Abstract: We prove that the law of the minimum \({m := {\rm min}_{t\in[0,1]}\xi(t)}\) of the solution \({\xi}\) to a one-dimensional stochastic differential equation with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets \({\{x \in C([0,1]) : {\rm inf} x \geq r\}}\) have finite perimeter with respect to the law \({\nu}\) of the solution \({\xi({\cdot})}\) in \({L^{2}(0,2)}\) . PubDate: 2019-03-20 DOI: 10.1007/s00032-019-00295-2

Abstract: Let C be a closed, convex and nonempty subset of a Banach space X. Let \({T : C \rightarrow X}\) be a nonexpansive inward mapping. We consider the boundary point map \({h_{C,T } : C \rightarrow \mathbb{R}}\) depending on C and T defined by \({h_{C,T} = {\rm max}\{\lambda \in [0,1] : [(1-\lambda)x + \lambda Tx] \in C\}}\) , for all \({x \in C}\) . Then for a suitable step-by-step construction of the control coefficients by using the function \({h_{C,T }}\) , we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} < \infty}\) and weak convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} = \infty}\) . PubDate: 2019-03-16 DOI: 10.1007/s00032-019-00293-4

Abstract: 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. PubDate: 2019-02-23 DOI: 10.1007/s00032-019-00291-6

Abstract: 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 locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety. PubDate: 2019-02-23 DOI: 10.1007/s00032-019-00292-5

Abstract: In this paper we prove existence and uniqueness results for a lower order perturbation of elliptic Dirichlet problems with a singular convection term in divergence form and L1 data. PubDate: 2019-02-15 DOI: 10.1007/s00032-019-00290-7

Abstract: We prove that nontrivial weak solutions of a class of fractional magnetic Schrödinger equations in \({\mathbb{R}^N}\) are bounded and vanish at infinity. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0283-3

Abstract: This paper concerns with the existence and regularity of solutions for the following Choquard type equation, $$-\Delta_u = \big(I_{\mu} * F(u)\big) f(u) {\rm in} \mathbb{R}^3, \quad \quad (P)$$ where \({I_\mu = \frac{1}{ x ^\mu}, 0 < \mu < 3}\) , is the Riesz potential, \({F(s)}\) is the primitive of the continuous function f(s), and \({I_{\mu} * F(u)}\) denotes the convolution of \({I_{\mu}}\) and F(u). By using the variational method, we prove that problem (P), in the zero mass case, possesses at least a nontrivial solution under certain conditions on f. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0289-x

Abstract: We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0288-y

Abstract: The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic p-Laplace operator with Robin boundary condition involving a function \({\beta}\) which in general is not constant. In particular we obtain sharp lower bounds in terms of the measure of the domain and we prove a monotonicity property of the eigenvalue with respect the set inclusion. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0286-0

Abstract: Let \({P_{\rm MAX}(d, s)}\) denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree d in \({\mathbb{P}^3}\) that is not contained in a surface of degree < s. A bound P(d, s) for \({P_{\rm MAX}(d, s)}\) has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family \({\mathcal{C}}\) of primitive multiple lines and we conjecture that the generic element of \({\mathcal{C}}\) has good cohomological properties. From the conjecture it would follow that \({P(d, s) = P_{\rm MAX}(d, s)}\) for d = s and for every \({d \geq 2s - 1}\) . With the aid of Macaulay2 we checked this holds for \({s \leq 120}\) by verifying our conjecture in the corresponding range. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0284-2

Abstract: In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the \({H^\infty}\) functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where \({e_{\ell}, {\ell} = 1, 2, 3}\) are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables \({x = (x_{1}, x_{2}, x_{3})}\) and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version \({T^{\alpha}, {\rm for} \alpha \in (0, 1)}\) , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0287-z

Abstract: We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details. PubDate: 2018-12-01 DOI: 10.1007/s00032-018-0285-1

