The Neumann problem for the Helmholtz equation in a starlike planar domain

The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.     

On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball

Some necessary and sufficient solvability conditions are obtained for the nonhomogeneous Neumann problem for a polyharmonic equation in the unit ball.     

On the arithmetic triangle arising from the solvability conditions for the Neumann problem

We study the arithmetic triangle arising from the solvability conditions for the Neumann problem for the polyharmonic equation in the unit ball. Recurrence relations for the elements of this triangle are obtained.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

On the number of interior peak solutions for a singularly perturbed Neumann problem

We consider the following singularly perturbed Neumann problem: where Δ = Σ ?²/?x is the Laplace operator, ? > 0 is a constant, Ω is a bounded, smooth domain in ?^N with its unit outward normal ν, and f is superlinear and subcritical. A typical f is f(u) = u^p where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N ? 2) when N ≥ 3. We show that there exists an ?₀ > 0 such that for 0 < ? < ?₀ and for each integer K bounded by where α_{N, Ω, f} is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for α_{N, Ω, f} is also given.) As a consequence, we obtain that for ? sufficiently small, there exists at least [α_{N, Ωf}/?^N (|ln ?|)^N] number of solutions. Moreover, for each m ∈ (0, N) there exist solutions with energies in the order of ?^N?m. © 2006 Wiley Periodicals, Inc.     

On the spectrum of the Steklov problem in a domain with a peak

We consider the spectral Steklov problem in a domain with a peak on the boundary. It is shown that the spectrum on the real nonnegative semi-axis can be either discrete or continuous depending on the sharpness of the exponent.     

On the spectrum of the Robin problem in a domain with a peak

A formally self-adjoint Robin-Laplace problem in a peak-shaped domain is considered. The associated quadratic form is not semi-bounded, which is proved to lead to a pathological structure of the spectrum of the corresponding operator. Namely, the residual spectrum of the operator itself and the point spectrum of its adjoint cover the whole complex plane. The operator is not self-adjoint, and the (discrete) spectrum of any of its self-adjoint extensions is not semi-bounded.     

The Neumann problem for the planar Stokes system

The Neumann problem for the Stokes system is studied on bounded and unbounded domains with Ljapunov boundary (i.e. of class ${{\mathcal C}^{1,\alpha }}$ ) in the plane. We construct a solution of this problem in the form of appropriate potentials and reduce the problem to an integral equation systems on the boundary of the domain. We determine a necessary and sufficient condition for the solvability of the problem. Then we study the direct integral equation method and prove that a solution of the corresponding integral equation can be obtained by the successive approximation.     

On the solvability problem for equations with a single coefficient

It is proved that the general solvability problem for equations in a free group is polynomially reducible to the solvability problem for equations of the formw(x ₁, ...,x _n)=g, whereg is a coefficient, i.e., an element of the group, andw(x ₁, ...,x _n) is a group word in the alphabet of unknowns. We prove the NP-completeness of the solvability problem in a free semigroup for equations of the formw(x ₁,...,x _n)=g, wherew(x ₁,...,x _n) is a semigroup word in the alphabet of unknowns andg is an element of a free semigroup. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 832–845, June, 1996. I wish to express my deep gratitude to S. I. Adyan and A. A. Razborov for the discussion of the present paper and for valuable remarks concerning the exposition. This research was partially supported by the International Science Foundation under grant MUV000.     

On a local in time solvability of the Neumann problem of quasilinear hyperbolic parabolic coupled systems

We prove a local in time existence theorem of classical solutions to some coupled system of quasilinear hyperbolic equations and quasilinear parabolic equations with Neumann boundary condition. This coupled system contains a non-linear thermoelastic equation as an important physical example.     

On a problem of Neumann

A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains many Fano subplanes. The number of planes of order less than $n$ previously known to contain a Fano subplane was $O(logn)$, whereas the number of planes of order less than $n$ that our theorem applies to is not bounded above by any polynomial in $n$.     

On a boundary-value problem solvability condition

We prove a sufficient condition for the solvability of a two-point problem.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 247–250, February, 1973.     

The Neumann problem for a mixed-type equation in a rectangular domain

We establish a uniqueness criterion for the Neumann problem for a mixed-type equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We construct a solution as the sum of a series of eigenfunctions of the corresponding Sturm-Liouville problem.     

On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.     

Neumann problem for a quasilinear elliptic equation in a varying domain

We investigate the Neumann problem for a nonlinear elliptic operator of Leray–Lions type in ${\Omega }^{(s)}=\Omega \backslash F^{(s)}$, $s=1,2,\dots$, where Ω is a domain in ${\mathbf{R}}^n$$(n?3)$, $F^{(s)}$ is a closed set located in the neighborhood of a $(n?1)$-dimensional manifold Γ lying inside Ω. We study the asymptotic behavior of $u^{(s)}$ as $s\to \infty$, when the set $F^{(s)}$ tends to Γ. To cite this article: M. Sango, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.