L p estimates of boundary integral equations for some nonlinear boundary value problems

Summary Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in theH ^1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend toL ^p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that inH ^1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method inL ^p space then follow.This research was performed while the second author was visiting professor at the University of Delaware, spring 1989     

Spectral theory and iterative methods for the Maxwell system in nonsmooth domains

We study spectral properties of boundary integral operators which naturally arise in the study of the Maxwell system of equations in a Lipschitz domain Ω ? ?³. By employing Rellich‐type identities we show that the spectrum of the magnetic dipole boundary integral operator (composed with an appropriate projection) acting on L²(?Ω) lies in the exterior of a hyperbola whose shape depends only on the Lipschitz constant of Ω. These spectral theory results are then used to construct generalized Neumann series solutions for boundary value problems associated with the Maxwell system and to study their rates of convergence (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

We prove that the standard second‐kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign‐definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so‐called combined potential, or combined field, operator.) This coercivity result yields k‐explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical‐asymptotic methods developed recently). Coercivity also gives k‐explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise‐polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation. © 2015 Wiley Periodicals, Inc.     

On Singular Integral Operators with Bounded Measurable Coefficients

Necessary and sufficient analytical conditions are determined for a singular integral operator of the form aP + bQ with bounded measurable coefficients to be a ?-operator on L_p(Γ) for all 1 < p < ∞. where Γ is a closed Lyapunov curve.     

Divergence L 2-Coercivity Inequalities

This paper derives sharp L ²-coercivity inequalities for the divergence operator on bounded Lipschitz regions in ? ⁿ . They hold for fields in H(div,Ω) that are orthogonal to N(div). The optimal constants in the inequality are defined by a variational principle and are identified as the least eigenvalue of a nonstandard boundary value problem for a linear biharmonic type operator. The dependence of the optimal constant under dilations of the region is described and a generalization that involves weighted surface integrals is also proved. When n = 2, this also yields a similar coercivity result for the curl operator.     

Regularized Combined Field Integral Equations

Summary. Many boundary integral equations for exterior boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an L²()-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods.     

Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs

Suppose L is a second-order elliptic differential operator in ℝ^d and D is a bounded, smooth domain in ℝ^d. Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent: (α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D; (β) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and (γ) Γ is a removable boundary singularity for the equation Lu = u^α in D; that is, if u ≥ 0 and Lu = u^α in D and if u = 0 on ∂D \ Γ, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder 𝒬 = ℝ₊ × D. Let Γ be a compact set on the lateral boundary of 𝒬. We show that the following three properties are equivalent: (a) Γ is 𝒢-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in 𝒬; (b) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and (c) Γ is a removable lateral singularity for the equation u˙ + Lu = u^α in 𝒬; that is, if u ≥ 0 and u˙ + Lu = u^α in 𝒬 and if u = 0 on ∂𝒬 \ Γ and on {∞} × D, then u = 0. © 1998 John Wiley & Sons, Inc.     

Modified boundary integral formulations for the Helmholtz equation

In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations.     

On the Characterization of the Intermediate Space in Generalized Factorizations

The study of systems of singular integral equations of CAUCHY type, of TOEPLITZ and WIENER -HOPF operators leads to the question of existence and representation of generalized factorizations of matrix functions Φ in [L^P(Γ, σ)]^m. This yields a corresponding factorization of the basic multiplication or translation invariant operator A = A_CA₊, respectively, which can be seen as a splitting of a bounded into unbounded operators. The present paper is devoted to the study of the nature of the induced intermediate space Z = im A₊ =im A^?1, in particular, for Γ = ? and Φ ε ζ[C^β(?)]^{m × m} which is of special interest in certain applications. As we know, this implies detailed results about the structure and the explicit asymptotic behaviour of solutions of boundary and transmission problems near singular points with a relation also to eigenvalue problems which result from the classical series expansion approach or from the Mellin symbol calculus (see [13]).     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 1: The radon integral operators

The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non-smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double-layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above-mentioned jump relations are solvable by the Fredholm theory; the double- (for Dirichlet) and the single- (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems.     