Authors:Boutechebak Souraya; Azeb Ahmed Abdelaziz Abstract: We consider a mathematical model which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. We model the material’s behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition. We derive variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field, the damage field and the adhesion field. We prove the existence of a unique weak solution to the problem. The proof is based on arguments of time dependent variational inequalities, parabolic inequalities, differential equations and fixed point. PubDate: 2018-05-29 DOI: 10.1007/s00032-018-0282-4

Authors:Lucio Boccardo; Stefano Buccheri; Giuseppa Rita Cirmi Abstract: We give a self-contained and simple approach to prove the existence and uniqueness of a weak solution to a linear elliptic boundary value problem with drift in divergence form. Taking advantage of the method of continuity, we also deal with the relative dual problem. PubDate: 2018-05-19 DOI: 10.1007/s00032-018-0281-5

Authors:Ayman Shehata Abstract: This study deals with the convergence properties of Struve matrix functions within complex analysis. Certain new classes of matrix differential recurrence relations, matrix differential equations, the various families of integral representations and integrals obtained here are believed to be new in the theory of Struve matrix functions, and the several properties of the modified Struve matrix functions are also included. Finally, we investigate the operational rules which yield a different view of the expansion formulae for Struve and modified Struve matrix functions. PubDate: 2018-05-15 DOI: 10.1007/s00032-018-0280-6

Authors:Giuseppe Maria Coclite; Lorenzo di Ruvo Abstract: We consider a generalized Ostrovsky–Hunter equation and the corresponding generalized Ostrovsky one with nonlinear dispersive effects. For the first equation, we study the well-posedness of entropy solutions for the Cauchy problem. For the second equation, we prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the first one. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method. PubDate: 2018-05-11 DOI: 10.1007/s00032-018-0278-0

Authors:M.L. Carvalho; J.V. Goncalves; E.D. Silva; C.A.P. Santos Abstract: In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in \({W^{1, \Phi}_{loc} (\Omega)}\) to the problem $$\left\{\begin{array}{ll}-\Delta_{\Phi}u=f(x,u)\, {\rm in}\, \Omega,\\u > 0\, {\rm in} \, \Omega, u = 0 \,{\rm on}\, \partial\Omega,\end{array}\right.$$ where f has \({\Phi}\) -sublinear growth. In our main results the function f(x, u) may be singular at u = 0 and the nonlinear term \({f(x, t)/t^{\ell-1}, t > 0}\) is strictly decreasing for a suitable \({\ell > 1}\) . Under different kind of boundary conditions we prove an improvement for the classical Brézis-Oswald and Díaz-Sáa’s results in Orlicz- Sobolev framework for singular nonlinearities as well. Some results discussed here are news even for Laplacian or p-Laplacian operators. PubDate: 2018-05-11 DOI: 10.1007/s00032-018-0279-z

Authors:Giovany M. Figueiredo; Marcos T. O. Pimenta Abstract: In this work we state and prove versions of some classical results, in the framework of functionals defined in the space of functions of bounded variation in \({\mathbb{R}^N}\) . More precisely, we present versions of the Radial Lemma of Strauss, the compactness of the embeddings of the space of radially symmetric functions of BV ( \({\mathbb{R}^N}\) ) in some Lebesgue spaces and also a version of the Lions Lemma, proved in his celebrated paper of 1984. As an application, we get existence of a nontrivial bounded variation solution of a quasilinear elliptic problem involving the 1−Laplacian operator in \({\mathbb{R}^N}\) , which has the lowest energy among all the radial ones. This seems to be one of the very first works dealing with stationary problems involving this operator in the whole space. PubDate: 2018-03-21 DOI: 10.1007/s00032-018-0277-1

Authors:Alex M. Batista; Marcelo F. Furtado Abstract: We deal with the equation $$-\left( 1+\int\nolimits_{\mathbb{R}^3} \nabla u ^2 dx\right)\Delta u + V(x)u=a(x) u ^{p-1}u,\quad x\in {\mathbb{R}}^3,$$ with p ∈ (3, 5). Under some conditions on the sign-changing potentials V and a we obtain a nonnegative ground state solution. In the radial case we also obtain a nodal solution. PubDate: 2018-01-31 DOI: 10.1007/s00032-018-0276-2