delta_{\infty}u(x):=\inf_{y\sim x}u(y)+\sup_{y\sim x}u(y)-2u(x)=f(x). \end{equation*} For the homogeneous case…
For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much…
The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the…
We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set…
We consider the focusing $L^2$-supercritical Schr\"odinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the…
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…
The aim of this paper is to analyse the Dirichlet-Neumann operator in axially symmetric conical domains. We provide a constructive treatment of the generic singularity at the vertex by using a new coordinate system that maps the conical…
Recently our understanding of black holes in D-spacetime dimensions, as solutions of the Einstein equation, has advanced greatly. Besides the well established spherical black hole we have now explicitly found other species of topologies of…
We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this…
We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $\Omega$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We…
In the present article, solvability in Sobolev spaces is investigated for a class of degenerate stochastic integro-differential equations of parabolic type. Existence and uniqueness is obtained, and estimates are given for the solution.
In this paper we take a new approach to a proof of existence and uniqueness of solutions for the 3D-Navier-Stokes equations, which leads to essentially the same proof for both bounded and unbounded domains and for homogeneous or…
Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
This paper is concerned with inverse source problems for the time-dependent Lam\'e system in an unbounded domain corresponding to the exterior of a bounded cavity or the full space $\R^3$. If the time and spatial variables of the source…
In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions…