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In this work, a two-dimensional (2D) and three-dimensional (3D) integrated topology optimization (TO) parallel-computing framework, named TopADD (TOPology optimization for Arbitrary Design Domains), is developed to deal with topology optimization problems with arbitrary design domains. The parallel-computing framework is an extended work of the initial parallel-computing framework developed by Aage et al. (Struct Multidiscip Optim 51(3): 565–572, 2015). The extension is threefold: (a) a 2D implementation has been incorporated into the framework to achieve seamless switching between 2D and 3D dimensions; (b) an efficient voxelizer that can initialize complex geometries into the design domains for topology optimization is developed; and (c) besides the compliance minimization problem, two other physics have been considered: the compliant mechanism and the heat conduction problems. Additionally, the computational efficiency of the proposed framework has been examined. Compared to the other frameworks in the literature, the proposed work has superior efficiency in both computational time and memory usage. Lastly, the proposed topology optimization framework’s compatibility with additive manufacturing (AM) has been demonstrated by exporting and printing the final optimized parts without postprocessing.

Color image restoration is an ill-posed problem, and regularization is necessary. In this paper, we first formulate the color image restoration problem into a constrained minimization problem that minimizes a quadratic functional and subject to a constraint that the total variation of the color image is less than a given parameter δ. The advantages of the constrained minimization problem over the traditional unconstrained one is that the parameter δ has an obvious physical meaning and is easy to select. However solving the constrained minimization problem is generally more difficult than solving the corresponding unconstrained form. We propose an effective alternating direction method of multipliers for color image restoration by using the structure of the problem. We prove the convergence of the method in detail. Experimental results demonstrate that the proposed method is feasible and much more effective for color image restoration.

Solving large-scale l1 minimization problem is very important and has attracted much attentions in recent years. In this paper, we present an efficient method for solving this problem. To this end, we first reformulate this problem as a nonnegative constrained minimization problem. Then we propose a modulus-based iterative method to solve the nonnegative constrained minimization problem. Convergence analysis and the choice of near optimal parameter are presented. Experimental results are given to illustrate feasibility and effectiveness of our proposed method.

In this paper, we consider the semidifferentiable case of an interval-valued minimization problem and establish sufficient optimality conditions and Wolfe type as well as Mond–Weir type duality theorems under semilocalE-preinvex functions. Furthermore, we present saddle-point optimality criteria to relate an optimal solution of the semidifferentiable interval-valued programming problem and a saddle point of the Lagrangian function.

In this paper, we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics. More precisely, for any given nonnegative integer k, by using a minimization argument, we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem, and show, by a deformation lemma and Miranda’s theorem, that the minimizer is the desired solution.

Given a nonconvex minimization problem where the objective function is nonlinear and twice differentiable. To gain more information about the objective function, it is essential to obtain all its s...

International audience; In this paper, we analyze the relevance of the use of the shallow water model and the Boussinesq model to simulate tsunamis generated by a landslide. In a first part, we determine if the two models are able to reproduce waves generated by a landslide. Each model has drawbacks but it seems that it is possible to use them together to improve the simulations. In a second part we try to recover the landslide displacement from the generated wave. This problem is formulated as a minimization problem and we limit the number of parameters to determine assuming that the bottom can be well described by an empirical law.; Dans cet article, nous analysons la pertinence de l'utilisation de modèles intégrés suivant la verticale tels que le modèle de Saint-Venant et le modèle de Boussinesq pour simuler les tsunamis générés par un glissement de terrain. Dans une première partie, nous déterminons si les deux modèles sont capables de reproduire les vagues générées par un glissement de terrain. Chaque modèle présente des inconvénients mais il semble qu'il soit possible de les utiliser ensemble pour améliorer les simulations. Dans une deuxième partie, nous essayons de retrouver le déplacement du glissement de terrain à partir de la vague générée. Ce problème a été formulé comme un problème de minimisation et nous limitons le nombre de paramètres à déterminer en supposant que le fond peut être bien décrit par une loi empirique.

