Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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4-Harmonic functions and beyond
Full text linkThe family of partial differential equations −∆4u−ε∆∞u = 0 (ε > 0) is studied in a bounded domain Ω for given boundary data. We show that for each ε > 0 the problem has a unique viscosity solution which is exactly the (4+ε)-harmonic map with the given boundary data. We also explore the connections between the solutions of these problems and infinity harmonic and 4-harmonic maps by studying the limiting behavior of the solutions as ε → ∞ and ε → 0+, respectively
Stabilization for small mass in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with density-dependent sensitivity: balanced case
Full text linkThis paper deals with the problem of the quasilinear parabolic--elliptic--elliptic attraction-repulsion chemotaxis system with q = p and χα −ξγ = 0:
1. u =∇·((u+1)m−1∇u−χu(u+1)p−2∇v+ξu(u+1)q−2∇w),
2. 0=∆v+αu−βv,
3. 0 = ∆w+γu−δw
in a bounded domain Ω ⊂ Rn (n ∈ N) with smooth boundary ∂ Ω, where m, p, q ∈ R, χ , ξ , α , β , γ , δ > 0 are constants. In the case that m ̸= 1, p ̸= 2 and q ̸= 2 boundedness and finite-time blow-up have been classified by the sizes of and the sign of χ α − ξ γ (Z. Angew.\ Math.\ Phys.; 2022; 73; 61), where the critical case χ α − ξ γ = 0 has been excluded. The purpose of this paper is to prove boundedness and stabilization in the case χα−ξγ =0
Suitable Radon measure for nonlinear Dirichlet boundary p(u)-Laplacian problem
Full text linkThis paper is devoted to the study of nonlinear homogeneous Dirichlet boundary p(u)-laplacian problem. Existence, uniqueness and structural stability results of weak solutions are obtained by approximation method and convergent sequences in terms of Young measures
Time discretization of a nonlocal phase-field system with inertial term
Full text linkTime discretizations of phase-field systems have been studied. For example, a time discretization and an error estimate for a parabolic-parabolic phase-field system have been studied by Colli--K. [Commun. Pure Appl. Anal. 18 (2019)]. Also, a time discretization and an error estimate for a simultaneous abstract evolution equation applying parabolic-hyperbolic phase field systems and the linearized equations of coupled sound and heat flow have been studied (see K. [ESAIM Math. Model. Numer. Anal.54 (2020), Electron. J. Differential Equations 2020, Paper No. 96]). On the other hand, although existence, continuous dependence estimates and behavior of solutions to nonlocal phase-field systems with inertial terms have been studied by Grasselli--Petzeltov\\u27a--Schimperna [Quart. Appl. Math. 65 (2007)], time discretizations of these systems seem to be not studied yet. In this paper we focus on employing a time discretization scheme for a nonlocal phase-field system with inertial term and establishing an error estimate for the difference between continuous and discrete solutions
Tangent quadrics in real 3-space
Full text linkWe examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of polynomial equations, also in the space of complete quadrics, and we solve them using certified numerical methods. Our aim is to show that Schubert’s problems are fully real
Computing a minimal resolution over the Steenrod algebra
Full text linkWe describe an algorithm that allows to compute a minimal resolution of the Steenrod algebra. The algorithm has built-in knowledge about vanishing lines for the cohomology of sub Hopf algebras of the Steenrod algebra which makes it both faster and more economical than the generic approach
Existence of entropy solutions for nonlinear elliptic problem having large monotonicity in weighted Orlicz-Sobolev spaces
Full text linkWe prove an existence result of entropy solution for a class of nonlinear elliptic problems of Leray-Lions type with large monotonicity condition in the framework of weighted Orlicz-Sobolev spaces and with right hand side f ∈ L1(Ω) 
Entropy solution for nonlinear degenerate elliptic problem with Dirichlet-type boundary condition in weighted Sobolev spaces
Full text linkIn this paper, we prove the existence and uniqueness results of entropy solution to a class of nonlinear degenerate elliptic problem with Dirichlet-type boundary condition and L1 data. The main tool used here is the regularization approach combined with theory of weighted Sobolev spaces
Topological indices for the antiregular graphs
Full text linkWe determine some classical distance-based and degree-based topo- logical indices of the connected antiregular graphs (maximally irregular graphs). More precisely, we obtain explicitly the k-Wiener index, the hyper-Wiener index, the degree distance, the Gutman index, the first, sec- ond and third Zagreb index, the reduced first and second Zagreb index, the forgotten Zagreb index, the hyper-Zagreb index, the refined Zagreb index, the Bell index, the min-deg index, the max-deg index, the symmet- ric division index, the harmonic index, the inverse sum indeg index, the M-polynomial and the Zagreb polynomial
Jordan algebras of symmetric matrices
Full text linkWe study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian