Unbounded Open Sets Research Articles

Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

The trichotomy between regular, semiregular, and strongly irregular boundary points for p-harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1<p<infty . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, p-harmonic measures, removability, and semibarriers.

Regularity of nonvanishing – at infinity or at the boundary – solutions of the defocusing nonlinear Schrödinger equation

Considering the defocusing nonlinear Schrödinger equation (NLSE) in generic (bounded or unbounded) open sets for n = 1, 2, and 3, we prove the regularity of weak, non-vanishing solutions at infinity or at the boundary of U. Our approach is based on suitably defined extension operators, along with a priori estimates for regular functions, under certain assumptions on the smoothness of the boundary. The results cover physically significant classes of solutions, as dark-solitons and compacton waveforms, when the notion of such solutions is extended in higher-dimensional set-ups.

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators L in RN and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author).

Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE

Let O be the basin of attraction of a given equilibrium of a dynamical system, whose solution is the law of large numbers limit of the solution of a Poissonian SDE as the size of the population tends to +∞. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of Day [J. Math. Anal. Appl. 147 (1990) 134–153] who studied the same problem for an ODE with a small Brownian perturbation. For that purpose, we will use the large deviations principle for the Poissonian SDE reflected at the boundary of O, studied in our recent work Pardoux and Samegni [Stoch. Anal. Appl. 37 (2019) 836–864]. The main motivation of this work is the extension of the results concerning the time of exit from the set O established in Kratz and Pardoux [Vol. 2215 of Lecture Notes in Math.. Springer (2018) 221–327] and Pardoux and Samegni [J. Appl. Probab. 54 (2017) 905–920] to unbounded open sets O. This is done in sections 4.2.5 and 4.2.7 of Britton and Pardoux [Vol. 2255 of Lecture Notes in Math. Springer (2019) 1–120], see also The SIR model with demography subsection below.

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

Maximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In the present notes, based on a joint work with prof. E. Lanconelli, we consider a class of sub-elliptic operators L in R^N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian(proved by Deny, Hayman and Kennedy) and to the sub-Laplacians on homogeneous Carnot groups (proved by Bonfiglioli and Lanconelli).

Intrinsic Ultracontractivity and Ground State Estimates of Non-local Dirichlet forms on Unbounded Open Sets

In this paper we consider a large class of symmetric Markov processes $${X=(X_t)_{t\ge0}}$$ on $${\mathbb{R}^d}$$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $${\alpha}$$ -stable-like type and with large jumps of super-exponential decay. Let $${D\subset \mathbb{R}^d}$$ be an open (not necessarily bounded and connected) set, and $${X^D=(X_t^D)_{t \ge 0}}$$ be the killed process of X on exiting D. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $${(P^{D}_t)_{t\ge0}}$$ of XD. When D is a horn-shaped region, we further obtain two-sided estimates of ground state in terms of jumping kernel of X and the reference function of the horn-shaped region D.

Псевдоспектр оператора конвенции-диффузии с переменным членом реакции

In this paper, we study the spectrum of non-self-adjoint convection-diffusion operator with a variable reaction term defined on an unbounded open set Ω of R^n. Our idea is to build a family of operators that have the same convection-diffusion-reaction formula, but which will be defined on bounded open sets 〖{Ω_η}〗_(η∈]0,1[) of R^n. Based on the relationships that link this family to Ω, we obtain relations between the spectrum and the pseudospectrum. We use the notion of the pseudospectrum to build relationships between convection-diffusion operator and its restrictions to bounded domains. Using these relationships we are able to find the spectrum of our operator in R^+. Also, the techniques developed to obtain the spectrum allow us to study the properties of the spectrum of this operator when we go to the limit as the reaction term tends to zero. Indeed, we show a spectral localization result for the same convection-diffusion-reaction operator when a perturbation is carried on the reaction term and no longer on the definition domain.

Commutators of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent

Let $\Omega \subset \mathbb{R}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces $\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ and the vanishing generalized weighted Morrey spaces $V\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\varphi(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ and $V\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$. This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces. Especially, it gives a unified result for generalized Morrey spaces and variable Morrey spaces which currently gained a lot of attentions from researchers in theory of function spaces.

Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 &lt; p &lt; ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities

We deal with Dirichlet systems involving the mean curvature operator in Minkowski spaceM(w)=div(∇w1−|∇w|2) in a ball in RN. Using the fixed point index and the lower and upper solutions method, we first obtain the existence of positive solutions for a class of differential systems with a singular φ-Laplacian, subjected to homogeneous mixed boundary conditions. The main application of the general results concerns Dirichlet systems with Lane–Emden type nonlinearities in a superlinear case, depending on two parameters λ1,λ2. So, we prove that for such a system there exists a continuous curve Γ which separates the first quadrant in two disjoint unbounded open sets O1 and O2 such that the system has zero, at least one or at least two radial positive solutions, according to (λ1,λ2)∈O1, (λ1,λ2)∈Γ or (λ1,λ2)∈O2.

