Consider the two-dimensional autonomous systems of differential equations

$$ \dot{x}=-y+\lambda x+P(x,y),\qquad\dot{y}=x+\lambda y+Q(x,y), $$

where $\lambda$ is a real constant and $P$ and $Q$ are $\mathcal{C}^{\infty}$-functions of order greater than or equal to two. These systems, so-called centre-focus-type systems, have either a centre or a focus at the origin. In this work, we give necessary and sufficient conditions of isochronicity using normal forms. We characterize the systems which have either an isochronous centre or an isochronous focus at the origin by means of the existence of a commutator of the field. Moreover, we prove that the maximum order of a weak isochronous focus for quadratic systems is two, and that for systems with cubic nonlinearities is three.

In this paper, we are concerned with the existence of unbounded orbits of the mapping

\begin{align*} \theta_1&=\theta+2\pi+\frac{1}{\rho}\mu(\theta)+o(\rho^{-1}), \\ \rho_1&=\rho+c-\mu'(\theta)+o(1),\quad\rho\to\infty, \end{align*}

where $c$ is a constant and $\mu(\theta)$ is $2\pi$-periodic. Assume that $c\not=0$, that $\mu(\theta)$ is non-negative (or non-positive) and that $\mu(\theta)$ has finitely many degenerate zeros in $[0,2\pi]$. We prove that every orbit of the given mapping tends to infinity in the future or in the past for sufficiently large $\rho$. On the basis of this conclusion, we further prove that the equation $x''+f(x)x'+V'(x)+\phi(x)=p(t)$ has unbounded solutions provided that $V$ is an isochronous potential at resonance and $F(x)$ ($F(x)=\int_0^xf(s)\,\mathrm{d} s$) and $\phi(x)$ satisfy some limit conditions. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this equation.

The usual asymptotic expansion for the solutions of an elliptic linear problem with oscillatory periodic coefficients is known to not be accurate near the boundary. In order to obtain a better approximation it is necessary to add to this expansion a boundary-layer term. This term has been obtained by other authors in the case of a plane boundary, such that its normal is proportional to some period. We consider the case where the normal is arbitrary.špace{-8pt}

In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like $k^{-7/6}$ as $k\to0$. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function $k^{-7/6}$ that we obtained in an earlier paper.

The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix

$$ \begin{pmatrix} -\dfrac{\mathrm{d}}{\mathrm{d} x}\rho(x) \dfrac{\mathrm{d}}{\mathrm{d} x}+q(x) & \dfrac{\mathrm{d}}{\mathrm{d} x}\dfrac{\beta(x)}{x} \\[12pt] -\dfrac{\beta(x)}{x}\dfrac{\mathrm{d}}{\mathrm{d} x} & \dfrac{m(x)}{x^2} \end{pmatrix} $$

is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarantee that the operator is semi-bounded from below are derived. It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential operator is given by the range $\text{range}((m\rho-\beta^2)/\rho x^2)$ in the case where the quasi-regularity conditions are not satisfied.

This work is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form

$$ \mathcal{L}u+|u|^{q-1}u \leq\mathcal{L}v+|v|^{q-1}v, $$

where $q>0$ is a given real number and $\mathcal{L}$ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form given by the relation

$$ \mathcal{L}=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\bigg[a_{ij}(x)\frac{\partial}{\partial x_j}\bigg]. $$

We assume that $n\geq2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots,n$, are measurable bounded functions on $\mathbb{R}^n$ such that $a_{ij}(x)=a_{ji}(x)$ and that the corresponding quadratic form is non-negative. The results obtained in this work were announced by the author in 2005.

Let $H_n$, $n\ge3$, be the space of all $n\times n$ Hermitian matrices. Assume that a map $\phi:H_n\to H_n$ preserves commutativity in both directions (no linearity or bijectivity of $\phi$ is assumed). Then $\phi$ is a unitary similarity transformation composed with a locally polynomial map possibly composed by the transposition. The same result holds for injective continuous maps on $H_n$ preserving commutativity in one direction only. We give counter-examples showing that these two theorems cannot be improved or extended to the infinite-dimensional case.

We prove the convergence in certain weighted spaces in momentum space of eigenfunctions of $H=T-\lambda V$ as the energy goes to an energy threshold. We do this for three choices of kinetic energy $T$, namely the non-relativistic Schrödinger operator, the pseudorelativistc operator $\sqrt{-\Delta+m^2}-m$, and the Dirac operator.

The oldest competition for an optimal (area-maximizing) shape was won by the circle. But if the fixed perimeter is measured by the line integral of $|\mathrm{d} x|+|\mathrm{d} y|$, a square would win. Or if the boundary integral of $\max(|\mathrm{d} x|,|\mathrm{d} y|)$ is given, a diamond has maximum area. For any norm in $\mathbb{R}^2$, we show that when the integral of $\|(\mathrm{d} x,\mathrm{d} y)\|$ around the boundary is prescribed, the area inside is maximized by a ball in the dual norm (rotated by $\pi/2$).

This ‘isoperimetrix' was found by Busemann. For polyhedra it was described by Wulff in the theory of crystals. In our approach, the Euler–Lagrange equation for the support function of $S$ has a particularly nice form. This has application to computing minimum cuts and maximum flows in a plane domain.

Let $\varOmega$ be a domain in $\mathbb{R}^N$ (possibly unbounded), $N\geq2$, $1<p<\infty$, and let $V\in L_{\mathrm{loc}}^\infty(\varOmega)$. Consider the energy functional $\mathcal{Q}_V$ on $C_{\mathrm{c}}^\infty(\varOmega)$ and its Gâteaux derivative $\mathcal{Q}_V^\prime$, respectively, given by

$$ \mathcal{Q}_V(u)\eqdef\frac{1}{p}\int_\varOmega(|\nabla u|^p+V|u|^p)\,\mathrm{d} x,\qquad\mathcal{Q}_V^\prime(u)= \mathrm{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u, $$

for $u\in C_{\mathrm{c}}^\infty(\varOmega)$. Assume that $\mathcal{Q}_V>0$ on $C_{\mathrm{c}}^{\infty}(\varOmega)\setminus\{0\}$ and that $\mathcal{Q}_V$ does not have a ground state (in the sense of a null sequence for $\mathcal{Q}_V$ that converges in $L_{\mathrm{loc}}^p(\varOmega)$ to a positive function $\varphi\in C_{\mathrm{loc}}^1(\varOmega)$, a ground state). Finally, let $f\in\mathcal{D}^\prime(\varOmega)$ be such that the functional $u\mapsto \mathcal{Q}_V(u)-\langle u,f\rangle:C_{\mathrm{c}}^\infty(\varOmega)\to\mathbb{R}$ is bounded from below. Then the equation

$$ \mathcal{Q}_V^\prime(u)=f $$

has a solution $u_0\in W^{1,p}_{\mathrm{loc}}(\varOmega)$ in the sense of distributions. This solution also minimizes the functional $u\mapsto\mathcal{Q}_V^{**}(u)-\langle u,f\rangle:C_{\mathrm{c}}^\infty(\varOmega)\to \mathbb{R}$, where $\mathcal{Q}_V^{**}$ denotes the bipolar ($\varGamma$-regularization) of $\mathcal{Q}_V$ and $\mathcal{Q}_V^{**}$ is the largest convex, weakly lower semicontinuous functional on $C_{\mathrm{c}}^\infty(\varOmega)$ that satisfies $\mathcal{Q}_V^{**}\leq\mathcal{Q}_V$. (The original energy functional $\mathcal{Q}_V$ is not necessarily convex.)