Bounded Convex Domain Research Articles

Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and determine the value of γ∈(0,∞]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the nth minimal worst case error for Lq(M)-approximation for all 1≤q≤∞.Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ<∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

Optimal convex domains for the first curl eigenvalue in dimension three

We prove that there exists a bounded convex domain Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class C 1 , 1 C^{1,1} ). Existence results for uniformly Hölder optimal domains in a box (that is, contained in a fixed bounded domain D ⊂ R 3 D\subset \mathbb {R}^3 ) are also presented.

Finite element analysis of extended Fisher-Kolmogorov equation with Neumann boundary conditions

This paper delves into the numerical analysis of the extended Fisher-Kolmogorov (EFK) equation within open bounded convex domains R⊂Rd, where d≤3. Two distinct finite element schemes are introduced, namely the semi-discrete and fully-discrete finite element approximations. The existence and uniqueness of solutions are established for both the semi-discrete and fully-discrete finite element approximations. Error bounds are investigated across different scenarios, including comparisons between the semi-discrete and exact solutions, as well as between the semi-discrete and fully-discrete solutions, along with the fully-discrete and exact solutions. An effective algorithm has been proposed to solve the nonlinear system resulting from the fully-discrete finite element approximation at each time step. The paper also provides computed numerical error results and showcases a variety of numerical experiments to further illustrate and support the findings.

The sharp Hardy–Moser–Trudinger inequality in dimension 𝑛

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ sup u ∈ W 0 1 , n ( B n ) , ∫ B n | ∇ u | n d x − ( 2 ( n − 1 ) n ) n ∫ B n | u | n ( 1 − | x | 2 ) n d x ≤ 1 ∫ B n e ( 1 − β n ) α n | u | n n − 1 | x | − β d x ≤ C \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right )^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any β ∈ [ 0 , n ) \beta \in [0,n) , n ≥ 3 n\geq 3 where α n = n ω n − 1 1 n − 1 \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and ω n − 1 \omega _{n-1} is the surface area of the n − 1 n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green’s functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324].

On the application of one modulus inequality to the mapping theory

The authors study mappings that satisfy some estimate of the distortion of the modulus of families of paths. Under certain conditions on the domains between which the mappings act, we established that, these mappings are Hölder logarithmic continuous at the boundary points. It is known that, the Hölder continuity is established for many classes of mappings, say quasiconformal and quasiregular mappings. In this regard, it is possible to point to the classical distortion estimates by Martio-Rickman-Väisälä type, as well as the estimates related to the modern classes of mappings with finite distortion. In particular, V.I. Ryazanov together with R.R. Salimov and E.O. Sevost'yanov established local distortion estimates for plane and spatial mappings under FMO condition, or under the Lehto-type integral condition. Recently, the second co-author have obtained Hölder logarithmic continuity for the studied class at points of the unit sphere. This article considers the situation of similar mappings of different domains, not only the unit sphere. Namely, we consider mappings between quasiextremal distance domains (QED-domains) and convex domains. Note that, quasiextremal distance domains introduced by Gehring and Martio are structures in which the modulus of families of paths is metrically related to the diameter of sets. Also, convex domains are involved in the formulation of the main result; we consider mappings that surjectively act onto them. In addition, the article contains the formulations and proofs for some other results on this topic. We consider several more cases in detail, in particular when: 1) the definition domain is a domain with a locally quasiconformal boundary, and the image domain is a bounded convex domain; 2) the definition domain is a regular domain in the sense of prime ends, and the image domain is a bounded convex domain; 3) the mapping acts between the QED-domain and the bounded convex domain and has a fixed point. In all three cases, the mapping is Hölder logarithmic continuous; moreover, in case 2), which refers to prime ends, logarithmic continuity should also be understood in terms of prime ends.