Traditional diffraction images without a specific migration kernel for promoting focusing abilities may cause confusion to seismic interpretation because diffraction images may show a finite-array response of diffracted/scattered waves. Because diffractors are discontinuous and sparsely distributed, a least-squares diffraction-imaging method is formulated by solving a hybrid L1-L2 norm minimization problem that imposes a sparsity constraint on diffraction images. It uses two different forward modeling operators for reflections and diffractions and L2 and L1 regularizations for penalizing the amplitudes of the reflection and diffraction images, respectively. A classic Kirchhoff diffraction demigration operator is implemented on an initial diffraction image model to synthesize diffracted/scattered waves. A Kirchhoff reflection demigration operator, formulated by considering the local reflection slopes and a cosine attenuation weighting function, is implemented on an initial reflection image to synthesize the reflected waves. A modified alternating direction approach of multipliers is developed for iteratively solving this minimization problem to create diffraction images and their separated diffractions. The depths and local reflection slopes of the reflection images are fixed during this iteration. To alleviate the energy leakage between diffractions and reflections, after performing the plane-wave destruction method on the conventional migration data, its estimated reflection image and residual image are provided as the initial reflection and diffraction images, respectively. Our method can remove steep-slope reflections, increase the focusing power of the diffractions, and eliminate noise. Two numerical experiments demonstrate its capability of separating and imaging small-scale discontinuities and inhomogeneities. The exposed geologic structures in the tunnel of field coal mining further illustrate this method’s potential in ascertaining hidden faults, edges, and collapsed columns. A safety warning should be definitely required if a mining working surface is advancing these hidden geologic disasters because an emergency of water bursting or gas leakage may happen.

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.

We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

In this paper a new nonlinear two-dimensional shell model of Naghdi’s type is formulated for shells which middle surface is parameterized by a $W^{1, \infty}$ function. Therefore the model inherently contains undeformed geometries with corners, so the model also includes models of junctions of nonlinear shells. Deformation of the shell is described by a pair $(\boldsymbol{\psi}, {\mathbf{R}})$ of independent unknowns, where $\boldsymbol{\psi}$ is the deformation of the middle surface and ${\mathbf{R}}$ is a function with value in rotations that describes rotation of the shell cross-section. The model is formulated as the minimization problem for the total energy functional that includes flexural, membrane, shear and drill energies differently scaled with respect to the thickness of the shell. We relate the new model for smooth enough undeformed geometry to the known shell models in two ways. First we restrict the proposed model on two particular subsets of admissible functions and obtain exactly the flexural shell model and a perturbation of the Koiter shell model. More important, we consider asymptotics, using $\Gamma$ –convergence, of the proposed model with respect to the thickness as a small parameter in the membrane and flexural regime and obtain exactly the nonlinear membrane and flexural shell model obtained as $\Gamma$ –limits starting from nonlinear three–dimensional elasticity. In that way we link rigorously the proposed model with the nonlinear three–dimensional elasticity.

In this work, we introduce a useful and efficient calculation method for solving linear fractional pantograph delay equations (FPDEs). The proposed method is primarily dependent on the least-squares approximation technique. After embedding the problem into a minimization problem, it solves the Lagrange multiplier method. The convergence analysis is theoretically proved. Finally, some numerical examples singled out to show the usefulness and capability of the suggested approach.

In this paper, we consider eigenvalue problems of a second-order measure differential equation. We first study some general theories regarding the completely continuity of eigenvalues, the variational characterizations of eigenvalues, and the oscillating properties of eigenfunctions. Then, we solve the following minimization problem: when the m-th Neumann eigenvalue is given, to find explicitly what measures will have the minimal total variation. As applications of this minimization problem, we will solve some extremal problems of eigenvalues and construct some optimal classes of non-degenerate measures for the second-order measure differential equations.

This paper studies the online machine minimization problem, where the jobs have real release times, uniform processing times and a common deadline. We investigate how the lookahead ability improves the performance of online algorithms. Two lookahead models are studied, that is, the additive lookahead and the multiplicative lookahead. At any time t, the online algorithm knows all the jobs to be released before time $$t+L$$ (or $$\beta \cdot t$$ ) in the additive (or multiplicative) lookahead model. We propose a $$\frac{e}{\alpha (e-1)+1}$$ -competitive online algorithm with the additive lookahead, where $$\alpha = \frac{L}{T} \le 1$$ and T is the common deadline of the jobs. For the multiplicative lookahead, we provide an online algorithm with a competitive ratio of $$\frac{\beta e}{(\beta -1) e +1}$$ , where $$\beta \ge 1$$ . Lower bounds are also provided for both of the two models, which show that our algorithms are optimal for two extreme cases, that is, $$\alpha = 0$$ (or $$\beta = 1$$ ) and $$\alpha = 1$$ (or $$\beta \rightarrow \infty $$ ), and remain a small gap for the cases in between. Particularly, for $$\alpha = 0$$ (or $$\beta = 1$$ ), the competitive ratio is e, which corresponds to the problem without lookahead. For $$\alpha = 1$$ (or $$\beta \rightarrow \infty $$ ), the competitive ratio is 1, which corresponds to the offline version (with full information).

This paper is aimed at the derivation of a discrete data smoothing function for the discrete Dirichlet condition in a regular grid on the surface of a spheroid. The method employed here is the local L2 seminorm minimization, through Euler Lagrange method, for the Beltrami operator. The method results in a weighted average of the surrounding points in a template based on the first order Taylor expansion of the unknown function under consideration. The coefficients of the weighted average are calculated and used to smooth the Geoid height data in Iran, derived from the EGM2008 geopotential model, Geodesy and Geodynamics, 11 (4)

This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain. The existence of the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a constructed auxiliary variational problem. And the H2 estimate of the type II divergence-free part of the minimizers is established by div-curl-gradient type estimates of vector fields.

Consider a regenerative storage process with a nondecreasing Lévy input (subordinator) such that every cycle may be split into two periods. In the first (off), the output is shut off and the workload accumulates. This continues until some stopping time. In the second (on), the process evolves like a subordinator minus a positive drift (output rate) until it hits the origin. In addition, we assume that the output rate of every on period is a random variable, which is determined at the beginning of this period. For example, at each period, the output rate may depend on the workload level at the beginning of the corresponding busy period. We derive the Laplace–Stieltjes transform of the steady-state distribution of the workload process and then apply this result to solve a steady-state cost minimization problem with holding, setup and output capacity costs. It is shown that the optimal output rate is a nondecreasing deterministic function of the workload level at the beginning of the corresponding on period.

In this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of ( f , g ) {(f,g)} -generalized k-strictly pseudononspreading mappings in a CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} space. We obtain a strong convergence of the proposed algorithm to the aforementioned problems in a complete CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} space. Furthermore, we give an application of our result to a nonlinear Volterra integral equation and a numerical example to support our main result. Our results complement and extend many recent results in literature.

We consider the functional $\int_\Omega g(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(\Omega)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous., Comment: 35 pages. arXiv admin note: text overlap with arXiv:1307.4442

In this paper, we deal with the optimization method of ridge parameters in a generalized ridge regression by minimizing a model selection criterion (MSC). The optimization methods based on minimizations of generalized C p criterion and GCV criterion are fast because the minimizers of the two criteria can be derived as closed forms. Although the methods are fast, they do not work well when the number of explanatory variables is larger than the sample size. Methods based on minimizations of other MSCs may work well even if the number of explanatory variables is larger than the sample size and are not fast because the minimizers of the MSCs have not been derived as closed forms. Even though a minimization problem of MSC cannot be solved analytically, we propose a fast optimization algorithm to minimize MSC by specifying the small number of minimizer candidates. By conducting numerical examinations, we compare performances of the optimization methods based on the minimizations of MSCs.

In this paper we consider the unconstrained minimization problem of a smooth function in ${\mathbb{R}}^n$ in a setting where only function evaluations are possible. We design a novel randomized derivative-free algorithm --- the stochastic three points (STP) method --- and analyze its iteration complexity. At each iteration, STP generates a random search direction according to a certain fixed probability law. Our assumptions on this law are very mild: roughly speaking, all laws which do not concentrate all measure on any halfspace passing through the origin will work. For instance, we allow for the uniform distribution on the sphere and also distributions that concentrate all measure on a positive spanning set. Given a current iterate $x$, STP compares the objective function at three points: $x$, $x+\alpha s$ and $x-\alpha s$, where $\alpha>0$ is a stepsize parameter and $s$ is the random search direction. The best of these three points is the next iterate. We analyze the method STP under several stepsize selection schemes (fixed, decreasing, estimated through finite differences, etc). We study non-convex, convex and strongly convex cases.

We consider the class of completely monotone probability mass functions (pmf) from a statistical perspective. An element in this class is known to be a mixture of geometric pmfs, a consequence of the celebrated Hausdorff Theorem. We show that the complete monotone least squares estimator exists, is strongly consistent and converges weakly to the truth at the n -rate. Furthermore, we fully describe its limit distribution as the unique solution of a well-posed minimization problem. Through a simulation study we assess the performance of the method under different scenarios.

