Variational Relation Problems Research Articles

Another look at Schrödinger equations with prescribed mass

In this paper, we investigate the existence of solutions for the nonlinear Schrödinger equation −Δu−λu=f(u) in RN with an L2-constraint in the Sobolev subcritical case when f possesses several weaker L2-supercritical conditions and in the Sobolev critical case when f(u)=μ|u|q−2u+|u|2⁎−2u with μ>0 and 2<q<2⁎=2NN−2 allowing to be L2-subcritical, L2-critical or L2-supercritical, where N≥3 is the dimension, λ∈R and f∈C(R,R). By establishing several new critical point theorems on a manifold, we introduce a new variational approach which enables us to weaken the previous L2-supercritical conditions in the Sobolev subcritical case and also to present an alternative scheme for all 2<q<2⁎, which is technically simpler compared to the Ghoussoub minimax principle [7] involving topological arguments, to construct bounded (PS) sequences on a manifold when the reaction performs as a mixed dispersion. In particular, the analysis developed in this paper also allows to control the energy level in the Sobolev critical case in a unified way whether N=3 or N≥4. Furthermore, this new approach can provide a new look at what were expected by Soave [13] and by Jeanjean-Le [10] and may it can be applied and modified to more related variational problems.

Uniform exponential contraction for viscous Hamilton–Jacobi equations

The well known phenomenon of exponential contraction for solutions to the viscous Hamilton-Jacobi equation in the space-periodic setting is based on the Markov mechanism. However, the corresponding Lyapunov exponent $\lambda(\nu)$ characterizing the exponential rate of contraction depends on the viscosity $\nu$. The Markov mechanism provides only a lower bound for $\lambda(\nu)$ which vanishes in the limit $\nu \to 0$. At the same time, in the inviscid case $\nu=0$ one also has exponential contraction based on a completely different dynamical mechanism. This mechanism is based on hyperbolicity of action-minimizing orbits for the related Lagrangian variational problem. In this paper we consider the discrete time case (kicked forcing), and establish a uniform lower bound for $\lambda(\nu)$ which is valid for all $\nu\geq 0$. The proof is based on a nontrivial interplay between the dynamical and Markov mechanisms for exponential contraction. We combine PDE methods with the ideas from the Weak KAM theory.

Directional variational principles and applications to the existence study in optimization

The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first provide a directional invariant point theorem. As direct consequences, we obtain several directional variational principles (with quite different formulations). Then, applying these results, we establish sufficient conditions for the existence of solutions for two general models of directional variational relation and inclusion problems. We also include corresponding consequences for particular cases.

Energy bounds of sign-changing solutions to Yamabe equations on manifolds with boundary

We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also established via the method of moving planes. This bound increases the energy range for which Palais–Smale sequences of related variational problem has a non-trivial weak limit.

Variational quantum algorithm with information sharing

We introduce an optimisation method for variational quantum algorithms and experimentally demonstrate a 100-fold improvement in efficiency compared to naive implementations. The effectiveness of our approach is shown by obtaining multi-dimensional energy surfaces for small molecules and a spin model. Our method solves related variational problems in parallel by exploiting the global nature of Bayesian optimisation and sharing information between different optimisers. Parallelisation makes our method ideally suited to the next generation of variational problems with many physical degrees of freedom. This addresses a key challenge in scaling-up quantum algorithms towards demonstrating quantum advantage for problems of real-world interest.

Calmness of the solution mapping of parametric variational relation problems

We consider a class of variational relation problems, depending on two parameters from metric spaces. The issue under investigation is the behaviour of the solution in a neighborhood of a fixed pair of parameters, more precisely, the H?lder calmness of the solution mapping. After establishing some sufficient condition for calmness in a general framework, we particularize the result for a variational inclusion and for an equilibrium problem.

Sets that maximize probability and a related variational problem

Let be a random variable of a Riemannian manifold. We assume that the C2‐probability density function of exists. This research addresses two variational questions. The first concerns sets that maximize their probability among those that have a fixed volume. We prove that such a set must have a probability density function that is constant along its boundary; equivalently, such a set must be a density level set. We also obtain the equations related to the maximization property (the stability of the solutions). The other variational problem is the inverse of the first question, namely which sets minimize their volume among those sets which have a predetermined probability? The solution of this problem will define a notion of a quantile set. We show that the solutions of both variational problems coincide (the critical point equation and the stability condition). As theoretical applications, we consider a decision‐making problem and fuzzy sets. As practical applications, we first explore how to locate a powerplant, and subsequently develop a model for a distribution of cheetahs.

New two types of semi-implicit viscosity iterations for approximating the fixed points of nonexpansive operators associated with contraction operators and applications

Motivated and inspired by the growing contribution with respect to iterative approximations from some researchers in the literature, we design and investigate two types of brand-new semi-implicit viscosity iterative approximation methods for finding the fixed points of nonexpansive operators associated with contraction operators in complete {operatorname{CAT}(0)} spaces and for solving related variational inequality problems. Under some suitable assumptions, strong convergence theorems of the sequences generated by the approximation iterative methods are devised, and a numerical example and some applications to related variational inequality problems are included to verify the effectiveness and practical utility of the convergence theorems. Our main results presented in this paper do not only improve, extend and refine some corresponding consequences in the literature, but also show that the additional variational inequalities, general variational inequality systems and equilibrium problems can be solved via approximation of the iterative sequences. Finally, we provide an open question for future research.

Conformal gradient fields on Finsler manifolds

In this note, some basic structural conditions that are imposed on a Finsler manifold as a consequence of existence of a conformal gradient field are presented. Then, after reiterating the definition of warped product Finsler manifolds and the constituents of its related variational problem, the correlation between geodesics of this warped product structure and geodesics of its constructing Finsler manifolds is studied.

