Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization of $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

Bernoulli’s Problem for the Infinity-Laplacian Near a Set with Positive Reach

We consider the exterior as well as the interior free-boundary Bernoulli problem associated with the infinity-Laplacian under a non-autonomous boundary condition. Recall that the Bernoulli problem involves two domains: one is given, the other is unknown. Concerning the exterior problem we assume that the given domain has a positive reach, and prove an existence and uniqueness result together with an explicit representation of the solution. Concerning the interior problem, we obtain a similar result under the assumption that the complement of the given domain has a positive reach. In particular, for the interior problem we show that uniqueness holds in contrast to the usual problem associated to the Laplace operator.

Acoustical Boundary Elements: Theory and Virtual Experiments

This paper presents an overview of basic concepts, features and difficulties of the boundary element method (BEM) and examples of its application to exterior and interior problems. The basic concepts of the BEM are explained firstly, and different methods for treating the non-uniqueness problem are described. The application of the BEM to half-space problems is feasible by considering a Green's Function that satisfies the boundary condition on the infinite plane. As a special interior problem, the sound field in an ultrasonic homogenizer is computed. A combination of the BEM and the finite element method (FEM) for treating the problem of acoustic-structure interaction is also described. Finally, variants of the BEM are presented, which can be applied to problems arising in flow acoustics.