Entire Euclidean Space Research Articles

VISCOSITY SOLUTIONS TO THE INFINITY LAPLACIAN EQUATION WITH SINGULAR NONLINEAR TERMS

Abstract In this paper, we study the singular boundary value problem $$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$ . We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$ . Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

Asymptotic behavior of solutions to the Yamabe equation with an asymptotically flat metric

We prove that any positive solution of the Yamabe equation on an asymptotically flat n-dimensional manifold of flatness order at least n−22 and n≤24 must converge at infinity either to a fundamental solution of the Laplace operator on the Euclidean space or to a radial Fowler solution defined on the entire Euclidean space. The flatness order n−22 is the minimal flatness order required to define ADM mass in general relativity; the dimension 24 is the dividing dimension of the validity of compactness of solutions to the Yamabe problem. We also prove such alternatives for bounded solutions when n>24.We prove these results by establishing appropriate asymptotic behavior near an isolated singularity of solutions to the Yamabe equation when the metric has a flatness order of at least n−22 at the singularity and n≤24, also when n>24 and the solution grows no faster than the fundamental solution of the flat metric Laplacian at the singularity. These results extend earlier results of L. Caffarelli, B. Gidas and J. Spruck, also of N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, when the metric is conformally flat, and work of C.C. Chen and C.S. Lin when the scalar curvature is a non-constant function with appropriate flatness at the singular point, also work of F. Marques when the metric is not necessarily conformally flat but smooth, and the dimension of the manifold is three, four, or five, as well as recent similar results by the second and third authors in dimension six.

On the continuum limit for the discrete nonlinear Schrödinger equation on a large finite cubic lattice

In this study, we consider the nonlinear Schödinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean space with simultaneous reduction in the lattice distance and expansion of the domain. Moreover, we obtain a precise global-in-time bound for the rate of convergence. Our proof heavily relies on Strichartz estimates on a finite lattice. A key observation is that, compared to the case of a lattice with a fixed size (Hong et al., 2021), the loss of regularity in Strichartz estimates can be reduced as the domain expands, depending on the speed of expansion. This allows us to address the physically important three-dimensional case.

Singular Moser–Trudinger Inequality Involving $$L^{n}$$ Norm in the Entire Euclidean Space

Singular Moser–Trudinger Inequality Involving $$L^{n}$$ Norm in the Entire Euclidean Space

The Universal Approximation Property

The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models’ potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture.

Concentration-compactness principle of singular Trudinger–Moser inequality involving N-Finsler–Laplacian operator

In this paper, suppose [Formula: see text] be a convex function of class [Formula: see text] which is even and positively homogeneous of degree 1. We establish the Lions type concentration-compactness principle of singular Trudinger–Moser Inequalities involving [Formula: see text]-Finsler–Laplacian operator. Let [Formula: see text] be a smooth bounded domain. [Formula: see text] be a sequence such that anisotropic Dirichlet norm[Formula: see text], [Formula: see text] weakly in [Formula: see text]. Denote [Formula: see text] Then we have [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the volume of a unit Wulff ball. This conclusion fails if [Formula: see text]. Furthermore, we also obtain the corresponding concentration-compactness principle in the entire Euclidean space [Formula: see text].

Kappa Regression: An Alternative to Logistic Regression

In this study, a new regression method called Kappa regression is introduced to model conditional probabilities. The regression function is based on Dombi’s Kappa function, which is well known in fuzzy theory. Here, we discuss how the Kappa function relates to the Logistic function as well as how it can be used to approximate the Logistic function. We introduce the so-called Generalized Kappa Differential Equation and show that both the Kappa and the Logistic functions can be derived from it. Kappa regression, like binary Logistic regression, models the conditional probability of the event that a dichotomous random variable takes a particular value at a given value of an explanatory variable. This new regression method may be viewed as an alternative to binary Logistic regression, but while in binary Logistic regression the explanatory variable is defined over the entire Euclidean space, in the Kappa regression model the predictor variable is defined over a bounded subset of the Euclidean space. We will also show that asymptotic Kappa regression is Logistic regression. The advantages of this novel method are demonstrated by means of an example, and afterwards some implications are discussed.