General theorems of the Knaster-Kuratowski- Mazurkiewicz type and applications to the existence study in optimization

ABSTRACT We establish theorems of the Knaster-Kuratowski-Mazurkiewicz (KKM) type on topological spaces with topologically-based generalized convex structures and apply them to the existence study for variational relation problems, variational inequalities, minimax inequalities and saddle points, and to alternative theorems. Our first main KKM-type theorem is a necessary and sufficient condition, not only a sufficient one as usual. The results improve and extend corresponding contributions in the literature in many aspects.

Dynamical behaviors of blowup solutions in trapped quantum gases: Concentration phenomenon

The current paper contemplates the blowup dynamics in trapped dipolar quantum gases. More precisely, employing the profile decomposition of bounded sequences in H˙1∩H˙12, we firstly construct related variational problems and derive two refined Gagliardo–Nirenberg inequalities. Secondly, a compactness lemma is utilized to prove that the blowup solutions with bounded H˙12 norm and bounded L3 norm absolutely concentrate at least a fixed amount.

Causal transport plans and their Monge–Kantorovich problems

ABSTRACTThis paper investigates causal optimal transport problems. Within this framework, primal attainments and dual formulations are obtained under standard hypothesis, for the related variational problems. Causal transport plans are intrinsically related to martingales by a preserving property. Specific concretizations yield primal problems equivalent to several classical problems of stochastic control, and of stochastic calculus; trivial filtrations yield usual problems of optimal transport.

Revisit the regularity theory of Schoen-Uhlenbeck

In this article, we give proofs of main results of the fundamental work of Schoen and Uhlenbeck on the regularity of energy minimizing harmonic maps. The current proofs may be more direct and intuitive, and can be applied for other related geometric variational problems as well.

New existence criteria of the solutions for a variational relation problem

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417–429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.

A non-conventional discontinuous Lagrangian for viscous flow

Drawing an analogy with quantum mechanics, a new Lagrangian is proposed for a variational formulation of the Navier–Stokes equations which to-date has remained elusive. A key feature is that the resulting Lagrangian is discontinuous in nature, posing additional challenges apropos the mathematical treatment of the related variational problem, all of which are resolvable. In addition to extending Lagrange's formalism to problems involving discontinuous behaviour, it is demonstrated that the associated equations of motion can self-consistently be interpreted within the framework of thermodynamics beyond local equilibrium, with the limiting case recovering the classical Navier–Stokes equations. Perspectives for applying the new formalism to discontinuous physical phenomena such as phase and grain boundaries, shock waves and flame fronts are provided.

Applications of the KKM Property to Coincidence Theorems, Equilibrium Problems, Minimax Inequalities and Variational Relation Problems

ABSTRACTIn this paper, we establish coincidence-like results in the case when the values of the correspondences are not convex. To do this, we define new type of correspondences, namely, properly quasi-convex-like correspondences. Further, we apply the obtained theorems to solve equilibrium problems and to establish a minimax inequality. In the last part of the paper, we study the existence of solutions for generalized vector variational relation problems. Our analysis is based on the applications of the KKM principle. We establish existence theorems involving new hypothesis and we improve the results of some recent papers.

Warped product Finsler manifolds from Hamiltonian point of view

In this paper, the Finslerian warped product structures are introduced as Hamiltonian formalism without restricting Finsler functions to be absolutely homogeneous. Afterwards, the constituents of the related variational problem and Finslerian connections of this warped product are obtained according to those of its constructing Finsler manifolds.

A non‐conventional discontinuous Lagrangian for viscous flow

AbstractAlthough many attempts for finding a variational formulation of Navier‐Stokes equations have been made, a Lagrangian for viscous flow has not been established yet. An auspicious suggestion was made by Scholle [1] by extending Seliger and Whitham's Lagrangian [2] with additional terms ending up with some partial success: on the one hand, the phenomenon ‘viscosity’ occurs in a qualitatively correct manner, on the other hand the equations of motion resulting from the variation of Hamilton's principle differ from Navier‐Stokes equations and therefore, their solutions reveal noticeable quantitative differences to those of Navier‐Stokes equations. In this paper the Lagrangian [1] is modified by applying an innovative idea by Anthony [3], motivated by the reformulation of the Lagrangian in terms of complex fields, which can also be understood as the inversion of Madelungs idea [4] of reformulating the complex Schrödinger's equation into a hydrodynamic form. The prize one has to pay is that the resulting Lagrangian is discontinuous and therefore the mathematical treatment of the related variational problem challenging. Furthermore, an additional parameter, ω0, has to be introduced. However, it is demonstrated that Navier‐Stokes equations are recovered by the limit ω0 → ∞, whereas the case of finite ω0 can be interpreted as a generalization towards non‐equilibrium thermodynamics [3]. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Variational relation problems in a general setting

In this paper, we consider variational relation problems involving a binary relation. The framework presented is more general than that in [J. Optim. Theory Appl. 138(2008) , 65–76] and in other recent papers which deal with this subject.

Existence Theorems for a Variational Relation Problem

ABSTRACTThe aim of this article is to establish new existence results for a certain type of variational relation problem studied by Agarwal et al. in a very recent article. This problem is a general model for several quasivariational inclusions and quasiequilibrium problems investigated in many recent articles. The main tools in proving the existence theorems are two well-known fixed point theorems (Eilenberg-Montgomery and Kakutany-Fan Glicksberg, respectively). In the last section, we prove that, applied in special cases, our results are different or stronger than several earlier results.