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Abstract: Abstract In this work, we are interested in studying deformations of the \(\sigma _2\) -curvature and the volume. For closed manifolds, we relate critical points of the total \(\sigma _2\) -curvature functional to the \(\sigma _2\) -Einstein metrics and, as a consequence of results of Gursky and Viaclovsky (Invent Math 145(2):251–278, 2001) and Hu and Li (Trans Am Math Soc 356(8):3005–3023, 2004), we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to \(\sigma _2\) -curvature which shows that the volume can be controlled by the \(\sigma _2\) -curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant \(\sigma _2\) -curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second-order properties of the volume functional, we show that there is a variation for which geodesic balls are indeed local minima in a natural direction. PubDate: 2023-02-01
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Abstract: Abstract A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra. PubDate: 2023-02-01
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Abstract: Abstract We consider a novel curvature flow for curves on the 2-dimensional light cone, contained in the 3-dimensional Minkowski space. We show that the solutions to the curvature flow (CF) for such curves are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperbolas are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant c from being the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. The curve is a soliton solution when \(c=0\) . We prove that for each vector v there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Considering non-trivial solutions to the CF, we prove that the corresponding solutions to the ICF may have at most three connected components. Moreover, at each end of such a curve the curvature is either unbounded or it tends to 0 or to the constant c. Explicitly given soliton solutions are included and some self-similar solutions on the light cone are visualized. PubDate: 2023-02-01
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Abstract: Abstract We give sufficient conditions for the well posedness in \({\mathcal {C}}^\infty\) of the Cauchy problem for third-order equations with time-dependent coefficients. PubDate: 2023-02-01
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Abstract: Generalized \(\Lambda\) -semiflows are an abstraction of semiflows with nonperiodic solutions, for which there may be more than one solution corresponding to given initial data. A select class of solutions to generalized \(\Lambda\) -semiflows is introduced. It is proved that such minimal solutions are unique corresponding to given ranges and generate all other solutions by time reparametrization. Special qualities of minimal solutions are shown. The concept of minimal solutions is applied to gradient flows in metric spaces and generalized semiflows. Generalized semiflows have been introduced by Ball. PubDate: 2023-02-01
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Abstract: Abstract We consider the following Hénon equation with critical growth: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = K( y )u^{\frac{N+2}{N-2}},u>0, &{}\hbox {in } B_{1}(0) \\ \displaystyle u =0,&{}\hbox {on } \partial B_{1}(0), \end{array}\right. } \end{aligned}$$ where \(B_{1}(0)\) is the unit ball in \({\mathbb {R}}^{N}\) , \(K:[0,1] \rightarrow {\mathbb {R}}^{+}\) is a bounded function and \(K''(1)\) exists. We prove a non-degeneracy result of the bubble solutions constructed in [24] via the local Pohozaev identities for \(N \ge 5\) . Then, as applications, by using reduction arguments combined with delicate estimates for the modified Green function and the error, we prove the new existence of infinitely many non-radial solutions, whose energy can be arbitrarily large. PubDate: 2023-02-01
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Abstract: Abstract A \(K^{\alpha}\) -translator is a surface in Euclidean space \({\mathbb {R}}^3\) that moves by translations in a spatial direction under the \(K^{\alpha}\) -flow, where K is the Gauss curvature and \(\alpha\) is a constant. We classify all \(K^{\alpha}\) -translators that are rotationally symmetric. In particular, we prove that for each \(\alpha\) there is a \(K^{\alpha}\) -translator intersecting orthogonally the rotation axis. We also describe all \(K^{\alpha}\) -translators invariant by a uniparametric group of helicoidal motions and the translators obtained by separation of variables. PubDate: 2023-02-01
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Abstract: Abstract We study the existence, nonexistence and qualitative properties of the solutions to the problem $$\begin{aligned} ({\mathcal {P}}) \quad \quad \left\{ \begin{aligned} (-\Delta )^s u -\theta \frac{u}{ x ^{2s}}&=u^p - u^q \quad \text {in }\,\, {\mathbb {R}}^N\\ u&> 0 \quad \text {in }\,\, {\mathbb {R}}^N\\ u&\in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N), \end{aligned} \right. \end{aligned}$$ where \(s\in (0,1)\) , \(N>2s\) , \(q>p\ge {(N+2s)}/{(N-2s)}\) , \(\theta \in (0, \Lambda _{N,s})\) and \(\Lambda _{N,s}\) is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole \(\mathbb {R}^N\) , and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space. PubDate: 2023-02-01
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Abstract: Abstract We provide a Hopf boundary lemma for the regional fractional Laplacian \((-\Delta )^s_{\Omega }\) , with \(\Omega \subset \mathbb {R}^N\) a bounded open set. More precisely, given u a pointwise or weak super-solution of the equation \((-\Delta )^s_{\Omega }u=c(x)u\) in \(\Omega\) , we show that the ratio \(u(x)/(\text {dist}(x,\partial \Omega ))^{2s-1}\) is strictly positive as x approaches the boundary \(\partial \Omega\) of \(\Omega\) . We also prove a strong maximum principle for distributional super-solutions. PubDate: 2023-02-01
