General Open Sets Research Articles

The Role of Topology and Capacity in Some Bounds for Principal Frequencies

We prove a lower bound on the sharp Poincaré–Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet–Laplacian. We also consider some limit situations, like the sharp Moser–Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

Unification of Generalized Open Sets with Respect to an Ideal

Unification of Generalized Open Sets with Respect to an Ideal

On fractional Hardy-type inequalities in general open sets

We show that, when sp > N, the sharp Hardy constant hs,p of the punctured space ℝN \ {0} in the Sobolev–Slobodeckiĭ space provides an optimal lower bound for the Hardy constant hs,p(Ω) of an open set Ω ⊂ ℝN. The proof exploits the characterization of Hardy’s inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler–Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of Ω. Moreover, we compute the limit of hs,p as s ↗ 1, as well as the limit when p ↗ ∞. Finally, we apply our results to establish a lower bound for the non-local eigenvalue λs,p(Ω) in terms of hs,p when sp > N, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p ↗ ∞.

Sobolev embeddings and distance functions

Abstract On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p \mathcal{D}^{{1,p}}_{0} into L q L^{q} and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 𝑝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 𝑝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 𝑝-Laplacian with sub-homogeneous right-hand side, as the exponent 𝑝 diverges to ∞. The case of first eigenfunctions of the 𝑝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.

Generalized Open Sets in Neutrosophic Soft Bitopological Spaces

Generalized Open Sets in Neutrosophic Soft Bitopological Spaces

Dynamics of the Taylor shift on Bergman spaces

The Taylor (backward) shift on Bergman spaces Ap(Ω) for general open sets Ω in the extended complex plane shows rich variety concerning its dynamical behaviour. Different aspects are worked out, where in the case p<2 a recent result of Bayart and Matheron plays a central role.

Representation of harmonic functions with respect to subordinate Brownian motion

In this article we prove a representation formula for non-negative generalized harmonic functions with respect to a subordinate Brownian motion in a general open set D⊂Rd. We also study oscillation properties of quotients of Poisson integrals and prove that oscillation can be uniformly tamed.

Some Studies on Fuzzy Generalized Open Sets

Inw thisw paper,w wew introduce importantw fuzzyw generalizedw openw setsw namelyw fuzzyw Generalizedw Regularw openw (briefly,w fuzzyw GR-open)w sets.w Thisw neww classw showsw strongerw propertiesw inw fuzzyw topologicalw spacesw andw also,w wew investigatew fuzzyw GR-neighbourhoodsw andw fuzzyw GR-continuityw properties.

On Regular Feebly Generalized-Open Sets and its Properties

On Regular Feebly Generalized-Open Sets and its Properties

Semi Generalized Open Sets and Generalized Semi Closed Sets in Topological Spaces

In this paper we introduce a new class of semi generalized open sets, generalized semi closed sets in topological spaces, and studied some of its basic properties. Moreover we define approximately semi generalized open sets and approximately generalized semi closed sets in topological spaces. Further we obtained some properties of closure, semi generalized open sets and generalized semi closed sets in topological spaces.

Uniqueness of the approximative trace

We study the approximative trace for individual elements in the Sobolev space $W^{1,p}(\Omega)$ for $1\le p\le\infty$. This notion of a trace was introduced for $p=2$ in [AtE11] in the setting of general open sets $\Omega\subset\mathbb{R}^d$. The approximative trace exhibits a curious nonuniqueness phenomenon. We provide a detailed analysis of this phenomenon based on methods of geometric measure theory and are able to give very weak geometric conditions that are sufficient for the uniqueness of the approximative trace. In particular, we prove that the approximative trace is unique on open sets with continuous boundary and on arbitrary connected domains in $\mathbb{R}^2$. Furthermore, we provide an example which shows that the uniqueness of the approximative trace depends on $p$. These results answer several open questions.

Applications of Pre*Generalized Open Sets

Applications of Pre*Generalized Open Sets

Semilinear Dirichlet problem for the fractional Laplacian

We give a general framework and results on existence, representation, and uniqueness of solutions to the semilinear problem for the fractional Laplacian with Dirichlet conditions on the complement and at the boundary of general open sets.

