Sign changing solutions of Poisson's equation

Let $\Omega$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partial\Omega,$ let $A$ be a measurable subset of $\Omega$ with measure $|A|$, and let $\gamma \in (0,1)$. We investigate whether the solution v_{\Om,A,\gamma} of $-\Delta v=\gamma{\bf 1}_{\Omega \setminus A}-(1-\gamma){\bf 1}_{A}$ with $v=0$ on $\partial \Omega$ changes sign. Bounds are obtained for $|A|$ in terms of geometric characteristics of \Om (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness of the boundary) such that {\rm essinf} v_{\Om,A,\gamma}\ge 0. We show that {\rm essinf} v_{\Om,A,\gamma}<0 for any measurable set $A$, provided |A| >\gamma |\Om|. This value is sharp. We also study the shape optimisation problem of the optimal location of $A$ (with prescribed measure) which minimises the essential infimum of v_{\Om,A,\gamma}. Surprisingly, if \Om is a ball, a symmetry breaking phenomenon occurs.Comment: 27 pages, 2 figures, various minor typos have been correcte

Torsional rigidity for regions with a Brownian boundary

Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $\Omega \subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in \Omega$ of the expected lifetime in $\Omega$ of a Brownian motion starting at $x$. In this paper we consider $\Omega = T^m \backslash \beta[0,t]$, the complement of the path $\beta[0,t]$ of an independent Brownian motion up to time $t$. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \backslash \beta [0,t]$ whose inradius is comparable to the largest inradius, while for $m=3$ most of $T^3 \backslash \beta [0,t]$ contributes. A similar result holds for $m \geq 4$ after the Brownian path is replaced by a shrinking Wiener sausage $W_{r(t)}[0,t]$ of radius $r(t)=o(t^{-1/(m-2)})$, provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of $\beta[0,t]$ in $R^3$ and $W_1[0,t]$ in $R^m$, $m \geq 4$, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on $T^m$, which has received a lot of attention in the literature in past years.Comment: 26 pages, 1 figur

Spectral Bounds for the Torsion Function

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in \Leb^2(\Omega), denoted by $\lambda(\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound \|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge 1 is sharp: for $m\in\{2,3,...\}$, and any $\epsilon>0$ we construct an open, bounded and connected set $\Omega_{\epsilon}\subset \R^m$ such that \|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon. An upper bound for $v_{\Omega}$ is obtained for planar, convex sets in Euclidean space $M=\R^2$, which is sharp in the limit of elongation. For a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary it is shown that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in \Leb^2(\Omega) is bounded away from $0$.Comment: 13 pages, 1 figur

Heat trace asymptotics with singular weight functions II

We study the weighted heat trace asymptotics of an operator of Laplace type with mixed boundary conditions where the weight function exhibits radial blowup. We give formulas for the first three boundary terms in the expansion in terms of geometrical data

On the torsion function with mixed boundary conditions

Let $D$ be a non-empty open subset of $\R^m,\,m\ge 2$, with boundary $\partial D$, with finite Lebesgue measure $|D|$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading asymptotic behaviour as $\varepsilon\downarrow 0$ of the $L^{\infty}$ norm of the torsion function with a Neumann boundary condition on $\partial D$, and a Dirichlet boundary condition on $\partial (\varepsilon K)$, in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that $D\setminus K$ is a non-trap domain.Comment: 9 page