Noncompact Boundary Research Articles

Qualitative properties of solutions of the Neumann problem for a higher-order quasilinear parabolic equation

The property of localization of perturbations is proved for a solution of an initial boundary-value Neumann problem in a regionD=Ωx, t>0, where Ω is a region in Rnwith a noncompact boundary.

Free Boundary Value Problems for the Stationary Navier-Stokes Equations in Domains with Noncompact Boundaries

A free boundary problem for an incompressible viscous fluid in a domain with noncornpact boundaries is considered; the upper boundary is to be determined by equilibrium conditions involving the fluid stress tensor and its surface tension. It is proved that if the data of the problem are regular, then the free boundary, the velocity vector and the pressure are regular. Furthermore the exponential decay of the solution is shown.

The Dirichlet problem with non-compact boundary

The Dirichlet problem with non-compact boundary

Normal solvability of elliptic boundary value problems on asymptotically flat manifolds

Normal solvability is shown for a class of boundary value problems on Riemannian manifolds with noncompact boundary using a concept of weighted pseudodifferential operators and weighted Sobolev spaces together with Lopatinski-Shapiro type boundary conditions. An essential step is to show that the standard normal derivative defined in terms of the Riemannian metric is in fact a weighted pseudodifferential operator of the considered class provided the metric is compatible with the symbols.

Stabilization of solutions of initial-boundary problems for quasilinear parabolic equations

Lower bounds are obtained for solutions of the initial-boundary Dirichlet problem for high order equations. Sharp bounds are also obtained for ess sup¦∇u(x, t)¦ of the Neumann initial-boundary problem for a second-ω order equation in D=Ωx(t >0), where (Rn, n ≥ 2 is a domain with noncompact convex boundary.

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 Ω ⊂Rn,n⩾2, whose boundary is not compact. The main condition on the domain is of the isoperimetric type.

Solvability of a certain problem on the plane motion of two viscous incompressible fluids with free noncompact boundaries

The solvability of a certain two-dimensional boundary-value problem for the system of the Navier-Stokes equations, describing the steady (partially common) motion of two heavy viscous incompressible capillary fluids with free noncompact boundaries, is proved.

Saint-Venant's principle on unbounded regions

SynopsisWe consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmèn-Lindelöf and Liouville.

Solvability of a problem with free noncompact boundary for a stationary navier-stokes system. I

Solvability of a problem with free noncompact boundary for a stationary navier-stokes system. I

Behavior of solutions of the Dirichlet problem for quasilinear divergent higher-order elliptic equations in unbounded domains

In this article we determine the energy apriori estimates of solutions of the Dirichlet problem in unbounded domains with noncompact boundaries. These estimates depend on the geometry of the boundary, and in the case of second-order equations, for a wide class of domains they coincide with those previously known [I, 2]. The method of obtaining these estimates is a new one, conceptually similar to the method used in [3, 4] to obtai n interior estimates of generalized solutions. On the basis of established energy estimates we prove alternative theorems of the type of the Phragmen--Lindel0f Theorem on the behavior of solutions at infinity. For linear equations, the first such theorems in terms of an interior diameter of the domain are given in [5]; in the same terms, for solutions of quasilinear divergent higher order elliptic equations, such a result is announced in [6]. We note further that for solutions of a polyharmonic equation in the case of some model domains, energy apriori estimates of the type of Saint-Venant's principle were established in [7].

Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier-Stokes system of equations in domains with noncompact boundaries

We consider the question of the possibility of approximation by solenoidal vectors from Co∞(Ω) of solenoidal vectors with finite Dirichlet integral, defined in a domain Ω, Ω⊂ℝ3, with some “exits” to infinity in the form of rotation bodies and vanishing on ∂Ω. A large class of domains is found for which such an approximation is impossible. It is shown that in these domains the formulation of the boundary problem for a stationary Navier-Stokes system of equations must include, besides the ordinary boundary conditions on ∂Ω and at infinity, the prescription of the flows of the velocity vector across certain “exits.”

Existence of solutions for the Navier-Stokes equations, having an infinite dissipation of energy, in a class of domains with noncompact boundaries

In the domain Ω ∋ ℝ3 with two exits at infinity, Ω1 = {x:∣x′∣ 2} and Ω2 = {x:0 2}, one investigates the stationary system of Navier-Stokes equations under the boundary condition . One proves existence and uniqueness theorems for the solution of this problem with an infinite Dirichlet integral, under a given flow of the velocity vector across the cross sections of the exits at infinity. As an example one considers the case whengi(t) ≡1.

Solvability of a problem on the plane motion of a viscous incompressible fluid with a free noncompact boundary

One considers the problem of the plane motion of a viscous incompressible fluid which fills partially a container V, bounded by the straight line ∑1 = {x:x2 = 0} and the contour (∂V∖∑1), consisting of two semilines l(1) = {x:x1<−1,x2 = h0} l(2) = {x:x1 = 0,x2⩾h0+1} joined by a smooth curvel(3). One assumes that the motion is due to a nonzero flow ℱ and by the motion of the lower wall ∑1 with a constant velocity R≠0. The unique solvability of this problem is proved for small R and ℱ.

Existence of axisymmetric solutions of the stationary system of Navier-Stokes equations in a class of domains with noncompact boundary

Existence of axisymmetric solutions of the stationary system of Navier-Stokes equations in a class of domains with noncompact boundary

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.

Three-dimensional solenoidal vectors

We consider the problem of the possibility of approximating solenoidal vectors from the Sobolev spaces\(\mathop W\limits^o{_\rho}{^1} (\Omega )\) by finite solenoidal vectors. The answer is positive if ΩcRn, n=2,3, is a strictly Lipschitz domain. We give examples of domains with noncompact boundaries for which such an approximation is not possible. We consider the auxiliary problem\(div \vec u = \varphi ,\vec u \in \mathop W\limits^o{_\rho}{^1} (\Omega )\) if the function τ e Lp(Ω) is given.

Scattering for Schrödinger operators in a class of domains with non-compact boundaries

Schrödinger operators in a class of domains with asymptotic cones are considered. A generalized Fourier transform representing the absolutely continuous part of the Schrödinger operator as multiplication by ¦ξ¦ 2 in the asymptotic cone is constructed. Wave operators relating the free Laplacian to Schrödinger operators are computed using the generalized Fourier transform. The wave operators relating Schrödinger operators acting in domains with the same asymptotic cone are computed and shown to be complete.

ON THE BEHAVIOR AT INFINITY OF SOLUTIONS OF SECOND ORDER ELLIPTIC EQUATIONS IN DOMAINS WITH NONCOMPACT BOUNDARY

In this paper the behavior at infinity of solutions of second order elliptic equations is studied. Here the solutions satisfy homogeneous Dirichlet conditions, Neumann conditions or periodicity conditions, in each case on the part of the boundary that belongs to some neighborhood of infinity. The authors obtain a priori estimates characterizing the behavior, as , of these solutions in domains with noncompact boundary, depending on the geometric properties of the domain and the behavior, again as , of the function on the right side of the equation. Bibliography: 13 titles.

On the solvability of boundary and initial-boundary value problems for the Navier-Stokes system in domains with noncompact boundaries

On the solvability of boundary and initial-boundary value problems for the Navier-Stokes system in domains with noncompact boundaries

The $n$-th order elliptic boundary problem for noncompact boundaries

The $n$-th order elliptic boundary problem for noncompact boundaries