A free boundary problem for discontinuous semilinear elliptic equations in convex ring

In this paper, we consider the following free boundary problem: ( P ) { Δ u = λ ϕ ( x ) ∑ i = 1 n H ( u − μ i ) in Ω = Ω 2 ∖ Ω ¯ 1 , u = 0 on ∂ Ω 2 , u = M on ∂ Ω 1 . The domain Ω is a convex ring where Ω 1 and Ω 2 are bounded convex domains in R N , N ≥ 2 , H is the Heaviside step function, λ , M , μ i ( i = 1 , . . , n ) are given positive real parameters and ϕ is a given function. We show that under suitable conditions, there exists a solution to problem (P) with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function ϕ which shows that there are more solutions.

THE MATHEMATICAL MODEL OF TWO-DIMENSIONAL ELECTROMAGNETHIC FIELD

The purpose of this work is to develop a mathematical model of two-dimensional electromagnetic field, i.e. to construct a system of axioms (assumptions) that we unconditionally accept, although they do not correspond to the real electromagnetic field, but approximate it quite well. A mathematical model of two-dimensional electromagnetic field is constructed based on four assumptions: 1. Boundedness: we assume that on an infinite plane there is a bounded convex domain, outside which the field is zero. 2. Existence of nodes: we assume that in the domain there is a finite set of points, called nodes. All calculations of potential values are carried out only at internal nodes․ 3. Existence of neighbors: we assume that all pairs of nodes are divided into two types: neighboring and non-neighboring, and only neighboring nodes interact. 4. Discretization: we assume that the magnetic permeability of the medium is discrete, i.e. is a piecewise constant function. It is shown that within our mathematical model, from Assumptions 1 – 4, it follows that the electromagnetic potential is necessarily a piecewise linear function. It is shown that within our mathematical model, minimization of the energy functional leads to a system of nonlinear equations, the number of equations is equal to the number of internal nodes. It is proved that within our mathematical model, from Assumptions 1 – 4, it follows that for the numerical solution of the resulting system of equations, the optimal mesh is the Delaunay triangulation. The results are illustrated by an example of numerical solution of a field problem for electromagnetic systems with magnetorheological fluid using the FEMM tool. It is shown that the FEMM application package corresponds to our model.

A characterization of bounded balanced convex domains in [formula omitted

In this paper, we investigate the characterization of balanced bounded convex domains in Cn in terms of the squeezing function. As an application, we provide a characterization of the polydisc in Cn.

Representation of analytic functions in bounded convex domains on the complex plane

The paper is devoted to entire functions of exponential type and regular growth. Exceptional sets are investigated outside of which these functions have estimates from below that asymptotically coincide with their estimates from above. An explicit construction of an exceptional set, which consists of disks with centers at zeros of the entire function, is indicated. The concept of a properly balanced set is introduced, which is a natural generalization of the concept of a regular set by B. Ya. Levin. It is proved that the zero set of an entire function is properly balanced if and only if each function analytic in the interior of the conjugate diagram of the entire function in question and continuous up to the boundary is represented by a series of exponential monomials whose exponents are zeros of this entire function. This result generalizes the classical result of A. F. Leont′ev on the representation of analytic functions in a convex domain to the case of a multiple zero set.

Global boundedness of the curl for a p‐curl system in convex domains

In this note, we study a nonlinear system involving the curl operator in a bounded convex domain in , which is derived from the steady‐state approximation for Bean critical‐state model for type‐ superconductors. We show the existence and establish the estimate of the curl of solutions to this system.

Hilbert Metric in the Unit Ball

The Hilbert metric between two points 𝑥, 𝑦 in a bounded convex domain 𝐺 is defined as the logarithm of the cross-ratio 𝑥, 𝑦 and the intersection points of the Euclidean line passing through the points 𝑥, 𝑦 and the boundary of the domain. Here, we study this metric in the case of the unit ball 𝔹𝑛. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.

Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎){−Δu=up−εinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN with N≥3, p=N+2N−2 and ε>0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.In general, the solutions of (⁎) may blow-up at multiple points a1,⋯,ak of Ω as ε→0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k=1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).