Strategic information transmission despite conflict

We analyze a standard Crawford and Sobel’s cheap talk game in a two dimensional framework, with uniform prior, quadratic preferences and a binary disclosure rule. Information might be credibly revealed by the Sender to the Receiver when players are able to strategically set aside their conflict. We exploit the few symmetries of the game parameters to derive multiple continua of equilibria, when varying the Sender’s bias over the entire euclidean space. In particular, credible information might be revealed whatever the bias. Then we show that the equilibria exhibited characterize the game’s full set of pure strategy equilibria.

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

AbstractWe prove a sharp Liouville type theorem for stable solutions of the equation urn:x-wiley:0025584X:media:mana201700260:mana201700260-math-0002on the entire Euclidean space , where and f is a continuous and nonnegative function in such that as , where and . Our theorem holds true for and is sharp in the case .

Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions

The main purpose of this paper is to establish sharp weighted Trudinger–Moser inequalities (Theorems 1.1, 1.2 and 1.3) and Caffarelli–Kohn–Nirenberg inequalities in the borderline case p=N (Theorems 1.5, 1.6 and 1.7) with best constants. Existence of extremal functions is also investigated for both the weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities. Radial symmetry of extremal functions for the weighted Trudinger–Moser inequalities are established (Theorem 1.4). Moreover, the sharp constants and the forms of the optimizers for the Caffarelli–Kohn–Nirenberg inequalities in some particular families of parameters in the borderline case p=N will be computed explicitly. Symmetrization arguments do not work in dealing with these weighted inequalities because of the presence of weights and the failure of the Polyá – Szegö inequality with weights. We will thus use a quasi-conformal mapping type transform and the corresponding symmetrization lemma to overcome this difficulty and carry out proofs of these results. As an application of the Caffarelli–Kohn–Nirenberg inequality, we also establish a weighted Moser–Onofri type inequality on the entire Euclidean space R2 (see Theorem 1.8).

Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space RN(N≥2). Precisely, if 0≤β<1 and 0<γ≤1−β, then there holds for any τ>0,supu∈W1,N(RN),∫RN(|∇u|N+τ|u|N)dx≤1⁡∫RN1|x|Nβ(eαNγ|u|NN−1−∑k=0N−2αNkγk|u|kNN−1k!)dx<∞, where αN=NωN−11/(N−1) and ωN−1 is the area of the unit sphere in RN. The above inequality is sharp in the sense that if γ>1−β, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β=0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case 0<β<1 and prove that for all τ>0, 0<β<1 and 0<γ≤1−β, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.

Sharp Sobolev type embeddings on the entire Euclidean space

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space \begin{document}${\mathbb R^n}$\end{document} , is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations

We consider elliptic differential operators on either the entire Euclidean space Rd or subsets consisting of a cube ΛL of integer length L. For eigenfunctions of the operator, and more general solutions of elliptic differential equations, we derive several quantitative unique continuation results. The first result is of local nature and estimates the vanishing order of a solution. The second is a sampling result and compares the L2-norm of a solution over a union of equidistributed δ-balls in space with the L2-norm on the entire space. In the case where the space Rd is replaced by a finite cube ΛL, we derive similar estimates. Particular features of our bound are that they are uniform as long as the coefficients of the operator are chosen from an appropriate ensemble, they are quantitative and explicit with respect to the radius δ, and they are L-independent and stable under small shifts of the δ-balls. Our proof applies to second order terms which have slowly varying coefficients on the relevant length scale. The results can also be interpreted as special cases of uncertainty relations, observability estimates, or spectral inequalities.

Pauli-Fierz Type Operators with Singular Electromagnetic Potentials on General Domains

We consider Dirichlet realizations of Pauli-Fierz type operators generating the dynamics of non-relativistic matter particles which are confined to an arbitrary open subset of the Euclidean position space and coupled to quantized radiation fields. We find sufficient conditions under which their domains and a natural class of operator cores are determined by the domains and operator cores of the corresponding Dirichlet-Schr\"{o}dinger operators and the radiation field energy. Our results also extend previous ones dealing with the entire Euclidean space, since the involved electrostatic potentials might be unbounded at infinity with local singularities that can only be controlled in a quadratic form sense, and since locally square-integrable classical vector potentials are covered as well. We further discuss Neumann realizations of Pauli-Fierz type operators on Lipschitz domains.

