Semicontinuity Research Articles

Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings: Decoupling approach revisited

We revisit the decoupling approach widely used (often intuitively) in nonlinear analysis and optimization and initially formalized about a quarter of a century ago by Borwein & Zhu, Borwein & Ioffe and Lassonde. It allows one to streamline proofs of necessary optimality conditions and calculus relations, unify and simplify the respective statements, clarify and in many cases weaken the assumptions. In this paper we study weaker concepts of quasiuniform infimum, quasiuniform lower semicontinuity and quasiuniform minimum, putting them into the context of the general theory developed by the aforementioned authors. Along the way, we unify the terminology and notation and fill in some gaps in the general theory. We establish rather general primal and dual necessary conditions characterizing quasiuniform ε-minima of the sum of two functions. The obtained fuzzy multiplier rules are formulated in general Banach spaces in terms of Clarke subdifferentials and in Asplund spaces in terms of Fréchet subdifferentials. The mentioned fuzzy multiplier rules naturally lead to certain fuzzy subdifferential calculus results. An application from sparse optimal control illustrates applicability of the obtained findings.

Syzygies, constant rank, and beyond

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.

Projected solutions for generalized quasivariational inequalities without compact constraint multimap

In this paper we study the existence of projected solutions for a class of generalized quasivariational inequalities in Banach spaces. Generalized quasivariational inequalities are defined by a principal operator and a constrained set determined by a multivalued map. We obtain existence results without assuming that either the constraint multimap or the principal operator are compact. We present two types of existence results: one is obtained considering the principal operator of class S+ and in the other we assume that the constrained multivalued map is condensing. Furthermore, for both these types of results, first the upper semicontinuity associated to pseudomonotonicity on the principal operator are required, then much weaker hypotheses of regularity and monotonicity, such us the upper sign continuity and the properly quasimonotonicity, are taken into account.

A categorical characterization of relative entropy on standard Borel spaces

We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere's category and we show convexity, lower semicontinuity and uniqueness.

A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique

In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.

Regularity of weak solutions for mixed local and nonlocal double phase parabolic equations

We study the mixed local and nonlocal double phase parabolic equation∂tu(x,t)−div(a(x,t)|∇u|q−2∇u)+Lu(x,t)=0 in QT=Ω×(0,T), where L is the nonlocal p-Laplace type operator. The local boundedness of weak solutions is proved by means of the De Giorgi-Nash-Moser iteration with the nonnegative coefficient function a(x,t) being bounded. In addition, when a(x,t) is Hölder continuous, we discuss the pointwise behavior and lower semicontinuity of weak supersolutions based on energy estimates and De Giorgi type lemma. Analogously, the corresponding results are also valid for weak subsolutions.

Multidegrees, prime ideals, and non-standard gradings

We study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of multidegrees in arbitrary positive multigraded settings. In these environments, we extend the notion of Cartwright-Sturmfels ideals by means of a standardization technique. Furthermore, we recover or extend important results in the literature, for instance: we provide a multidegree version of Hartshorne's result stating the upper semicontinuity of arithmetic degree under flat degenerations, and we give an alternative proof of Brion's result regarding multiplicity-free varieties.

Lagrangian methods for optimal control problems governed by multivalued quasi-hemivariational inequalities

The primary objective of this paper is to develop a general augmented Lagrangian approach for optimal control problems governed by multivalued quasi-hemivariational inequalities. By imposing general coercivity and monotonicity-type conditions, we establish multiple existence results for the optimal control problems. Furthermore, we establish a comprehensive characterization of the zero duality gap property for the primal-dual problems, utilizing the concept of lower semicontinuity of new perturbation functions. Our results represent a significant advancement over recent literature on the subject.

Multi-valued perturbations on stochastic evolution equations driven by fractional Brownian motions

We consider a stochastic evolution inclusion having deterministic multi-valued nonlinearity and fractional Brownian motion with nonlinear diffusion. We establish the nonemptiness and compactness of its solution set. After that, the upper semicontinuity with respect to random parameters and initial values of the corresponding solution map is proved. In particular, the results on nonemptiness and upper semicontinuity imply that the inclusion under consideration defines a multi-valued random dynamical system. Moreover, under an extra smooth assumption on the diffusion, it is demonstrated that the solution set has the topological structure of R δ -type.

Approximating properties of metric and generalized metric projections in uniformly convex and uniformly smooth Banach spaces

This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed and convex cones. Furthermore, the inverse images of the generalized metric projection are neither a convex set nor a cone. We also prove that the distance from a point to its projection on a convex set is a weakly lower semicontinuous function for all three notions of projections. We provide illustrating examples to highlight the different behavior of the three projections in Banach spaces.

Split Lipschitz-type functions

In recent years, Lipschitz-type functions have been studied in an extensive manner. Particularly, we have locally Lipschitz functions, uniformly locally Lipschitz functions and Lipschitz in the small functions, etc. To answer some questions related to upper semicontinuity, recently in 2019, a new class of functions, called split continuous, is defined which is weaker than that of continuous functions. In this article, we define ‘split’ versions of Lipschitz-type functions. Various interesting properties of these functions are also investigated.

