I E McGillivray | Searcharxiv

arXiv2011

The spectral shift function for planar obstacle scattering at low energy

Let $H$ signify the free non-negative Laplacian on $\mathbb{R}^2$ and $H_Y$ the non-negative Dirichlet Laplacian on the complement $Y$ of a nonpolar compact subset $K$ in the plane. We derive the low-energy expansion for the Krein spectral shift function (scattering phase) for the obstacle scattering system $\{\,H_Y,\,H\,\}$ including detailed expressions for the first three coefficients. We use this to investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to $K$.

math.SP↗

arXiv2011

Riesz-type inequalities and maximum flux exchange flow

Let $D$ stand for the open unit disc in $\mathbb{R}^d$ ($d\geq 1$) and $(D,\,\mathscr{B},\,m)$ for the usual Lebesgue measure space on $D$. Let $\mathscr{H}$ stand for the real Hilbert space $L^2(D,\,m)$ with standard inner product $(\cdot,\,\cdot)$. The letter $G$ signifies the Green operator for the (non-negative) Dirichlet Laplacian $-Δ$ in $\mathscr{H}$ and $ψ$ the torsion function $G\,χ_D$. We pose the following problem. Determine the optimisers for the shape optimisation problem \[ α_t:=\sup\Big\{(Gχ_A,χ_A):\,A\subseteq D\text{is open and}(ψ,χ_A)\leq t\,\Big\} \] where the parameter $t$ lies in the range $0<t<(ψ,1)$. We answer this question in the one-dimensional case $d=1$. We apply this to a problem connected to maximum flux exchange flow in a vertical duct. We also show existence of optimisers for a relaxed version of the above variational problem and derive some symmetry properties of the solutions.

math.AP↗