We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain \({\Omega} {\subset} \mathbb{R}^{d}, d \geq 1\). When \(\Omega = [0, 1]\), such graphs include the standard Toeplitz graphs and, for \(\Omega = [0, 1]^{d}\), the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices

This study focuses on the existence and concentration of ground state solutions for a class of fractional Kirchhoff–Schrödinger equations. We first study the problem $$\left\{ \begin{array}{ll} M ([u]^{2}_{s} + \int_{\mathbb{R}^{N}} V(x)u^{2}) ((-{\Delta})^{s}u + V (x)u) = \bar{c}u + f(u)\, {\rm in}\,\, \mathbb{R}^N,\\ u > 0, u\, {\in} \, {H}^{s} (\mathbb{R}^N),\end{array} \right.$$ where \(s \in (0

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.

This paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on \(R^n\) and to a family of Hamiltonian systems involving double well potentials will also be discussed.

We study the motion of discrete interfaces driven by ferromagnetic interactions on the two-dimensional triangular lattice by coupling the Almgren, Taylor and Wang minimizing movements approach and a discrete-to-continuum analysis, as introduced by Braides, Gelli and Novaga in the pioneering case of the square lattice. We examine the motion of origin-symmetric convex “Wulfflike” hexagons, i.e. origin-symmetric

We are interested in quasilinear problems as follows: $$ \left\{ \begin{array}{ll} -\Delta u -u \Delta (u^2)= -\lambda |u|^{q-2}u+|u|^{22^*-2}u+\mu g(x,u), \quad \mathrm{in}~ \Omega ,\\ u=0,\quad \mathrm{on}~ \partial \Omega , \end{array}\right. $$(p) where \(\Omega \subset \mathbb {R}^N \)is a bounded domain with regular boundary \(\partial \Omega , N\ge 3, \lambda , \mu > 0,1

We study the existence of positive supersolutions of nonlocal equations of type \((-\Delta)^{s} u + |\Delta u|^{q} = \lambda f(u)\) posed in exterior domains where the datum f can be comparable with \(u^p\) near the origin. We prove that the existence of bounded supersolutions depends on the values of p, q and s.

Let \(\mathbb{D}\) be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that \(k^{2} = 1\). This algebra is often referred to as the algebra of hyperbolic numbers. A function \(f : \mathbb{D} \rightarrow \mathbb{D}\) is called \(\mathbb{D}\)-holomorphic in a domain \(\Omega \subset \mathbb{D}\) if it admits derivative in the sense that \({\rm lim}_{h\rightarrow{0}}\fr

Regularization by noise for certain classes of fluid dynamic equations, a theme dear to Giuseppe Da Prato [23], is reviewed focusing on 3D Navier-Stokes equations and dyadic models of turbulence.

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes into account area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by

Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong & Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the intrinsic

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. There is nowadays a large literature on the evolution of superoscillations under Schrödinger equation with different type of potentials. In this paper, we study the evolution of superoscillations under the Klein-Gordon equation and we describe in precise mathematical terms in what sense

In this work, we study the existence of a weak solution to the following quasi linear elliptic problem involving the fractional p-Laplacian operator, a Hardy potential, and multiple critical Sobolev nonlinearities with singularities, $$(-{\Delta}_{p})^{s}u - {\mu}\frac{|u|^{p-2}u}{|x|^{ps}}=\frac{|u|^{p^{*}_{s}(\beta)-2_{u}}}{|x|^{\beta}}+\frac{|u|^{p^{*}_{s}{(\alpha)-2_u}}}{|x|^{\alpha}},$$ where

The author regrets to point out some errors in [1], even thought these do not influence the validity of the results obtained. The detailed corrections are in the following.

In this paper we investigate the existence of a positive vector solution for a class of non-linear strongly coupled Schrödinger system in \(\mathbb{R}^N\), which is non-autonomous and asymptotically linear at infinity. Using topological arguments, combined with sharp exponential decay estimates, we obtain a bound state solution when the ground state is not attained.

This paper concerns in building extremal curves with respect to the parameters \(\lambda,\mu \geq 0\) for existence and multiplicity of \(D^{1,2}(\mathbb{R}^N)\)-solutions for the multi-parameter elliptic system $$ \left\{\begin{array}{lll} -{\Delta}{u} = {\lambda}{w}(x)f_{1}(u)g_{1}(v)\quad {\rm in}\quad \mathbb{R}^N,\\ -{\Delta}{v} = {\mu}w(x)f_{2}(v)g_{2}(u)\quad {\rm in} \quad \mathbb{R}^N,\\ u

In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional dynamic debonding model describing a thin film peeled away from a substrate. The system underlying the process couples the (weakly damped) wave equation with a Griffith’s criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the

We study the triviality of the solutions of weighted superlinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a Liouville-type theorem for solutions bounded from below with nonnegative initial data, under an integral growth condition on the weight.

In this paper we study the Ostrovsky–Hunter equation for the case where the flux function f(x, u) may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (namely fx(x, u) is uniformly Lipschitz locally in u and fu(x, u) is uniformly bounded), then there exists a unique entropy solution. To show the existence, after proving some a priori

Through the use of linearized bundles, we prove the stability of tautological bundles over the symmetric product of a curve and of the kernel of the evaluation map on their global sections.

In this manuscript it is proved existence results for some singular problems involving an anisotropic operator. In the approach we combine sub-supersolutions, truncation arguments and the Schaefer’s Fixed Point Theorem [23]. In this work it is not used approximation arguments as in [33, 37]

We address the problem of understanding the dynamics around typical singular points of 3D piecewise smooth vector fields. A model Z0 in 3D presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: (i) Z0 has an invariant plane \(\pi_0\) filled up with periodic orbits (this means that the restriction \(Z_{0|\pi_0}\) is a center around the singularity);

In this paper, we show the existence and multiplicity of positive solutions of the fractional Kirchhoff system $$\begin{aligned} {\left\{ \begin{array}{ll}\mathcal {L}_{M}(u) = {\lambda }f(x)|u|^{q-2}{u} + \frac{2{\alpha }}{{\alpha }+{\beta }} |u|^{\alpha -2} \,u|v|^{\beta} &{}\quad \mathrm{in}\, \Omega ,\\ \mathcal {L}_M(v) = {\mu }g(x)|v|^{q-2}v + \frac{2{\beta }}{{\alpha }+{\beta }}|u|^{\alpha}

Fluid flows around an obstacle generate vortices which are difficult to locate and to describe. A variational formulation for a class of mixed and nonstandard boundary conditions on a smooth obstacle is discussed for the Stokes equations. Possible boundary data are then derived through separation of variables of biharmonic equations in a planar region having an internal concave corner. Explicit singular

We are interested in finding nontrivial solutions for a Hamiltonian elliptic system in dimension two involving a potential function which can be coercive and nonlinearities that have maximal growth with respect to the Trudinger–Moser inequality. To establish the existence of solutions, we use variational methods combined with Trudinger–Moser type inequalities in Lorentz–Sobolev spaces and a finite-dimensional

We discuss the difference between orthogonal polynomials on finite and infinite dimensional vectors spaces. In particular we prove an infinite dimensional extension of Favard lemma.

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration \({\pi : \mathbb{CP}^3 \rightarrow S^4}\). We prove three results about the topology of the twistor discriminant locus of an algebraic surface in \({\mathbb{CP}^3}\). First of all we prove that, with the exception of two special cases, the real dimension of the twistor discriminant