Nonexistence of Nontrivial Solutions with Decay Order for a Biharmonic P-Laplacian Equation and System

We use the Morawetz multiplier to show that there are no nontrivial solutions of certain decay order for a biharmonic equation with a p-Laplacian term and a system of coupled biharmonic equations with p-Laplacian terms in the entire Euclidean space.

Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$

In this paper, we apply the sharp Adams-type inequalities for the Sobolev space $W^{m,\frac{n}{m}}\left( \mathbb{R} ^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic equations in $\mathbb{R}^{2m}$ of the form \[ \left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }% \mathbb{R} ^{2m}. \] We study the existence of the nontrivial solutions when the nonlinear terms have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our approach is variational methods such as the Mountain Pass Theorem ([5]) without Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for the critical growth case. Moreover, using the regularity lifting by contracting operators and regularity lifting by combinations of contracting and shrinking operators developed in [14] and [11], we will prove that our solutions are uniformly bounded and Lipschitz continuous. Finally, using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum principle, we prove our positive solutions are radially symmetric and monotone decreasing about some point. This appears to be the first work concerning existence of nontrivial nonnegative solutions of the Bessel type polyharmonic equation with exponential growth of the nonlinearity in the whole Euclidean space.

Stationary Apollonian Packings

The notion of stationary Apollonian packings in the d-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for the growth durations and it is shown that the packing is space-filling with probability 1, in the sense that the Lebesgue measure of its complement is zero. Finally, the phenomenon is studied that grains arrange in clusters and properties related to percolation are investigated.

Numerical Integration of Discontinuous Functions in Many Dimensions

We consider the problem of numerically integrating functions with hyperplane discontinuities over the entire Euclidean space in many dimensions. We describe a simple process through which the Euclidean space is partitioned into simplices on which the integrand is smooth, generalising the standard practice of dividing the interval used in one-dimensional problems. Our procedure is combined with existing adaptive cubature algorithms to significantly reduce the necessary number of function evaluations and memory requirements of the integrator. The method is embarrassingly parallel and can be trivially scaled across many cores with virtually no overhead. Our method is particularly pertinent to the integration of Green's functions, a problem directly related to the perturbation theory of impurity models. In three spatial dimensions, we observe a speed-up of order 100 which increases with increasing dimensionality.

Extension of positive definite functions

Let Ω⊂Rn be an open, connected subset of Rn, and let F:Ω−Ω→C, where Ω−Ω={x−y:x,y∈Ω}, be a continuous positive definite function. We give necessary and sufficient conditions for F to have an extension to a continuous positive definite function defined on the entire Euclidean space Rn. The conditions are formulated in terms of existence of a unitary representations of Rn whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations.

Landau--Lifshitz--Slonczewski Equations: Global Weak and Classical Solutions

We consider magnetization dynamics under the influence of a spin-polarized current, given in terms of a spin-velocity field $\boldsymbol{v}$, governed by the following modification of the Landau--Lifshitz--Gilbert equation $\frac{\partial \boldsymbol{m}}{\partial t} + \boldsymbol{v} \cdot \nabla \boldsymbol{m}= \boldsymbol{m} \times (\alpha \, \frac{\partial \boldsymbol{m}}{\partial t} + \beta \, \boldsymbol{v} \cdot \nabla \boldsymbol{m} - \Delta \boldsymbol{m}),$ called the Landau--Lifshitz--Slonczewski equation. We focus on the situation of magnetizations defined on the entire Euclidean space $\boldsymbol{m}(t) : \mathbb{R}^3 \to \mathbb{S}^2$. Our construction of global weak solutions relies on a discrete lattice approximation in the spirit of Slonczevski's spin-transfer-torque model and provides a rigorous justification of the continuous model. Using the method of moving frames, we show global existence and uniqueness of classical solutions under smallness conditions on the initial data in terms of t...