Study of Non-Linear Impulsive Neutral Fuzzy Delay Differential Equations with Non-Local Conditions

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept.

A variational theory for integral functionals involving finite-horizon fractional gradients

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general varGamma -convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.

Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain

Abstract The model introduced in [45] in the framework of the theory on stress-driven rearrangement instabilities (SDRI) [3, 43] for the morphology of crystalline materials under stress is considered. As in [45] and in agreement with the models in [50, 55], a mismatch strain, rather than a Dirichlet condition as in [19], is included into the analysis to represent the lattice mismatch between the crystal and possible adjacent (supporting) materials. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of the restriction to a finite number m of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous results for epitaxially strained thin films and material cavities [6, 35, 34, 45]. Due to the lack of compactness and lower semicontinuity for the sequences of m-minimizers, i.e., minimizers among configurations with at most m connected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence of m-minimizers’ energies to the energy infimum as m → ∞ {m\to\infty} . Finally, regularity properties for the morphology satisfied by every minimizer are established.

Semiflows: Prolongation sets, point-transitive, topologically transitive

Let φ:T×X→X, or simply (T,X), be any topological semiflow on a space X with a topological semigroup T. We give some new results about prolongation sets for the semiflow (T,X). Also, we recall notions of point-transitivity and topological transitivity for (T,X) and give some examples to study the relations between them. Let X be a locally compact Hausdorff space and let (T,X) be point-transitive. Then Mγ={x∈X:γ(x)=min⁡(γ)} is dense in X for every T-invariant upper semicontinuous function γ:X→[0,∞). The converse does hold if (T,X) is a flow on the compact metric space (X,d). We show that the semiflow (T,X) on a Hausdorff space X with TX‾=X is topologically transitive if and only if every T-invariant function γ:X→[0,∞), that is continuous on a comeagre set, is constant on a comeagre set. We introduce a TP-point in a topological space X, which is weaker than an isolated point, and show that if X is a regular topological space with a TP-point p∈X and (T,X) is syndetically transitive, then p is a periodic point and X=Kp for some compact set K⊆T. Finally, we show that every expansive semiflow (T,[0,1]) is point-transitive if T is a right C-semigroup.

Risk concentration and the mean‐expected shortfall criterion

AbstractExpected shortfall (ES, also known as CVaR) is the most important coherent risk measure in finance, insurance, risk management, and engineering. Recently, Wang and Zitikis (2021) put forward four economic axioms for portfolio risk assessment and provide the first economic axiomatic foundation for the family of . In particular, the axiom of no reward for concentration (NRC) is arguably quite strong, which imposes an additive form of the risk measure on portfolios with a certain dependence structure. We move away from the axiom of NRC by introducing the notion of concentration aversion, which does not impose any specific form of the risk measure. It turns out that risk measures with concentration aversion are functions of ES and the expectation. Together with the other three standard axioms of monotonicity, translation invariance and lower semicontinuity, concentration aversion uniquely characterizes the family of ES. In addition, we establish an axiomatic foundation for the problem of mean‐ES portfolio selection and new explicit formulas for convex and consistent risk measures. Finally, we provide an economic justification for concentration aversion via a few axioms on the attitude of a regulator towards dependence structures.

A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

We present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type where Omega is a bounded open subset of {mathbb {R}}^N and W:Omega ,{times }, Omega ,{times }, {mathbb {R}}^{d times N}{times }, {mathbb {R}}^{d times N} rightarrow {mathbb {R}}.

Limiting Behavior of Random Attractors of Stochastic Supercritical Wave Equations Driven by Multiplicative Noise

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero.

Dynamics of the non-autonomous stochastic p-Laplacian parabolic problems on unbounded thin domains

This paper focuses on the dynamics of the non-autonomous stochastic p-Laplacian parabolic problems defined on unbounded thin domains. We first show that the tails of solutions of the equation are uniformly small outside a bounded domain, which is utilized to overcome the non-compactness of Sobolev embeddings on unbounded domains. We then prove the existence and uniqueness of random attractors for the equations defined on (n + 1)-dimensional unbounded thin domains and further establish the upper semi-continuity of attractors as the thin domains collapse onto the space Rn.

Stability of stochastic reaction-diffusion equation under random influences in high regular spaces

In this paper, we systematically study the high-order stability of the stochastic reaction-diffusion equation driven by additive noise as the noise intensity vanishes. First, with a general assumption on the nonlinear term, we obtain the convergence of solutions and upper semi-continuity of random attractors in L2(RN). Second, by using the nonlinear decomposition method, we technically establish the convergence of solutions in Lp(RN)∩H1(RN)(p>2), and therefore, the upper semi-continuity of random attractors is proved, where p is the growth exponent of the nonlinearity. Finally, by induction argument, we prove that the solution is uniformly bounded near the initial time in Lδ(RN) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.