On Solution Regularity of the Two-Dimensional Radiation Transfer Equation and Its Implication on Numerical Convergence

In this paper, we deal with the differential properties of the scalar flux ϕ ( x ) defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation. Estimates for the derivatives of ϕ ( x ) near the boundary of the domain are given based on Vainikko’s regularity theorem. The optimal pointwise error estimates in terms of the scalar flux are presented for the two classic finite difference methods: diamond difference (DD) and step difference (SD). Numerical results indicate the implication of the solution smoothness on the numerical convergence behavior.

Problems for generalized Monge–Ampère equations

AbstractThis paper deals with some Monge–Ampère type equations involving the gradient that are elliptic in the framework of convex functions. First, we show that such equations may be obtained by minimizing a suitable functional. Moreover, we investigate a P-function associated with the solution to a boundary value problem of our generalized Monge–Ampère equation in a bounded convex domain. It will be shown that this P-function attains its maximum value on the boundary of the underlying domain. Furthermore, we show that such a P-function is actually identically constant when the underlying domain is a ball. Therefore, our result provides a best possible maximum principles in the sense of L. E. Payne. Finally, in case of dimension 2, we prove that this P-function also attains its minimum value on the boundary of the underlying domain. As an application, we will show that the solvability of a Serrin’s type overdetermined problem for our generalized Monge–Ampère type equation forces the underlying domain to be a ball.

Superharmonic functions related to convexity estimates for a class of semilinear elliptic problems

In this paper, we consider homogeneous Dirichlet problems for a class of semilinear elliptic equations involving the Laplacian with power-type nonlinearities in bounded convex domains of Rn, and find some superharmonic functions which can be used to give convexity estimates for the solutions directly.

Carleson measures on convex domains with smooth boundary of finite type

AbstractFollowing M. Abate and A. Saracco's work on strongly pseudoconvex domains in , we characterize Carleson measures of in bounded convex domains in with smooth boundary of finite type. We also give examples of Carleson measures with uniformly discrete (with respect to the Kobayashi distance) sequences.

Non-existence of patterns for a class of weighted degenerate operators

A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this result asserts that stable patterns do not exist in convex domains.In this paper we consider a weighted version of the Laplace operator, where the weight may be singular or degenerate at the origin, and prove the nonexistence of patterns, extending the results by Casten-Holland and Matano to general weak solutions (not necessarily stable) and under a suitable assumption on the nonlinearity and the domain.Our results exhibit some intriguing behavior of the problem according to the weight and the geometry of the domain. Indeed, our main results follow from a geometric assumption on the second fundamental form of the boundary in terms of a parameter which describes the degeneracy of the operator. As a consequence, we provide some examples and show that nonexistence of patterns may occur also for non convex domains whenever the weight is degenerate.

Composition operators between Bergman spaces on infinite type convex domains in ℂ2

ABSTRACT We provide characterizations of boundedness and compactness for composition operators between Bergman spaces of smoothly bounded convex domains in C 2 admitting the so-called geometric F-type. Such geometric type contains all domains of finite type and many cases of infinite type in the sense of Range.

Example of a solution for Dorodnitzyn’s limit formula

In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain R⊂R2. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane R2. Under the same assumption, we can give an approximate solution u for the limit formula if |u|<<<1 such that: u(z)≅δ∗c∗z+625⋅12i0⋅4U23z4+o(z5),that corresponds to an approximate horizontal velocity component when a small parameter ϵ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, c>0, δ is the boundary layer’s height in Dorodnitzyn’s coordinates, U is the free-stream velocity at the upper boundary of the domain, and T0 is the absolute surface temperature.

Global weak solution for a chemotaxis Navier–Stokes system with [formula omitted]-Laplacian diffusion and singular sensitivity

In this paper, we consider a chemotaxis-Navier–Stokes system with p-Laplacian diffusion and singular sensitivity in a bounded convex domain Ω⊂R3 with smooth boundary. It is shown that under an appropriate hypothesis for p and δ, there exists a locally bounded global weak solution of the corresponding initial–boundary problem.