Multiplicative inequalities in domains with noncompact boundary

Exact embedding theorems of the multiplicative type are established for functions of Sobolev spaces defined in a domain R ⁿ,n2, whose boundary is not compact. The main condition on the domain is of the isoperimetric type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 260–268, February, 1992.     

Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.     

On the Helmholtz decomposition in general unbounded domains

It is well known that the Helmholtz decomposition of L^q-spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C²-type in if we replace the space L^q, 1 < q < ∞, by L² ∩ L^q for q > 2 and by L^q + L² for 1 < q < 2. In this paper, we generalize this new approach from the three-dimensional case to the n-dimensional case, n ≥ 2. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform C²-type. Received: 15 February 2006     

The Neumann problem and Helmholtz decomposition in convex domains

We show that the Neumann problem for Laplace's equation in a convex domain Ω with boundary data in Lp(∂Ω) is uniquely solvable for 1<p<∞. As a consequence, we obtain the Helmholtz decomposition of vector fields in Lp(Ω,Rd).     

Stable boundary element domain decomposition methods for the Helmholtz equation

In this paper, we present a stable boundary element domain decomposition method to solve boundary value problems of the Helmholtz equation via a tearing and interconnecting approach. A possible non-uniqueness of the solution of local boundary value problems due to the appearance of local eigensolutions is resolved by using modified interface conditions of Robin type, which results in a Galerkin boundary element discretization which is robust for all local wave numbers. Numerical examples confirm the stability of the proposed approach.     

Solvability of boundary and initial-boundary value problems for the Navier-Stokes equations in domains with noncompact boundaries

One considers boundary and Initial-boundary value problems for Navier-Stokes equations in domains with several “outlets” at infinity. It is shown that if one prescribes the limiting values of the pressure at infinity in those “outlets” which expand sufficiently fast, then these problems are solvable in the class of vectors with a finite Dirichlet integral.     

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficientnumerical methods for the solution of boundary integral equationsdefined on a non-smooth boundary. In particular the solutionof the Helmholtz equation in the exterior of a closed wedgeis studied. The analytical behaviour of the solution of theresulting boundary integral equation (with a non-compact operator)near the wedge is investigated. Numerical analysis of the collocationand iterated collocation method for the problem is presented.Graded meshes are used to reflect the ‘singular’behaviour of the analytical solution, as well as the degreeof the polynomial approximant, in order to yield results with‘optimal convergence rates’. Finally the convergenceanalysis of some modified two-grid iterative methods for thefast solution of the resulting linear systems is given and numericalresults are presented which agree with the theoretical predictions.     

关于集值上鞅分解式的注记

讨论了集值上鞅与支撑函数的一些性质,利用支撑函数研究了一般Banach空间上集值上鞅的Riesz分解定理,推广和改进了以往的结果。     

On the Helmholtz equation in a strip with penetrable boundary

Rostov State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 3, pp. 92–94, July–September, 1994.     

Solving the Helmholtz equation in some infinite domains

A numerical-analytical method is proposed to solve boundary-value problems for the Heimholtz equation in two-dimensional domains of complicated configuration.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 9–11, 1987.     

Linearly convex domains with singularities on the boundary

We present a topological classification of linearly convex domains with almost smooth boundary whose singularities lie in a hyperplane. We investigate sets with linearly convex boundary and the closures of linearly convex domains.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

Nodal domains and growth of harmonic functions on noncompact manifolds

Harmonic functions are studied on complete Riemannian manifolds. A decay estimate is given for bounded harmonic functions of variable sign. For unbounded harmonic functions of variable sign, relations are derived between growth properties and nodal domains. On Riemannian manifolds of nonnegative Ricci curvature, it has been conjectured that harmonic functions, having at most a given order of polynomial growth, must form a finite dimensional vector space. This conjecture is established in certain special cases.     

Remark on the anisotropic prescribed mean curvature equation on arbitrary domains

In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on ${x \in \varOmega \subset \mathbb {R}^n}In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on x ? \varOmega ì \mathbb Rⁿ{x \in \varOmega \subset \mathbb {R}^n}, the height u of the hypersurface M = graph u over \varOmega{\varOmega} and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of ?\varOmega{\partial \varOmega} that guarantees the existence of smooth solutions even for not necessarily convex domains.