Bengt Ove Turesson

Nonlinear potential theory and weighted Sobolev spaces

The book systematically develops nonlinear potential theory and the Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincare inequalities, Ma ...

An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

Abstract We present a modification of the alternating iterative method, which was introduced by Kozlov and Maz’ya, for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The reason for this modification is that the standard alternating iterative algorithm does not always converge for the Cauchy problem for the Helmholtz equation. The method is then implemented numerically using the finite difference method.

An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation

In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by ...

Riesz Potential Equations in Local Lp-spaces.

We consider the following equation for the Riesz potential of order one: Uniqueness is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove an existence result and derive an asymptotic formula for solutions near the origin. Our analysis is carried out in local L p -spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted L p -spaces and homogenous Sobolev spaces.

A fixed point theorem in locally convex spaces

For a locally convex space with the topology given by a family {p(┬; α)} α ∈ ω of seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set. We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu, Under some additional assumptions, one of which is the existence of a fixed point for the operator, we prove that there exists a fixed point of. For a class of elements satisfyingKn(p)u;┬))(α) → 0 asn → ∞, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.

Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

Abstract The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in ( H 1 / 2 ) ∗ spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well.

Single layer potentials on surfaces with small Lipschitz constants

This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi on R-N, w ...

Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation

Abstract The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann–Dirichlet iterations by the Robin–Dirichlet ones which repairs the convergence for less than the first Dirichlet–Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin–Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.