A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

On the solvability of a transmission problem for the Laplace operator with a dynamic boundary condition on a nonregular interface

In this paper, we consider a transmission problem for the Laplace operator when an interface contains angular points and time derivatives of the unknown functions enter in the transmission condition. We prove the existence of a unique solution in the weighted Hölder classes.     

On the second eigenvalue of the Laplace operator on a spherical band

In this paper we prove that the second eigenvalue of the Laplacian for a spherical band on the unit sphere has multiplicity 2. We also show that among all spherical bands of given fixed area less than the second eigenvalue is maximized at the band which is symmetrical with respect to the equator.

     

On the lowest eigenvalue of the Laplacian with Neumann boundary condition at a small obstacle

We study the lowest eigenvalue ${\lambda }_1(\epsilon )$ of the Laplacian $-\Delta$ in a bounded domain $\Omega \subset {\mathbb{R}}^d$, $d⩾2$, from which a small compact set $K_{\epsilon }\subset B_{\epsilon }$ has been deleted, imposing Dirichlet boundary conditions along $\mathrm{\partial }\Omega$ and Neumann boundary conditions on $\mathrm{\partial }{K}_{\epsilon }$. We are mainly interested in results that require minimal regularity of $\mathrm{\partial }{K}_{\epsilon }$ expressed in terms of a Poincaré condition for the domains $\Omega ⧹{\epsilon }^{-1}{K}_{\epsilon }$. We then show that ${\lambda }_1(\epsilon )$ converges to ${\Lambda }_1$, the first Dirichlet eigenvalue of $\Omega$, as $\epsilon \to 0$. Assuming some more regularity we also obtain asymptotic bounds on ${\lambda }_1(\epsilon )-{\Lambda }_1$, for $\epsilon$ small, where we employ an idea of [Burenkov and Davies, J. Differential Equations 186 (2002) 485–508].     

Basis properties of a problem for the Laplace operator on the square with spectral parameter in a boundary condition

We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane fixed on two sides, the load being distributed along one of the free sides. We study the completeness, minimality, and basis property of the system of eigenfunctions and establish conditions guaranteeing the equiconvergence of spectral expansions in this system and in a given basis.     

A condition for nonnegativity of an operator related to the Dirichlet operator

Let L₀ be a positive definite closed linear operator with domain of definition D(L₀) dense in the Hilbert space H; let(, ₁, ₂) be the positive boundary value space of the operator L₀ such that the restriction of L ₀ ^* to ker ₂ is the Friedrichs extension of the operator L₀. We establish a test for nonnegativity of an operator T of the form Ty=L ₀ ^* y+^*(₁–C)y, y D(T)= ker(₂+), where :H and C: are respectively a compact operator and a bounded nonnegative operator.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 30–33.     

On the uniform convergence of spectral expansions in a problem for the Laplace operator on the square with spectral parameter in the boundary condition

We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane. We establish conditions ensuring the uniform convergence of spectral expansions in the selected Riesz basis and in the entire system of eigen-functions.     

Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part

We study a chemotaxis system on bounded domain in two dimensions where the formation of chemical potential is subject to the Dirichlet boundary condition. For such a system the solution is kept bounded near the boundary and hence the blowup set is composed of a finite number of interior points. If the initial total mass is 8π and the domain is close to a disc then the solution exhibits a collapse in infinite time of which movement is subject to a gradient flow associated with the Robin function.     

Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities

In this paper, the asymptotic behavior of solutions u _ε of the Poisson equation in the ε-periodically perforated domain Ω_ε ? $ {{\mathbb{R}}^n} $ , n ≥ 3, with the third nonlinear boundary condition of the form ? _ν u _ε + ε^?γσ(x, u _ε) = ε ^?γ g(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order ε^α with α > 1 and any γ. Here, all types of asymptotic behavior of solutions u _ε , corresponding to different relations between parameters α and γ, are studied.     

On the smallest eigenvalue of the Stokes operator in a domain with fine-grained random boundary

This article deals with a problem arising in localization of the principal eigenvalue (PE) of the Stokes operator under the Dirichlet condition on the fine-grained random boundary of a domain contained in a cube of size t ? 1. The random microstructure is assumed identically distributed in distinct unit cubic cells and, in essence, independent. In this setting, the asymptotic behavior of the PE as t → ∞ is deterministic: it proves possible to find nonrandom upper and lower bounds on the PE which apply with probability that converges to 1. It was proved earlier that in two dimensions the nonrandom unilateral bounds on the PE can be chosen asymptotically equivalent, which implies the convergence in probability to a nonrandom limit of the appropriately normalized PE. The present article extends this result to higher dimensions.     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. The text was submitted by the authors in English.     

On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space

We study the Laplace equation in the half-space ${\mathbb{R}_{+}^{n}}$ with a nonlinear supercritical Robin boundary condition ${\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}$ on ${\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}$ , where n ≥ 3 and λ ≥ 0. Existence of solutions ${u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}$ is obtained by means of a fixed point argument for a small data $f \in {L^{d}(\mathbb{R}^{n-1})}$ . The indexes p, q are chosen for the norm ${\Vert\cdot\Vert_{E_{pq}}}$ to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. ${x\in\mathbb{R}^{n-1}}$ . When f is radially symmetric, u is invariant under rotations around the axis {x _n  = 0}. Moreover, in a certain L ^q -norm, we show that solutions depend continuously on the parameter λ ≥ 0.     

Homogenization of the boundary value problem for the laplace operator in a perforated domain with a rapidly oscillating nonhomogeneous Robin-type condition on the boundary of holes in the critical case

The asymptotic behavior of solutions to a boundary value problem in a domain periodically perforated by small holes with a rapidly oscillating nonhomogeneous Robin-type condition on their boundaries is investigated in the case of critical parameter values.