This paper deals with the analysis of the singularities arising from the solutions of the problem ${-}\,{\rm Curl\ } F=\mu $, where F is a 3 × 3 matrix-valued Lp-function ($1\les p<2$) and μ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that $F=\nabla u$, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3 and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3 and show that their boundaries can be written in term of the measure μ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.

Let n, $N \in {\open N}$ with $\Omega \subseteq {\open R}^n$ open. Given ${\rm H} \in C^2(\Omega \times {\open R}^N \times {\open R}^{Nn})$, we consider the functional 1

$${\rm E}_\infty (u,{\rm {\cal O}})\, : = \, \mathop {{\rm ess}\,\sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u,{\rm D}u),\quad u\in W_{{\rm loc}}^{1,\infty} (\Omega, {\open R}^N),\quad {\rm {\cal O}}{\Subset}\Omega.$$

$L^\infty $

2

$$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0,\hfill}\right.$$

$[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$

${\cal D}$

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$.

We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator $ P_{\lambda }:= -\Delta _{{\open H}^{N}} - \lambda $ where 0 ⩽ λ ⩽ λ1(ℍN) and λ1(ℍN) is the bottom of the L2 spectrum of $-\Delta _{{\open H}^{N}} $, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for the operator $P_{\lambda _{1}({\open H}^{N})}$. A different, critical and new inequality on ℍN, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$

The aim of this paper is to study Hamiltonian elliptic system of the form 0.1

$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$

0.2

$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$

This paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

Let Ω ⊂ ℝN be a bounded domain and δ(x) be the distance of a point x ∈ Ω to the boundary. We study the positive solutions of the problem Δu + (μ/(δ(x)2))u = up in Ω, where p > 0, p ≠ 1 and μ ∈ ℝ, μ ≠ 0 is smaller than the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper, we first give the complete picture of the radial solutions in balls. In particular, we establish for p > 1 the existence of a unique large solution behaving like δ−(2/(p−1)) at the boundary. In general domains, we extend the results of Bandle and Pozio and show that there exists a unique singular solutions u such that $u/\delta ^{\beta _-}\to c$ on the boundary for an arbitrary positive function $c \in C^{2+\gamma }(\partial \Omega ) \, (\gamma \in (0,1)), c \ges 0$. Here β− is the smaller root of β(β − 1) + μ = 0.

Global weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

Let p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative 𝔽p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, 𝔽).

In the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

We are interested in standing waves of a modified Schrödinger equation coupled with the Chern–Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the nonexistence for nontrivial solutions.

We provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

This paper is concerned with the travelling waves for a class of non-local dispersal non-cooperative system, which can model the prey-predator and disease-transmission mechanism. By the Schauder's fixed-point theorem, we first establish the existence of travelling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations, whose bounds are deduced from a precise analysis. Further, we characterize the minimal wave speed of travelling waves and obtain the non-existence of travelling waves with slow speed. Finally, we apply the general results to an epidemic model with bilinear incidence for its propagation dynamics.

We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.

We study the non-autonomously forced Burgers equation

$$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

We consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

In this paper we characterize the boundedness on the product of Sobolev spaces Hs(𝕋) × Hs(𝕋) on the unit circle 𝕋, of the bilinear form Λb with symbol b ∈ Hs(𝕋) given by

$${{\Lambda}_b} (\varphi, \psi): = \int_{\open T} {\left( {{{( - {\Delta })}^s} + I} \right)(\varphi \psi )(\eta )b(\eta ) {\rm d}\sigma (\eta ).}$$

We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm–Liouville problems. Via Γ-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm–Liouville operator.