Normalized solutions of (p, q)-Laplacian equations with mass subcritical growth

On the sub-supersolution method for [formula omitted]-Laplacian equations

We consider the following elliptic problem with mixed local and nonlocal operators: ( − Δ ) s 1 u + ( − Δ ) s 2 u + λ u = f ( u ) in R N , ∫ R N ∣ u ( x ) ∣ 2 d x = c , u ∈ H s 2 ( R N ) , \left\{\begin{array}{l}{\left(-\Delta )}^{{s}_{1}}u+{\left(-\Delta )}^{{s}_{2}}u+\lambda u=f\left(u)\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u\left(x)| }^{2}{\rm{d}}x=c,\hspace{1.0em}\\ u\in {H}^{{s}_{2}}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where 0 < s 1 < s 2 < 1 0\lt {s}_{1}\lt {s}_{2}\lt 1 , ( − Δ ) s i {\left(-\Delta )}^{{s}_{i}} is fractional Laplacian. We get a sharp description of the existence and nonexistence of the global minimizer on the mass constraint, which is called energy ground state. By using of concentration-compactness principle, we show that there exists a constant c 0 {c}_{0} such that there exists an energy ground state if c > c 0 c\gt {c}_{0} and there exists no energy ground state if 0 < c < c 0 0\lt c\lt {c}_{0} . Some almost critical conditions which determine c 0 = 0 {c}_{0}=0 or c 0 > 0 {c}_{0}\gt 0 are established. In the case c = c 0 c={c}_{0} , under certain conditions, we establish the existence result of energy ground state. Finally, by using the method of mountain pass type characterization, we show that any energy ground state is action ground state of the corresponding action functional.

We have an $\sf{M}\times\sf{N}$ real-valued arbitrary matrix $A$ (e.g., a dictionary) with $\sf{M}<\sf{N}$ and data $d$ describing the sought-after object with the help of $A$. This work provides an in-depth analysis of the (local and global) minimizers of an objective function ${\mathcal{F}}_d$ combining a quadratic data-fidelity term and an $\ell_0$ penalty applied to each entry of the sought-after solution, weighted by a regularization parameter $\beta>0$. For several decades, this objective has attracted a ceaseless effort to conceive algorithms approaching a good minimizer. Our theoretical contributions, summarized below, shed new light on the existing algorithms and can help in the conception of innovative numerical schemes. Solving the normal equation associated with any $\sf{M}$-row submatrix of $A$ is equivalent to computing a local minimizer $\hat u$ of ${\mathcal{F}}_d$. (Local) minimizers $\hat u$ of ${\mathcal{F}}_d$ are strict if and only if the submatrix, composed of those columns of $A$ whose indices form the support of $\hat u$, has full column rank. An outcome is that strict local minimizers of ${\mathcal{F}}_d$ are easily computed without knowing the value of $\beta$. Each strict local minimizer is linear in data. It is proved that ${\mathcal{F}}_d$ has global minimizers and that they are always strict. They are studied in more detail under the (standard) assumption that rank$(A)=\sf{M}<\sf{N}$. The global minimizers with $\sf{M}$-length support are seen to be impractical. Given $d$, critical values $\beta_{\sf{K}}$ for any ${\sf{K}}\leqslant\sf{M}-1$ are exhibited such that if $\beta>\beta_{\sf{K}}$, all global minimizers of ${\mathcal{F}}_d$ are ${\sf{K}}$-sparse. An assumption on $A$ is adopted and proved to fail only on a closed negligible subset. Then for all data $d$ beyond a closed negligible subset, the objective ${\mathcal{F}}_d$ for $\beta>\beta_{\sf{K}}$, ${\sf{K}}\leqslant\sf{M}-1$, has a unique global minimizer, and this minimizer is ${\sf{K}}$-sparse. Instructive small-size ($5\times 10$) numerical illustrations confirm the main theoretical results.

We address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion. The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

In any dimension [Formula: see text], for given mass [Formula: see text] and when the [Formula: see text] energy functional [Formula: see text] is coercive on the mass constraint [Formula: see text] we are interested in searching for constrained critical points at positive energy levels. Under general conditions on [Formula: see text] and for suitable ranges of the mass, we manage to construct such critical points which appear as a local minimizer or correspond to a mountain pass or a symmetric mountain pass level. In particular, our results shed some light on the cubic–quintic nonlinear Schrödinger equation in [Formula: see text].

Recently, nonconvex and nonsmooth models such as those using ‘norm’ have drawn much attention in the area of image restoration. This work investigates the local and global minimizers of the gradient regularized model with box constraints. There are four major ingredients. Firstly, we show that the set of local minimizers can be represented by solutions to some quadratic problems, which are independent of the fidelity parameter α. Based on this, every point satisfying the first-order necessary condition is a local minimizer. Secondly, any two local minimizers have different energy values under certain assumptions, implying the uniqueness of the global minimizer. Thirdly, there exists a uniform lower bound for nonzero gradients of the restored images. Finally, we show that the global minimizer set is piecewise constant in terms of α, and when A is of full column rank and α is large enough, the distance between the true image and the restored images is bounded by the noise level. The numerical examples perfectly demonstrate our theoretical analysis.

In this paper we introduce a motion planning method which uses an artificial potential field obtained by solving Laplace's differential equation. A potential field based on Laplace's equation has no minimal point; therefore, path planning is performed without falling into local minima. Furthermore, we propose an application of the motion planning method for recursive motion planning in an uncertain environment. We illustrate the robot motion generated by the proposed method with simulation examples.

We prove the local Hölder continuity of strong local minimizers of the stored energy functionalE(u)=∫Ωλ|∇u|2+h(det∇u)dx\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$E(u)=\\int_{\\Omega}\\lambda |\\nabla u|^{2}+h({\\rm det} \\nabla u)\\,{\\rm d}x$$\\end{document}subject to a condition of ‘positive twist’. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as {sto 0+}, linearly as {s to +infty}, and satisfies {h(s)=+infty} if {s leqq 0}. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed Hölder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term {int_{Omega} h({rm det} nabla u),{rm d}x} can have by analysing the regularity of local minimizers in the class of ‘shear maps’. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are Hölder continuous.

We study the finite element approximation of the solid isotropic material with penalization method (SIMP) for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local minimizer to the infinite-dimensional problem, we consider two popular regularization methods: W1,p-type penalty methods and density filtering. Previous results prove weak(-*) convergence in the space of the material distribution to a local minimizer of the infinite-dimensional problem. Notably, convergence was not guaranteed to all the isolated local minimizers. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element local minimizers that strongly converges to the minimizer in the appropriate space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding.

We consider a nonlocal isoperimetric problem defined in the whole space \mathbb R^N , whose nonlocal part is given by a Riesz potential with exponent \alpha \in (0, N –1) . We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L^1 -norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer. Finally we deduce that for small masses the ball is also the unique global minimizer, and that for small exponents a in the nonlocal term the ball is the unique minimizer as long as the problem has a solution.

We consider a nonlocal isoperimetric problem defined in the whole space $\R^N$, whose nonlocal part is given by a Riesz potential with exponent $\alpha\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the $L^1$-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.

The zero velocity curve and the zero velocity surface are examples of barrier structures that energetically separate the regions of possible and impossible motion. Our recent work deduced that states associated with the barrier structures globally minimize an energy function. The present paper explores the role of local energy minimizers in the planar circular restricted three-body problem. It is demonstrated that a structure, with which local energy minimizers associate, also plays the role of a barrier within a conditioned range that the local minimizers are regarded as the global minimizers. Such conditional barrier structures robustly exist and regulate the motion even in the high-energy regime, where the zero velocity curve ceases to exist.

Energy minimization is often the key point of solving problems in computer vision. For decades, many methods have been proposed (deterministic, stochastic,…). Some can only reach local minimum and others strong local minimum close to the optimal solution (global minimum). Since beginning of 21th century, minimization based on Graph theory have been generalized to find global minimum of multi-labeling problems. In this work, we study deterministic local minimization methods (Iterative Conditional Modes and Direct Descent Energy), and a stochastic global minimization with an improved Simulated Annealing algorithm. A new approach formulation to help local minimization to converge to a minimum closed to the global one is proposed. This method combines local and global energy constraints in an multiresolution way. We focus on stereo matching application. The improved Simulated Annealing proved to reach global minimum as good as Graph based minimization methods. Promising results of proposed local minimization methods are obtained on Middlebury Stereo database compare to global methods.

We investigate the structure of the local and global minimizers of the Huber loss function regularized with a sparsity inducing L0 norm term. We characterize local minimizers and establish conditions that are necessary and sufficient for a local minimizer to be strict. A necessary condition is established for global minimizers, as well as non-emptiness of the set of global minimizers. The sparsity of minimizers is also studied by giving bounds on a regularization parameter controlling sparsity. Results are illustrated in numerical examples.

The conformal mapping technique has long been used to obtain exact solutions to Laplace's equation in two-dimensional domains with awkward geometries. However, a major limitation of the technique is that it is only directly compatible with Dirichlet and zero-flux Neumann boundary conditions. It would be useful to have a means of adapting the technique to handle more general boundary conditions, for example, Robin or nonlinear flux conditions. Boundary tracing is an unconventional method for tackling boundary value problems with generic flux boundary conditions, where one takes a known solution to the field equation and seeks new boundaries satisfying the prescribed boundary condition. In this paper, we adapt boundary tracing for compatibility with conformal mapping to produce a new prescription for studying Laplace's equation coupled with general flux boundary conditions. We illustrate the procedure via two simple examples involving heat transfer. In both cases, we demonstrate how to construct infinite families of nontrivial domains in which the solution to the chosen flux boundary value problem is exactly equal to a selected harmonic function.