Multidimensional Ball Research Articles

Kernel well-posedness and computation by power series in backstepping output feedback for radially-dependent reaction–diffusion PDEs on multidimensional balls

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction–diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach (in the spirit of the method of Frobenius for ordinary differential equations), finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

Dirichlet Problem for the Laplace Equation in the Hyperoctant of a Multidimensional Ball

Dirichlet Problem for the Laplace Equation in the Hyperoctant of a Multidimensional Ball

On the ratio of total masses of species to resources for a logistic equation with Dirichlet boundary condition

We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li [1] showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto [14] showed that the supremum is infinity in the multi-dimensional ball. In this paper, we prove that the same results still hold true for the Dirichlet problem, and our proof is based on the sub-super solution method. Moreover, we show the differences of the solutions corresponding to the maximizing sequence between the one- and two-dimensional domains.

Holmgren’s Problem for the Laplace Equation in the Hyperoctant of a Multidimensional Ball

Holmgren’s Problem for the Laplace Equation in the Hyperoctant of a Multidimensional Ball

On the asymptotics of the principal moments of inertia of a convex body in the isotropic state

We recall Bourgain’s conjecture on the upper bound for the main moments of inertia of a multidimensional body in the isotropic state. We also recall the currently known bounds obtained by Bourgain and Klartag. Then we evaluate the moments of inertia of the multidimensional ball and of the multidimensional cube, and state a conclusive comment corroborating the Bourgain’s conjecture.

On the asymptotics of the principal moments of inertia of a convex body in the isotropic state

We recall Bourgain's conjecture on the upper bound for the main moments of inertia of a multidimensional body in the isotropic state. We also recall the currently known bounds obtained by Bourgain and Klartag. Then we evaluate the moments of inertia of the multidimensional ball and of the multidimensional cube, and state a conclusive comment corroborating the Bourgain's conjecture.

ФУНКЦИИ ГРИНА НЕКОТОРЫХ КРАЕВЫХ ЗАДАЧ ДЛЯ БИГАРМОНИЧЕСКИХ ОПЕРАТОРОВ И ИХ КОРРЕКТНЫЕ СУЖЕНИЯ

In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

GREEN'S FUNCTIONS AND CORRECT RESTRICTIONS OF THE POLYHARMONIC OPERATOR

In this paper, for completeness of presentation, we give explicitly the Green's functions for the classical problems - Dirichlet, Neumann, and Robin for the Poisson equation in a multidimensional unit ball. There are various ways of constructing the Green's function of the Dirichlet problem for the Poisson equation. For many types of areas, it is built explicitly. Recently, there has been renewed interest in the explicit construction of Green's functions for classical problems. The Green's function of the Dirichlet problem for a polyharmonic equation in a multidimensional ball is constructed in an explicit form, and for the Neumann problem the construction of the Green's function remains an open problem. The paper gives a constructive way of constructing the Green's function of Dirichlet problems for a polyharmonic equation in a multidimensional ball. Finding general well-posed boundary value problems for differential equations is always an urgent problem. In this paper, we briefly outline the theory of restriction and extension of operators and describe well-posed boundary value problems for a polyharmonic operator.

On the unboundedness of the ratio of species and resources for the diffusive logistic equation

<p style='text-indent:20px;'>Concerning a class of diffusive logistic equations, Ni [<xref ref-type="bibr" rid="b1">1</xref>,Abstract] proposed an optimization problem to consider the supremum of the ratio of the <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> norms of species and resources by varying the diffusion rates and the profiles of resources, and moreover, he gave a conjecture that the supremum is <inline-formula><tex-math id="M2">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> in the one-dimensional case. In [<xref ref-type="bibr" rid="b1">1</xref>], Bai, He and Li proved the validity of this conjecture. The present paper shows that the supremum is infinity in a case when the habitat is a multi-dimensional ball. Our proof is based on the sub-super solution method. A key idea of the proof is to construct an <inline-formula><tex-math id="M3">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> unbounded sequence of sub-solutions.

Singular perturbations of Laplace operator and their resolvents

An inner closed (without boundary) smooth manifold of a lower dimension is cut from a multidimensional ball. In this region, invertible restrictions of the Laplace operator are well defined. In particular, the well-posed non-smooth Bitsadze-Samarskiĭ problem for the Laplace equation is defined. Moreover, we obtain M.G. Krein's formula for the trace of the difference of resolvents of the studied operators. We prove assertions on the spectrum of the non-smooth Bitsadze-Samarskiĭ problem.