For the rank regularized minimization problem, we introduce several classes of stationary points by the problem itself and its equivalent reformulations including the mathematical program with an e...

In this article, some new vectorial versions of Takahashi’s nonconvex minimization theorem, which involve algebraic notions instead of topological notions, are established. A nonlinear separation theorem, which extends the result derived by Gerth and Weidner (JAMA 67:297–320, 1990) to general linear spaces (not necessarily endowed with a topology), is proved. Some examples, in order to illustrate and compare the results of this article with the corresponding known results from the literature, are provided.

The notions of upper and lower global directional derivatives are introduced for dealing with nonconvex and nonsmooth optimization problems. We provide calculus rules and monotonicity properties for these notions. As a consequence, new formulas for the Dini directional derivatives, radial epiderivatives and generalized asymptotic functions are given in terms of the upper and lower global directional derivatives. Furthermore, a mean value theorem, which extend the well-known Diewert’s mean value theorem for radially upper and lower semicontinuous functions, is established. We also provide necessary and sufficient optimality conditions for a point to be a local and/or global solution for the nonconvex minimization problem. Finally, applications for nonconvex and nonsmooth mathematical programming problems are also presented.

In this paper, we are interested in standing waves of the nonlinear Schrodinger equation coupled with the Maxwell equation. Our aim is to formulate the orbital stability of standing waves in the full gauge invariant form. For this purpose, we study a new constraint minimization problem.

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding comm...

The hydra number is a new graph parameter related to some minimization problem for Horn formulas in propositional logic. The hydra number of a graph G = ( V , E ) is the smallest number of hyperarcs of the form u v → w in a directed hypergraph H = ( V , F ) such that if u v ∈ E , then all vertices in V are reachable from { u , v } in H and if u v ⁄ ∈ E , then no other vertex apart from u and v is reachable from { u , v } . Reachability is defined by forward chaining, a standard marking procedure. In this paper we answer negatively a question posed in Sloan et al. (2017) concerning an anticipated formula for the hydra number of disconnected graphs. On the positive side, we show that the expression which appears in this formula is an upper bound for the hydra number of these graphs.

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme involves the knowledge of the shape gradient which is established through the minimax formulation. We illustrate the feasibility of the proposed method and highlight its advantage over the classical setting of tracking the Neumann data through several numerical examples.

In this paper, we use the gate condition on two multivalued k-demicontractive mappings to approximate a common solution of a finite family of monotone inclusion problem and fixed point problem in CAT(0) space. Furthermore, we propose a Halpern-type proximal point algorithm and prove its strong convergence to a common solution of a finite family of monotone inclusion problems and fixed point problem for two multivalued k-demicontractive mappings in a complete CAT(0) space. We also applied our result to the problem of finding a common solution of a finite family of minimization problem and fixed point problem in CAT(0) space. Finally, numerical experiments of our result are presented to further show its applicability.

In this paper, we construct a novel algorithm for the split common fixed point problem for two demicontractive operators in Hilbert spaces. By using inertial self-adaptive algorithms, we obtain strong convergence results for finding a solution of the split common fixed point problems. Applications to solving the split minimization problem and the split feasibility problem are included. Our results extend and generalize many previously known results in this research area. Moreover, numerical experiments are supplied to demonstrate the convergence behavior and efficiency of the proposed algorithm.

The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.

Modeling and analyzing high-dimensional data has become a common task in various fields and applications. Often, it is of interest to learn a function that is defined on the data and then to extend its values to newly arrived data points. The Laplacian pyramids approach invokes kernels of decreasing widths to learns a given dataset and a function defined over it in a multi-scale manner. Extension of the function to new values may then be easily performed. In this work, we extend the Laplacian pyramids technique to model the data by considering two-directional connections. In practice, kernels of decreasing widths are constructed on the row-space and on the column space of the given dataset and in each step of the algorithm the data is approximated by considering the connections in both directions. Moreover, the method does not require solving a minimization problem as other common imputation techniques do, thus avoids the risk of a non-converging process. The method presented in this paper is general and may be adapted to imputation tasks. The numerical results demonstrate the ability of the algorithm to deal with a large number of missing data values. In addition, in most cases, the proposed method generates lower errors compared to existing imputation methods applied to benchmark dataset.

In this article, we study mixed equilibrium problems, and present algorithms and convergence theorems for the proposed algorithms, like proximal gradient method, Tikhonov regularization method, Man...