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Abstract: Abstract We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type \((n+\ell ,k-\ell )\) . While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension. PubDate: 2023-02-01
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Abstract: Abstract In this paper, we derive general bounds for the number of rational points on a cubic surface defined over \({\mathbb {F}}_q\) , which constitute an extension of a result due to Weil. Exploiting these bounds, we are able to give a complete characterization of the intersections between the Norm–Trace curve over \({\mathbb {F}}_{q^3}\) and the curves of the form \(y=ax^3+bx^2+cx+d\) , generalizing a previous result by Bonini and Sala and providing more detailed information about the weight spectrum of one-point AG codes arising from such curve. PubDate: 2023-02-01
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Abstract: Abstract A positive radial singular solution for \(\Delta u+f(u)=0\) with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem \(\Delta u+\lambda f(u)=0\) on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., \(u^p\log u\) , \(\exp (u^p)\) and \(\exp (\cdots \exp (u)\cdots )\) as well as \(u^p\) and \(e^u\) . Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. PubDate: 2023-02-01
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Abstract: Abstract In the spirit of Berestycki and Lions (Arch. Rational Mech. Anal., 82: 313–345, 1983), we prove the existence of saddle-type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u + u= \big (I_\alpha *F(u)\big )F'(u)\qquad \text { in }\;\mathbb {R}^N \end{aligned}$$ where \(N\ge 2\) and \(I_\alpha\) is the Riesz potential of order \(\alpha \in (0,N)\) . Given a finite Coxeter group G with rank \(k\le N\) , we construct a G-groundstate uniformly with lowest energy amongst G-saddle solutions for the Choquard equation in a noncompact setting. Moreover, if \(F'\) is odd and has constant sign on \((0,+\infty )\) , then every G-groundstate maintains signed on the fundamental domain of the corresponding Coxeter group and receives opposite signs on any two adjacent regions so that nodal domains of G-groundstate are of cone shapes demonstrating Coxeter’s symmetric configurations in \(\mathbb {R}^N\) . These results further complete the variational framework in constructing sign-changing solutions for the Choquard equation but still require a quadratic or super-quadratic growth on F near the origin. PubDate: 2023-02-01
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Abstract: Abstract In this paper, we consider the Cauchy problem for the two-component Fornberg–Whitham system on the real line and study the issue of uniform dependence on initial data for this equation. We prove that the solution map of this problem cannot be uniformly continuous in Sobolev spaces \(H^s({\mathbb {R}})\times H^{s-1}({\mathbb {R}})\) for \(s > {\frac{3}{2}}\) . PubDate: 2023-02-01
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Abstract: Abstract In this paper we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates \((r, \theta , z)\) , we investigate the shape of possibly sign-changing solutions whose derivative in \(\theta\) vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of \(\theta\) or strictly monotone with respect to \(\theta\) . In the special case of a planar domain, the result holds in a circular sector as well as in an annular one, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case where its opening is larger than \(\pi\) . PubDate: 2023-02-01
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Abstract: Abstract Rota–Baxter operators for groups were recently introduced by Guo, Lang, and Sheng. Bardakov and Gubarev showed that with each Rota–Baxter operator, one can associate a skew brace. Skew braces on a group G can be characterised in terms of certain gamma functions from G to its automorphism group \({{\,\mathrm{Aut}\,}}(G)\) that are defined by a functional equation. For the skew braces obtained from a Rota–Baxter operator, the corresponding gamma functions take values in the inner automorphism group \({{\,\mathrm{Inn}\,}}(G)\) of G. In this paper, we give a characterisation of the gamma functions on a group G, with values in \({{\,\mathrm{Inn}\,}}(G)\) , that come from a Rota–Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota–Baxter operator. For gamma functions that can be obtained from a Rota–Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits. PubDate: 2023-02-01
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Abstract: Abstract Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated q-system. Using this result, we provide the first construction of a family of \(\mathbb F_{q^m}\) -linear MRD codes of length 2m that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting. PubDate: 2023-02-01
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Abstract: Abstract We study symmetries of specific left-invariant sub-Riemannian structure with filtration (4, 7) and their impact on sub-Riemannian geodesics of corresponding control problem. We show that there are two very different types of geodesics, they either do not intersect the fixed point set of symmetries or are contained in this set for all times. We use the symmetry reduction to study properties of geodesics. PubDate: 2023-02-01
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Abstract: Abstract The main result of this paper is to study the local deformations of Calabi-Yau \(\partial \bar{\partial }\) -manifold that are co-polarised by the Gauduchon metric by considering the subfamily of co-polarised fibres by the class of Aeppli/De Rham-Gauduchon cohomology of Gauduchon metric given on the central fibre. In the latter part, we prove that the p-SKT h- \(\partial \bar{\partial }\) -property is deformation open by constructing and studying a new notion called hp-Hermitian symplectic (hp-HS) form. PubDate: 2023-02-01
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