Related papers: A solution to the Pompeiu problem

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove…

In this paper, we simplify and extend the results of \cite{GZ} to include the case in which $\Om =\R^3$. Let ${[L^2({\mathbb{R}}^3)]^3}$ be the Hilbert space of square integrable functions on ${\mathbb {R}}^3 $ and let ${\mathbb…

In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and…

We prove that for all $n\in \mathbb{N}$, there exists a constant $C_{n}$ such that for all $d \in \mathbb{N}$, for every row contraction $T$ consisting of $d$ commuting $n \times n$ matrices and every polynomial $p$, the following…

For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are…

We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in $C^{2,\alpha}(B_R)\cap C(\overline{B_R})$ for the inhomogeneous $\infty$-Bilaplacian…

We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the…

We say that a PDE in a Riemannian manifold $M$ is geometric if,$\ $whenever $u$ is a solution of the PDE on a domain $\Omega$ of $M$, the composition $u_{\phi}:=u\circ\phi$ is also solution on $\phi^{-1}\left( \Omega\right) $, for any…

We address the homogeneous in space and time (ST-homogeneous) G\"{o}del-type metrics within the cubic galileon theory, a particular class of generalized galileon theories. We check the consistency of such spacetimes for a physically…

In this paper we show that steady states $u$ of the pressureless Euler equation which belong to $L^3_{loc}(\mathbb{R}^2,\mathbb{R}^2)$ are shear flows. This is achieved by combining results of degenerate Monge-Amp\`ere-type equations with…

We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic solutions of such a problem with a normalization condition and set-theoretic constraints on their complex…

We study stable solutions of the following nonlinear system $$ -\Delta u = H(u) \quad \text{in} \ \ \Omega$$ where $u:\mathbb R^n\to \mathbb R^m$, $H:\mathbb R^m\to \mathbb R^m$ and $\Omega$ is a domain in $\mathbb R^n$. We introduce the…

The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations…

The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the…

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for…

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex…

In this paper we consider the fully nonlinear parabolic free boundary problem $$ \left\{\begin{array}{ll} F(D^2u) -\partial_t u=1 & \text{a.e. in}Q_1 \cap \Omega\\ |D^2 u| + |\partial_t u| \leq K & \text{a.e. in}Q_1\setminus\Omega,…

Let $M$ be a smooth $4$-manifold underlying some del Pezzo surface of degree $d \geq 6$. We consider the smooth Nielsen realization problem for $M$: which finite subgroups of $\text{Mod}(M) = \pi_0(\text{Homeo}^+(M))$ have lifts to…

We study global convex solutions of the Monge-Amp\`ere equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n, \] where $\mu \not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a…

The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ \Delta u+g(u)= \mu \phi _1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on…