Cauchy Fluxes and Gauss–Green Formulas for Divergence-Measure Fields Over General Open Sets

We establish the interior and exterior Gauss-Green formulas for divergence-measure fields in $L^p$ over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing discontinuities and singularities. The method, based on a distance function, allows to give a representation of the interior (resp. exterior) normal trace of the field on the boundary of any given open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth approximations of the open set. In the particular case of open sets with continuous boundary, the approximating smooth sets can explicitly be characterized by using a regularized distance. We also show that any open set with Lipschitz boundary has a regular Lipschitz deformable boundary from the interior. In addition, some new product rules for divergence-measure fields and suitable scalar functions are presented, and the connection between these product rules and the representation of the normal trace of the field as a Radon measure is explored. With these formulas at hand, we introduce the notion of Cauchy fluxes as functionals defined on the boundaries of general bounded open sets for the rigorous mathematical formulation of the physical principle of balance law, and show that the Cauchy fluxes can be represented by corresponding divergence-measure fields.

On Peng's type maximum principle for optimal control of mean-field stochastic differential equations with jump processes

In this paper, we investigate the Peng's type optimal control problems for stochastic differential equations of mean-field type with jump processes. The coefficients of the system contain not only the state process but also its marginal distribution through their expected values. We assume that the control set is a general open set that is not necessary convex. The control variable is allowed to enter into both diffusion and jump terms. We extend the maximum principle of Buckdahn et al. (2011) to jump case.

Viability of an open set for stochastic control systems

The problem of compatibility of a stochastic control system and a set of constraints – the so called viability property – has been widely investigated during the last three decades. Given a stochastic control system, the question is to characterize sets A such that for any initial condition in A there exists a control ensuring that the associated stochastic process remains forever almost surely in A (this is called the viability property of A). When A is closed and the dynamics is continuous, the viability property has been characterized in the literature through several equivalent geometric conditions involving A, the drift and the diffusion of the control system. In this article we give a necessary and sufficient condition involving the boundary of an open set A ensuring the viability property of A, whenever A has a C2,1 boundary and the dynamics are Lipschitz. If moreover a classical convexity condition on the control dynamics holds true, we show that the viability of an open set A is equivalent to the viability of its closure. This last result is rather surprising, because several very elementary examples in the deterministic framework show that, in general, there is no such equivalence for a general open set A. We will also discuss examples illustrating that the above equivalence is wrong when either the boundary of A does not have enough regularity, or the dynamics are not Lipschitz continuous.

On Peng's type maximum principle for optimal control of mean-field stochastic differential equations with jump processes

In this paper, we investigate the Peng's type optimal control problems for stochastic differential equations of mean-field type with jump processes. The coefficients of the system contain not only the state process but also its marginal distribution through their expected values. We assume that the control set is a general open set that is not necessary convex. The control variable is allowed to enter into both diffusion and jump terms. We extend the maximum principle of Buckdahn et al. (2011) to jump case.

First exit from an open set for a matrix-exponential Lévy process

We study the first exit from a general open set for a one-dimensional Lévy process, where the Lévy measure is proportional to a two-sided matrix-exponential distribution. Under appropriate conditions on the Lévy measure, we obtain an explicit solution for the joint distribution of the first-exit time and the position of the Lévy process upon first exit, in terms of the zeros and poles of the corresponding Laplace exponent. The present result complements several earlier works on the use of exit sets for Lévy processes with algebraically similar Laplace exponents, where exits from open intervals are the main focus.

A comparison principle for the porous medium equation and its consequences

We prove a comparison principle for the porous medium equation in more general open sets in \mathbb R^{n+1} than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

A Stochastic Maximum Principle for General Mean-Field Systems

In this paper we study the optimal control problem for a class of general mean-field stochastic differential equations, in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not necessary convex, and the coefficients are only continuous on the control variable without any further regularity or convexity. We validate the approach of Peng (SIAM J Control Optim 2(4):966–979, 1990) by considering the second order variational equations and the corresponding second order adjoint process in this setting, and we extend the Stochastic Maximum Principle of Buckdahn et al. (Appl Math Optim 64(2):197–216, 2011) to this general case.