The Dirichlet,Neumann and Mixed Boundary Value Problems of the Theory of Elasticity in n-Dimensional Domains with Boundaries Containing Closed Cuspidal Edges

The Dirichlet, Neumann and mixed problems of statics of the theory of elasticity for anisotropic homogeneous media are studied in n-dimensional (n ≧ 2) domains with boundaries containing closed cuspidal edges. Theorems of the existence and uniqueness of solutions of these problems in the Besov and Bessel potential spaces are obtained. Smoothness of solutions in a closed domain occupied by elastic medium is investigated.     

On the Stokes and Navier-Stokes System for Domains with Noncompact Boundary in Lq-spaces

We consider the L^q-theory of weak solutions of the Stokes and Navier-Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IRⁿ, and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut-off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q =n/2, n ≥ 3, to estimate the nonlinearity.     

The Nonclassical Boundary-contact Problems of Elasticity for Homogeneous Anisotropic Media

The existence and uniqueness of solutions of the nonclassical boundary-contact problems (i.e., problems with a contact on some part of the boundaries) of elasticity for homogeneous anisotropic media are investigated in Besov and Bessel potential spaces using methods of potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

The hp-Version of the Boundary Element Method in ℝ3 The Basic Approximation Results

This paper deals with the basic approximation properties of the h–p version of the boundary element method (BEM) in ℝ³. We extend the results on the exponential convergence of the h–p version of the boundary element method on geometric meshes from problems in polygonal domains to problems in polyhedral domains. In 2D elliptic boundary value problems the solutions have only corner singularities whereas in 3D problems they contain additional edge and corner-edge singularities. The solutions of the corresponding boundary integral equations inherit those singularities. The detailed investigations in our analysis take care of the various types of those singularities. While edge singularities can be analysed using standard one-dimensional approximation results the corner-edge singularities demand a new analysis. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Sommerfeld Half-plane Problems with Higher Order Boundary Conditions

Sommerfeld-type diffraction problems for a half-plane with arbitrary n-th order generalized impedance boundary conditions arc examined in a Sobolev space setting. The corresponding boundary-transmission problems for the two dimensional Helmholtz equation are shown to be well-posed in a family of Sobolev spaces with finite energy norms, through a reduction to equivalent systems of boundary integral equations of Wiener-Hopf type in [L₂⁺ (IR)]². Formulas for the solutions as well as the so-called edge conditions arc obtained for any n, by explicit canonical generalized factorization of the presymbols of the associated Wiener-Hopf operators.     

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \mathbb Rⁿ{\mathbb R^n} . The solutions take values in ordered Banach spaces E, e.g. E=\mathbb R^N{E=\mathbb R^N} ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.     

A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn,p (D)

We discuss the construction of a polyanalytic function Φ of order n on a simple bounded domain D. The function satisfies n prescribed generalized Riemann-Hilbert boundary conditions on the boundary ?D and n generalized jump conditions on a simple closed smooth contour γ contained in D. The boundary conditions are transformed into n classical Riemann-Hilbert problems and the n jump conditions into n Riemann problems of conjugation for some 2n holomorphic functions. These transformed problems are solved using the standard methods from the literature.     

Boundary Value Problems with Compatible Boundary Conditions

If Y is a subset of the space ℝⁿ × ℝⁿ, we call a pair of continuous functions U, V Y-compatible, if they map the space ℝⁿ into itself and satisfy Ux · Vy ≥ 0, for all (x, y) ∈ Y with x · y ≥ 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P _δ is obtained. Then passing to the limits as δ tends to zero the so-obtained accumulation points are solutions of the problem.     

Tug-of-War and Infinity Laplace Equation with Vanishing Neumann Boundary Condition

We study a version of the stochastic “tug-of-war” game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.     

The boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of the boundaries

The existence and uniqueness of solutions of the boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of their boundaries are investigated in the Besov and Bessel potential classes using the methods of the potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

Boundary integral equations and maximum modulus estimates for the Stokes system

A maximum modulus estimate for the Stokes system in bounded domains of ℝⁿ (n ≥ 2) is established via methods of hydrodynamical potential theory. The method is based on the unique solvability of the boundary integral equations' system resulting from the double layer potential ansatz together with a projection onto the normal field on the boundary. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient _n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.     

A new technique of integral representations in ?n

A new technique of integral representations in ℂ ⁿ , which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the ∂-equations on strictly pseudoconvex domains in ℂ ⁿ are obtained. These new formulas are simpler than the classical ones, especially the solutions of the ∂-equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in ℂ ⁿ so that all corresponding formulas are simplified.     

Boundary value problems for the Laplacian in convex and semiconvex domains

We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in Rn, when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel-Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition).     

Boundary integral equations for mixed Dirichlet,Neumann and transmission problems

We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.     

Traveling Wave Solutions to the Free Boundary Incompressible Navier-Stokes Equations

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n\ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n\ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well-known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

On singularities of solution of Maxwell's equations in axisymmetric domains with conical points

Boundary value problems (BVP) in three‐dimensional axisymmetric domains can be treated more efficiently by partial Fourier analysis. Partial Fourier analysis is applied to time‐harmonic Maxwell's equations in three‐dimensional axisymmetric domains with conical points on the rotation axis thereby reducing the three dimensional BVP to an infinite sequence of 2D BVPs on the plane meridian domain Ω_a?? of . The regularity of the solutions u _n (n∈?₀:={0, 1, 2,…}) of the two dimensional BVPs is investigated and it is proved that the asymptotic behaviour of the solutions u _n near an angular point on the rotation axis can be characterized by singularity functions related to the solutions of some associated Legendre equations. By means of numerical experiments, it is shown that the solutions u _n for n∈?₀\{1} belong to the Sobolev space H² irrespective of the size of the solid angle at the conical point. However, the regularity of the coefficient u ₁ depends on the size of the solid angle at the conical point. The singular solutions of the three dimensional BVP are obtained by Fourier synthesis. Copyright © 2006 John Wiley & Sons, Ltd.     

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lamé equations in three‐dimensional axisymmetric domains subjected to non‐axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement û , the body force f̂ ϵ (L₂)³ and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three‐dimensional boundary value problem to an infinite sequence of two‐dimensional boundary value problems, whose solutions û _n (n = 0, 1, 2,…) are the Fourier coefficients of û . This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients û _n are proved, which are important for further numerical treatment, e.g. by the finite‐element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two‐dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients û _n and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.     

Boundary-contact problems for domains with edge singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.     

On the Continuous Dependence of the Exterior Dirichlet Problem on the Boundary

We consider a sequence of exterior domains D_j,j∈ℕ₀, and assume that the boundaries ∂D_j converge to ∂D₀ with respect to the Hausdorff distance. We investigate solutions to the exterior Dirichlet problem for the Laplace equation and for the Helmholtz equation in these domains. Assuming convergence of the boundary data and D_j⊂D₀, j∈ℕ, then, by essentially using the method of Perron, we show that the solutions in the domains D_j converge to the solution in the domain D₀ with respect to the maximum norm. We prove the same result in case that the requirement D_j⊂D₀,j∈ℕ, is replaced by an equicontinuity property of all barrier functions to all boundary points. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 707–716 (1997)