Approximation mittels Linearkombinationen von Fundamentallösungen elliptischer Differentialoperatoren

Let Ω ? Rⁿ be a bounded domain with the smooth boundary Γ, L an elliptic differential operator of order 2m with constant coefficients, E a fundamental solution of L and B = (B₁, …, B_m) a normal system of boundary operators on Γ. Furtehr let (yk)Γ₁ ? Rⁿ/Ω be a sequence of points and D_j(Γ) (j = 1, …, m) suitable function spaces over Γ (e.g. C^s-spaces or SOBOLEV spaces). It is investigated, under which conditions on the sequence (yk)^x₁ ? Rⁿ/Ω the set      

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Superdiffusions and removable singularities for quasilinear partial differential equations

To every second-order elliptic differential operator L and to every number α ϵ (1, 2] there is a corresponding measure-valued Markov process X called the (L, α)-superdiffusion. Suppose that Γ is a closed set in R^d. It is known that the following three statements are equivalent: (α) the range of X does not hit Γ; (β) if u ≥ 0 and Lu = u^α in R^d\Γ, then u = 0 (in other words, Γ is a removable singularity for all solutions of equation Lu = u^α); (γ) Cap_2,α′(Γ) = 0 where 1/α + 1/α′ = 1 and Cap_γ,q is the so-called Bessel capacity. The equivalence of (β) and (γ) was established by Baras and Pierre in 1984 and the equivalence of (α) and (β) was proved by Dynkin in 1991. In this paper, we consider sets Γ on the boundary ∂D of a bounded domain D and we establish (assuming that ∂D is smooth) the equivalence of the following three properties: (a) the range of X in D does not hit Γ (b) if u ≥ 0 and Lu = u^α in D, and if u → 0 as x → α ϵ ∂D\Γ, then u = 0; (c) Cap^∂_2/α,α′(Γ) = 0 where Cap^∂_γ-qis the Bessel capacity on ∂D. This implies positive answers to two conjectures posed by Dynkin a few years ago. (The conjectures have already been confirmed for α = 2 and L = Δ in a recent paper of Le Gall.) By using a combination of probabilistic and analytic arguments we not only prove the equivalence of (a)-(c) but also give a new, simplified proof of the equivalence of (α)-(γ). The paper consists of an Introduction (Section 1) and two parts, probabilistic (Sections 2 and 3) and analytic (Sections 4 and 5), that can be read independently. An important probabilistic lemma, stated in the Introduction, is proved in the Appendix. © 1996 John Wiley & Sons, Inc.     

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ? be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ?[Γ] is not a weakly factorial domain.     

Some properties of Van Koch’s curves

We investigate the properties of an integral operator T with a Cauchy kernel. The operator acts from L ^∞(Γ, μ), where Γ is a Van Koch curve, to the space of functions ? → ?. We prove that the range of T is nontrivial and lies in the space AC(Γ) of functions continuous in ?, vanishing at ∞, and analytic outside Γ. We also show that T is injective and compact while satisfying some special functional equation. These results may be regarded as a natural continuation of our research on the problem of AC-removability of quasiconformal curves whose solution was announced in [1] for the first time and supplemented later with some other properties of Van Koch’s curves [2, 3]. In this paper the problem is discussed in a more general setting and, in particular, all important details lacking in [1] are given. Some open problems are formulated.     

Fractional powers of operators corresponding to coercive problems in Lipschitz domains

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ? 2, and let L be a second-order matrix strongly elliptic operator in Ω written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation Lu = f, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well. We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary Γ = ?Ω or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem.     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Fredholm Theory for a Class of Singular Integral Operators with Carleman Shift and Unbounded Coefficients

A criterion for the Fredholmness of singular integral operators with Carleman shift in L_P(Γ) is obtained, where Γ is either the unit circle or the real line. The approach allows to consider unbounded coefficients in a class related to that of quasicontinuous functions. Applications to Wiener-Hopf-Hankel type operators and operators with linear fractional Carleman shift on IR are included.     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

The trace class conjecture without the K-finiteness assumption

Let G be the group of real points of a reductive algebraic ℚ-group satisfying the same assumptions as in [5], Chapter I, and let Γ be a discrete subgroup of G. Let R_Γ be the right regular representation of G in L²(Γ\G). We prove in this Note that, for any integrable rapidly decreasing function ƒ on G, the restriction of R_Γ(ƒ) to the discrete spectrum of R_Γ is a trace class operator.