We introduce concepts of interval-valued univex mappings, consider optimization conditions for interval-valued univex functions for the constrained interval-valued minimization problem, and show examples for the illustration purposes.

Recently, the l p -norm regularization minimization problem ( P p λ ) has attracted great attention in compressed sensing. However, the l p -norm ‖x‖ p p in problem ( P p λ ) is nonconvex and non-Lipschitz for all p ∈ ( 0 , 1 ) , and there are not many optimization theories and methods proposed to solve this problem. In fact, it is NP-hard for all p ∈ ( 0 , 1 ) and λ > 0 . In this paper, we study one modified l p -norm regularization minimization problem to approximate the NP-hard problem ( P p λ ) . Inspired by the good performance of Half algorithm in some sparse signal recovery problems, an iterative thresholding algorithm is proposed to solve our modified l p -norm regularization minimization problem ( P p , 1 ∕ 2 , ϵ λ ) . Numerical results on some sparse signal recovery problems show that our algorithm performs effectively in finding the sparse signals compared with some state-of-art methods.

The greater the number of devices on a network, the higher load in the network, the more chance of a collision occurring, and the longer it takes to transmit a message. The size of load can be identified by measuring the network occupancy, hence it is desirable to minimize the latter. In this paper, we present an approach for network occupancy minimization by optimizing the packing process while satisfying multiple constraints. We formulate the minimization problem as a bin packing problem and we implement a modification of the Best-Fit Decreasing algorithm to find the optimal solution. The approach considers grouping signals that are sent to different destinations in the same package. The analysis is done on a medium-sized plant model, and different topologies are tested. The results show that the proposed solution lowers the network occupancy compared to a reference case.

The forward–backward algorithm is a splitting method for solving convex minimization problems of the sum of two objective functions. It has a great attention in optimization due to its broad application to many disciplines, such as image and signal processing, optimal control, regression, and classification problems. In this work, we aim to introduce new forward–backward algorithms for solving both unconstrained and constrained convex minimization problems by using linesearch technique. We discuss the convergence under mild conditions that do not depend on the Lipschitz continuity assumption of the gradient. Finally, we provide some applications to solving compressive sensing and image inpainting problems. Numerical results show that the proposed algorithm is more efficient than some algorithms in the literature. We also discuss the optimal choice of parameters in algorithms via numerical experiments.

In this paper, we propose a multiphase image segmentation method via solving the min-cut minimization problem under the multigrid method framework. At each level of the multigrid method for the min-cut problem, we first transfer it to the equivalent form, e.g., max-flow problem, then actually solve the dual of the max-flow problem. Particularly, a classical multigrid method is used to solve the sub-minimization problems. Several outer iterations are used for the multigrid method. The proposed idea can be used for general min-cut/max-flow minimization problems. We use multiphase image segmentation as an example in this work. Extensive experiments on simulated and real images demonstrate the efficiency and effectiveness of the proposed method.

In this paper, a new self-adaptive step size algorithm to approximate the solution of the split minimization problem and the fixed point problem of nonexpansive mappings was constructed, which combined the proximal algorithm and a modified Mann’s iterative method with the inertial extrapolation. The strong convergence theorem was provided in the framework of Hilbert spaces and then proven under some suitable conditions. Our result improved related results in the literature. Moreover, some numerical experiments were also provided to show our algorithm’s consistency, accuracy, and performance compared to the existing algorithms in the literature.

We study the gauge cooling technique for the complex Langevin method applied to the computation in lattice quantum chromodynamics. We propose a new solver of the minimization problem that optimizes the gauge, which does not include any parameter in each iteration, and shows better performance than the classical gradient descent method especially when the lattice size is large. Two numerical tests are carried out to show the effectiveness of the new algorithm.

The quasi-Newton equation is the very foundation of an assortment of the quasi-Newton methods for optimization minimization problem. In this paper, we deriving a new quasi-Newton equation based on the second-order Taylor’s series expansion. The global convergence is established underneath suitable conditions and numerical results are reported to show that the given algorithm is more effective than those of the normal BFGS method.

In this paper, we propose the split of modified variational inequality problems (SMVIP), by using the concept of the modified generalized system of variational inequalities (MGSV). Then, we prove the strong convergence theorem for solving fixed point problems of nonlinear mappings and two variational inequality problems and solving the SMVIP. Applying our main result, we prove strong convergence theorems of the split minimization problem and the split variational inequality problem. In support of our main result, a numerical example is also presented.

