Boundary Conical Point Research Articles

Existence of the first eigenvalue of the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere

In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

The Dirichlet problem in a cone for second order elliptic quasi-linear equation with the p-Laplacian

We study the behavior near the boundary conical point of weak solutions to the Dirichlet problem for elliptic quasi-linear second-order equation with the p-Laplacian and the strong nonlinearity on the right side.

The Behavior of Solutions to the Dirichlet Problem for Second Order Elliptic Equations with Variable Nonlinearity Exponent in a Neighborhood of a Conical Boundary Point

We study the Dirichlet problem for the p-Laplacian in a conical domain with the homogeneous boundary condition on the lateral surface of a cone with vertex at the origin. We assume that the variable exponent p = p(x) is separated from 1 and ∞ and denote by Ω the intersection of the cone with the unit (n − 1)-dimensional sphere. We prove that (i) if p satisfies the Lipschitz condition and ∂Ω is of class C 2+β, then the solution to the Dirichlet problem is O(|x| λ ) in a neighborhood of the origin, where λ is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the p(0)-Laplacian and (ii) if p satisfies the Holder condition, p(0) = 2, and ∂Ω is of class C 1+β, then the solution to the Dirichlet problem is O(|x| λ0) in a neighborhood of the origin, where λ 0 is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the Laplace operator. Bibliography: 18 titles.

Behaviour of strong solutions to the degenerate oblique derivative problem for elliptic quasi-linear equations in a neighbourhood of a boundary conical point

We study the behaviour of strong solutions to the degenerate oblique derivative problem for quasi-linear second-order elliptic equations in a neighbourhood of the boundary conical point of a bounded domain.

Almgren-type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations

A monotonicity approach to the study of the asymptotic behaviour near corners of solutions to semilinear elliptic equations in domains with a conical boundary point is discussed. The presence of logarithms in the first term of the asymptotic expansion is excluded for boundary profiles sufficiently close to straight conical surfaces.

On the degenerate oblique derivative problem for elliptic second-order equation in a domain with boundary conical point

We study the behaviour of strong solutions to the degenerate oblique derivative problem for linear second-order elliptic equation in a neighbourhood of the boundary conical point of a bounded domain.

The Friedrichs-Wirtinger type inequality and its application to the transmission problem in a conical domain

We formulate the new Friedrichs-Wirtinger type inequality with the sharp constant and apply it to the investigation of the behavior of weak solutions to the transmission problem for linear elliptic divergence second order equations in a neighborhood of the boundary conical point. We establish the precise rate of decreasing of the solution. Mathematics subject classification (2010): 35J25, 35J60, 35J85, 35B65.

The transmission problem for quasi-linear elliptic second order equations in a conical domain. I, II

We investigate the behavior of weak solutions to the transmission problem for quasi-linear elliptic divergence second order equations in a neighborhood of the boundary conical point. In Part I we study the problem for quasi-linear equation with semi-linear principal part. In Part 2 we study the problem for general quasi-linear elliptic divergence second order equations.

On the Structure of asymptotics of the solution of a second-order elliptic equation in a neighborhood of an angular point

The behavior of solutions of elliptic equations in neighborhoods of angular and conical boundary points has been well studied; the asymptotics of these solutions has been constructed. In the present paper, we propose a new approach to constructing asymptotic decompositions in a neighborhood of an angular boundary point, which allows us to describe the structure of these asymptotics in a relatively simple and illustrative way.

Best possible estimates of solutions to the Robin boundary value problem for linear elliptic nondivergence second-order equations in a neighborhood of the conical point

We investigate the behavior of strong solutions to the Robin boundary value problem for linear elliptic nondivergence second-order equations in a neighborhood of the boundary conical point. We establish precise exponent of the solution decreasing rate.

Dirichlet problem for linear elliptic equations degenerating at a conical boundary point

Dirichlet problem for linear elliptic equations degenerating at a conical boundary point

Domain sensitivity analysis of the acoustic far‐field pattern

AbstractWe consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd.

The Stokes resolvent in 3D domains with conical boundary points: nonregularity in $L^p$-spaces

It is shown that solutions of the 3D Stokes resolvent problem in domains with conical boundary points, under homogeneous Dirichlet boundary conditions, do not satisfy the usual resolvent estimate in $L^p$-spaces if $p $ is close to infinity or close to $1.$

Asymptotic behaviour of solutions of the first boundary-value problem for strongly hyperbolic systems near a conical point at the boundary of the domain

An existence and uniqueness theorem for generalized solutions of the first initial-boundary-value problem for strongly hyperbolic systems in bounded domains is established. The question of estimates in Sobolev spaces of the derivatives with respect to time of the generalized solution is discussed. It is shown that the smoothness of generalized solutions with respect to time is independent of the structure of the boundary of the domain but depends on the coefficients of the right-hand side. Results on the smoothness of the generalized solution and its asymptotic behaviour in a neighbourhood of a conical boundary point are also obtained.

Local solvability and regularity results for a class of semilinear elliptic problems in nonsmooth domains

We consider a class of semilinear elliptic problems in two- and three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Frechet differentiable. Applying the local invertibility theorem we prove that the solution of the semilinear problem has the same asymptotic behaviour near the conical points as the solution of the linearized problem if the norms of the given right hand sides are small enough. Estimates for the difference between the solution of the semilinear and of the linearized problem are derived.

Estimates of generalized solutions of the Dirichlet problem for quasilinear elliptic equations of the second order in a domain with conical boundary point

We obtain a priori estimates for generalized second derivatives (in the Sobolev weighted norm) of solutions of the Dirichlet problem for the elliptic equation $$\frac{d}{{dx_i }}a_i (x,u,u_x ) + a(x,u,u_x ) = 0,x \in G,$$ in the neighborhood of a conical boundary point of the domain G. We give an example that demonstrates that the estimates obtained are almost exact.

L^p-theory for the stokes system in 3d domains with conical boundary points

L^p-theory for the stokes system in 3d domains with conical boundary points

On quasilinear elliptic equations in domains with conical boundary points.

On quasilinear elliptic equations in domains with conical boundary points.

Über das Verhalten der Lösungen der Maxwellschen Randwertaufgabe in Gebieten mit Kegelspitzen

AbstractWe study the behavior of the solution E of the Maxwell's boundary value problem ∇ × ∇ × E + λE = F, n × E|r = 0 in domains Ω which have conical boundary points. In a neighbourhood K(R) = B(a,R) ∩ Ω of a singular boundary point a the field E is expanded using a theorem of N. Weck. It e.g. turns out that the solution lies in H1(3)(K(R)) if K(R) is convex.