On two topologies that were suggested by Zeeman

The class of Zeeman topologies on spacetimes in the frame of relativity theory is considered to be of powerful intuitive justification, satisfying a sequence of properties with physical meaning, such as the group of homeomorphisms under such a topology is isomorphic to the Lorentz group and dilatations, in Minkowski spacetime, and to the group of homothetic symmetries in any curved spacetime. In this article, we focus on two distinct topologies that were suggested by Zeeman as alternatives to his fine topology, showing their connection with two orders, ie, a timelike and a (noncausal) spacelike one. For the (noncausal) spacelike order, we introduce a partition of the null cone, which gives the desired topology invariantly from the choice of the hyperplane of partition. In particular, we observe that these two orders induce topologies within the class of Zeeman topologies, while the two suggested topologies by Zeeman himself are intersection topologies of these two order topologies (respectively) with the manifold topology. We end up with a list of open questions and a discussion, comparing the topologies with bounded against those with unbounded open sets and their possible physical interpretation.

Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces

This article is divided into two parts. In the first part we show that a set E has locally finite s -perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global s -minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e., local and global), showing that in bounded open sets with Lipschitz boundary they coincide. Conversely, in general this is not true in unbounded open sets, where a global s-minimal set may fail to exist (we provide an example in the case of a cylinder \Omega \times \mathbb R ).

Periodic solutions for indefinite singular perturbations of the relativistic acceleration

Using the Leray–Schauder degree, we study the existence of solutions for the following periodic differential equation with relativistic acceleration and singular nonlinearity:where μ &gt; 1 and the weight h: [0, T] → ℝ is a continuous sign-changing function. There are no a priori estimates on the set of positive solutions (a condition used in general to apply the Leray–Schauder degree), and we prove that no solution of the equation appears on the boundary of an unbounded open set during the deformation to an autonomous problem.

The Perron method for varvec{p}-harmonic functions in unbounded sets in mathbf {R}^n and metric spaces

The Perron method for solving the Dirichlet problem for p-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a p-Poincaré inequality, 1<p<infty . The upper and lower (p-harmonic) Perron solutions are studied for open sets, which are assumed to be p-parabolic if unbounded. It is shown that continuous functions and quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and lower Perron solutions coincide), that the Perron solution agrees with the p-harmonic extension, and that Perron solutions are invariant under perturbation of the function on a set of capacity zero.

Periodic solutions to second-order indefinite singular equations

The efficient conditions guaranteeing the existence of a T-periodic solution to the second order differential equationu″=h(t)g(u) are established in the paper. Here, g is a positive and decreasing function which has a strong singularity at the origin, and the weight h∈L(R/TZ) is a sign-changing function. The obtained results have the form of relation between the multiplicities of the zeroes of the weight function h and the order of the singularity of the nonlinear term. The approach is based on Leray–Schauder degree theory, proving that no T-periodic solution of a certain homotopy appears on the boundary of an unbounded open set during the deformation to an autonomous problem.

Unsplittable coverings in the plane

A system of sets forms an m-fold covering of a set X if every point of X belongs to at least m of its members. A 1-fold covering is called a covering. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for sphere packings as well as by the planar sensor cover problem. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body C, there exists a constant m=m(C) such that every m-fold covering of the plane with translates of C splits into 2 coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every m, an unsplittable m-fold covering of the plane with translates of any open convex body C which has a smooth boundary with everywhere positive curvature. Somewhat surprisingly, unbounded open convex sets C do not misbehave, they satisfy the conjecture: every 3-fold covering of any region of the plane by translates of such a set C splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: shift-chains. We also show that there is a constant c>0 such that, for any positive integer m, every m-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by at mostc2m/2 sets.

Martin boundary of unbounded sets for purely discontinuous Feller processes

Abstract In this paper, we study the Martin kernels of general open sets associated with inaccessible points for a large class of purely discontinuous Feller processes in metric measure spaces. Let D be an unbounded open set. Infinity is accessible from D if the expected exit time from D is infinite, and inaccessible otherwise. We prove that under suitable assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from D. Similar results are also proved for finite boundary points of D.

On Burenkov's extension operator preserving Sobolev–Morrey spaces on Lipschitz domains

We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n‐dimensional Euclidean space.

Well-posedness in weighted Sobolev spaces for elliptic equations of Cordes type

In this paper we prove some weighted \(W^{2,2}\)-a priori bounds for a class of linear, elliptic, second-order, differential operators of Cordes type in certain weighted Sobolev spaces on unbounded open sets \(\varOmega \) of \(\mathbb {R}^{n},\,n\ge 2\). More precisely, we assume that the leading coefficients of our differential operator satisfy the so-called Cordes type condition, which corresponds to uniform ellipticity if \(n=2\) and implies it if \(n\ge 3\), while the lower order terms are in specific Morrey type spaces. Here, our analytic technique mainly makes use of the existence of a topological isomorphism from our weighted Sobolev space, denoted by \(W^{2,2}_s(\varOmega )\) (\(s\in \small \mathbb {R}\)), whose weight is a suitable function of class \(C^2(\bar{\varOmega })\), to the classical Sobolev space \(W^{2,2}(\varOmega )\), which allow us to exploit some well-known unweighted a priori estimates. Using the above mentioned \(W^{2,2}_s\)-a priori bounds, we also deduce some existence and uniqueness results for the related Dirichlet problems in the weighted framework.

Porous medium equation and fast diffusion equation as gradient systems

We show that the Porous Medium Equation and the Fast Diffusion Equation, \(\dot u - \Delta {u^m} = f\), with m ∈ (0, ∞), can be modeled as a gradient system in the Hilbert space H−1(Ω), and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ⊆ ℝn and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.