In this work, we reformulate the method presented in App. Opt. 53:2297 (2014) as a constrained minimization problem using the augmented Lagrangian method. First we introduce the new method and then describe the numerical solution, which results in a simple algorithm. Numerical experiments with both synthetic and real fringe patterns show the accuracy and simplicity of the resulting algorithm.

We propose a novel projected gradient-like method to solve a minimization problem with simple constraints. The new search direction is consistent with the first-order optimality condition for the simple constrained optimization problem, and makes the search step-size easier to determine. Convergence properties of the method are analyzed. Numerical experiments are presented to illustrate the performance and application of the proposed method.

This paper is concerned with constraint minimizers of an \begin{document}$L^2-$\end{document} critical minimization problem (1) in \begin{document}$\mathbb{R}^N$\end{document} ( \begin{document}$N≥ 1$\end{document} ) under an \begin{document}$L^2-$\end{document} subcritical perturbation. We prove that the problem admits minimizers with mass \begin{document}$ρ^\frac{N}{2}$\end{document} if and only if \begin{document}$0≤ρ for \begin{document}$b≥0$\end{document} and \begin{document}$0 for \begin{document}$b , where the constant \begin{document}$b$\end{document} comes from the coefficient of the perturbation term, and \begin{document}$Q$\end{document} is the unique positive radically symmetric solution of \begin{document}$Δ u(x)-u(x)+u^{1+\frac{4}{N}}(x) = 0$\end{document} in \begin{document}$\mathbb{R}^N$\end{document} . Furthermore, we analyze rigorously the concentration behavior of minimizers as \begin{document}$ρ \nearrow ρ^*$\end{document} for the case where \begin{document}$b>0$\end{document} , which shows that the concentration rates are determined by the subcritical perturbation, instead of the local profiles of the potential \begin{document}$V(x)$\end{document} .

We consider a minimization problem for the variational functional associated with a Gross–Pitaevskii equation arising in the study of an attractive Bose–Einstein condensate. Under an ellipse-shaped trapping potential, that is, the bottom of the trapping potential is an ellipse, we prove that any minimizer of the minimization problem blows up at one of the endpoints of the major axis of the ellipse if the parameter associated to the attractive interaction strength approaches a critical value.

In this paper, we consider \begin{document}$ L^2 $\end{document} constrained minimization problem for a modified Gross-Pitaevskii equation with higher order interactions in \begin{document}$ \mathbb{R}^2 $\end{document} . By using an auxiliary functional and some detailed energy estimates, the blow-up behavior of ground state for the modified Gross-Pitaevskii equation was obtained under different parameter regimes. Our conclusion extends some results of [ 3 ,Theorem 3.4].

The existence and nonexistence of the minimizer of the $$L^2$$-constraint minimization problem $$e(\alpha ):=\inf \{ E(u)|\ u\in H^1(\mathbb {R}^N),\ \left\| u\right\| _{L^2(\mathbb {R}^N)}^2=\alpha \}$$ are studied. Here, $$\begin{aligned} E(u):=\frac{1}{2}\int _{\mathbb {R}^N}|\nabla u|^2+V(x) |u|^2dx-\int _{\mathbb {R}^N}F(|u|)dx, \end{aligned}$$$$V(x)\in C(\mathbb {R}^N)$$, $$0\not \equiv V(x)\le 0$$, $$V(x)\rightarrow 0$$ ($$|x|\rightarrow \infty $$) and $$F(s) = \int _0^s f(t) dt $$ is a rather general nonlinearity. We show that there exists $$\alpha _0\ge 0$$ such that $$e(\alpha )$$ is attained for $$\alpha >\alpha _0$$ and $$e(\alpha )$$ is not attained for $$0 0$$.

We consider variational integral functionals ∫ Ω g ( x , u ( x ) , D u ( x ) ) d x , where Ω is a bounded open subset in R n and the integrand g ( x , s , ξ ) = f ( x , ξ ) + b ( x ) s is not subjected to any growth condition from above neither satisfies any regularity assumption, growing linearly with respect to the s -variable. Their interest relies on the fact that they are strictly connected with optimal transport problems with congestion effects. The aim of this paper is to find sufficient conditions on the boundary datum u ∗ in order to obtain global and explicit estimates for the solutions of the following minimization problem min ∫ Ω g ( x , u ( x ) , D u ( x ) ) d x ; u ∈ u ∗ + W 0 1 , 1 ( Ω ) .