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We study the dependence of the continuity constants for the regularized Poincare and Bogovskiu{i} integral operators acting on differential forms defined on a domain $Omega$ of $mathbb{R}^n$. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) $L^2(Omega,Lambda^ell)$ to $H^1(Omega,Lambda^{ell-1})$, $ell in {1, ldots, n}$. For domains $Omega$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(Omega)/diam(B)$. A program on how to develop estimates for higher order Sobolev norms is presented. The results are extended to certain classes of unions of star shaped domains.
This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement te
We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(Gamma)$ to itself.
Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,eta)$ are elements of $C^{r}_{*}S^{m}_{1,delta}$ classes that have limited regularity in the $x$ variable.
We obtain new local smoothing estimates for the Euclidean wave equation on $mathbb{R}^{n}$, by replacing the space of initial data by a Hardy space for Fourier integral operators. This improves the bounds in the local smoothing conjecture for $pgeq 2