Complexity of Methods for Approximating Convex Compact Bodies by Double Description Polytopes and Complexity Bounds for a Hyperball

A comparative analysis of the complexity of approaches to the approximation of convex compact bodies by double description polytopes is provided as applied to a ball. A complexity bound for the Estimate Refinement method is obtained in the case of approximation of a multidimensional ball.

Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is analyzed. When applied to convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. In previous studies, for the approximation of a multidimensional ball, the convergence rates of the method were estimated in terms of the number of faces of all dimensions and the cardinality of the facial structure (the norm of the f-vector) of the constructed polytope was shown to have an optimal rate of growth. In this paper, the asymptotic convergence rate of the method with respect to faces of all dimensions is compared with the convergence rate of best approximation polytopes. Explicit expressions are obtained for the asymptotic efficiency, including the case of low dimensions. Theoretical estimates are compared with numerical results.

Asymptotic properties of the estimate refinement method in polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is considered. In the approximation of convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. The properties of the method are examined as applied to the polyhedral approximation of a multidimensional ball. As vertices of approximating polytopes, the method is shown to generate a deep holes sequence on the surface of the ball. As a result, previously obtained combinatorial properties of convex hulls of the indicated sequences, namely, the convergence rates with respect to the number of faces of all dimensions and the optimal growth of the cardinality of the facial structure (of the norm of the f-vector) can be extended to such polytopes. The combinatorial properties of the approximating polytopes generated by the estimate refinement method are compared to the properties of polytopes with a facial structure of extremal cardinality. It is shown that the polytopes generated by the method are similar to stacked polytopes, on which the minimum number of faces of all dimensions is attained for a given number of vertices.

Representation of Green’s function of the Neumann problem for a multi-dimensional ball

Representation of the Green’s function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function, we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that the Green’s function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel at and .

Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality

The problem of polyhedral approximation of a multidimensional ball is considered. It is well known that the norm of the f-vector (the maximum number of faces of all dimensions) of an approximating polytope grows at least as fast as O(δ(1 − d)/2), where δ is the Hausdorff deviation and d is the space dimension. An iterative method, namely, the deep holes method is used to construct metric nets. As applied to the problem under study, the method sequentially supplements the vertex set of the polytope with its deep holes in the metric on the ball surface (i.e., with points of the surface that are farthest away from the vertices of the polytope). It is shown that the facet structure cardinality of the constructed polytope has an optimal growth rate. It is also shown that the number of faces of all dimensions in the approximating polytopes generated by the method is asymptotically proportional to the number of their vertices. Closed-form expressions for the constants are obtained, which depend only on the dimension of the space, including the case of high dimensions. For low dimensions (d ranging from 3 to 5), upper bounds for the growth rate of the number of faces of all dimensions are obtained depending on the accuracy of the approximation.

Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction–diffusion equations

We discuss free boundary problems modeling the diffusion of invasive or new species in a multi-dimensional ball or annulus, where unknown functions are population density and the outer boundary representing the spreading front of the species. We define spreading and vanishing to describe the asymptotic behaviors of radially symmetric solutions. The main purpose is to study the underlying principle to determine spreading and vanishing for the free boundary problems of general reaction–diffusion equations. We also focus on the problems with a logistic or bistable reaction term to show dichotomy results, vanishing speed and sufficient conditions for spreading or vanishing.

The Poisson Operator for Orthogonal Polynomials in the Multidimensional Ball

In this paper we define the Poisson operator related to an orthonormal system on the multidimensional ball and we analyze some weighted inequalities for this operator in mixed norm spaces.

Calculation of the Volume of Multidimensional Cones and Balls Using Cavalieri's Principle

Formulas for the volume of multidimensional cones and balls are deduced based on Cavalieri's principle and basic properties of the Lebesgue measure (without using integration).

Decomposition of spaces of distributions induced by tensor product bases

Rapidly decaying kernels and frames (needlets) in the context of tensor product Jacobi polynomials are developed based on several constructions of multivariate C∞ cutoff functions. These tools are further employed to the development of the theory of weighted Triebel–Lizorkin and Besov spaces on [−1,1]d. It is also shown how kernels induced by cross product bases can be constructed and utilized for the development of weighted spaces of distributions on products of multidimensional ball, cube, sphere or other domains.

A rearrangement formula for a singular Cauchy-Szegö integral in a ball from ℂ n

We obtain an analog of the Poincare-Bertrand formula for a singular Cauchy-Szego integral in a multidimensional ball. We understand the principal value of the integral in the Cauchy sense. The obtained formula differs from that of Poincare-Bertrand for the Cauchy integral in a complex plane.