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Michiel van den Berg

Publications and source records attributed to Michiel van den Berg.

At least 19 recordsLinked to original sources
arXiv2026

On the gradient of the torsion function for elongated cylinders

Let $(0,L)\times E_a\subset \R^{d+1}$ denote the cylinder of length $L$, and base $E_a\subset \R^{d},$ where $E_a$ is an open ellipsoid with semi-axes $a=(a_1,...,a_d)$. Let $w_{(0,L)\times E_a}$ denote its torsion function. Heat equation tools are used to show that (i) if $E_a$ is has small eccentricity and $L$ is sufficiently large, then the maxima of $|\nabla w_{(0,L)\times E_a}|$ are located at the centres $(0,0)$ and $(L,0)$ respectively, (ii) if $E_a$ has large eccentricity and $L$ is sufficiently large, then the maxima are located on the lateral side of the cylinder.

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arXiv2025

On functionals involving the $p$-capacity and the $q$-torsional rigidity

Upper bounds are obtained for the $p$-capacity of compact sets in $\R^d$, with $d \ge 2$ and $1<p<d$. Upper and lower bounds are obtained for the product of $p$-capacity and powers of the $q$-torsional rigidity over the collection of all non-empty, open, bounded and convex sets in $\R^d$ with either a perimeter constraint, or a measure constraint, or a combination of perimeter and measure constraints. For some range of parameters we identify the ball as the unique (up to homotheties) maximiser or minimiser respectively.

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arXiv2025

Qualitative properties of the heat content

We obtain monotonicity and convexity results for the heat content of domains in Riemannian manifolds and in Euclidean space subject to various initial temperature conditions. We introduce the notion of a strictly decreasing temperature set, and show that it is a sufficient condition to ensure monotone heat content. In addition, in Euclidean space, we construct a domain and an initial condition for which the heat content is not monotone, as well as a domain and an initial condition for which the heat content is monotone but not convex.

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arXiv2025

On localisation of eigenfunctions of the Laplace operator

We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and (ii) localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions $κ$-localise in $L^2$.

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arXiv2024

On the torsion function for simply connected, open sets in $\R^2$

For an open set $\Om \subset \R^2$ let $λ(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown that there exist {$η_1>0,η_2>0$} such that { (i) $\|w_{\Om}\|_{\infty} λ(\Om)\ge 1+η_1$ for any non-empty, open, simply connected set $\Om\subset \R^2$ with $\lb(\Om) >0$, (ii) $\|w_{\Om}\|_1λ(\Om)\le {(1-η_2)}|\Om|$ for any non-empty, open, simply connected set $\Om\subset\R^2$ with finite measure $|\Om|$.

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arXiv2024

On some isoperimetric inequalities for the Newtonian capacity

Upper bounds are obtained for the Newtonian capacity of compact sets in $\R^d,\,d\ge 3$ in terms of the perimeter of the $r$-parallel neighbourhood of $K$. For compact, convex sets in $\R^d,\,d\ge 3$ with a $C^2$ boundary the Newtonian capacity is bounded from above by $(d-2)M(K)$, where $M(K)>0$ is the integral of the mean curvature over the boundary of $K$ with equality if $K$ is a ball. For compact, convex sets in $\R^d,\,d\ge 3$ with non-empty interior the Newtonian capacity is bounded from above by $\frac{(d-2)P(K)^2}{d|K|}$ with equality if $K$ is a ball. Here $P(K)$ is the perimeter of $K$ and $|K|$ is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained. An upper bound is obtained for expected Newtonian capacity of the Wiener sausage in $\R^d,\,d\ge 5$ with radius $\varepsilon$ and time length $t$.

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arXiv2021

Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to peak ratio $E(Ω)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $Ω$ with finite measure in Euclidean space $\R^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if $Ω_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises, and $E(Ω_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

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arXiv2021

Intrinsic ultracontractivity for domains in negatively curved manifolds

Let $M$ be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets $D$ in $M$ for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(D)$, and the supremum of the torsion function for $D$ are comparable with the square of the capacitary width for $D$ if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel for finite scale.

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arXiv2021

Localisation for the torsion function and the strong Hardy inequality

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.

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arXiv2020

On capacity and torsional rigidity

We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.

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arXiv2020

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.

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arXiv2020

Sign changing solutions of Poisson's equation

Let $Ω$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partialΩ,$ let $A$ be a measurable subset of $Ω$ with measure $|A|$, and let $γ\in (0,1)$. We investigate whether the solution $v_{\Om,A,γ}$ of $-Δv=γ{\bf 1}_{Ω\setminus A}-(1-γ){\bf 1}_{A}$ with $v=0$ on $\partial Ω$ 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 ${\rm essinf} v_{\Om,A,γ}\ge 0$. We show that ${\rm essinf} v_{\Om,A,γ}<0$ for any measurable set $A$, provided $|A| >γ|\Om|$. 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 $v_{\Om,A,γ}$. Surprisingly, if $\Om$ is a ball, a symmetry breaking phenomenon occurs.

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arXiv2019

On the relations between principal eigenvalue and torsional rigidity

We consider the problem of minimising or maximising the quantity $λ(Ø)T^q(Ø)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $λ(Ø)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H^1_0(Ø)$, while $T(Ø)$ is the torsional rigidity of $Ø$. The optimisation problem above is considered in the class of {\it all domains} $Ø$, in the class of {\it convex domains} $Ø$, and in the class of {\it thin domains}. The full Blaschke-Santaló diagram for $λ(Ø)$ and $T(Ø)$ is obtained in dimension one, while for higher dimensions we provide some bounds.

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arXiv2019

Heat flow from polygons

We study the heat flow from an open, bounded set $D$ in $\R^2$ with a polygonal boundary $\partial D$. The initial condition is the indicator function of $D$. A Dirichlet $0$ boundary condition has been imposed on some but not all of the edges of $\partial D$. We calculate the heat content of $D$ in $\R^2$ at $t$ up to an exponentially small remainder as $t\downarrow 0$.

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arXiv2018

On the $L^p$ norm of the torsion unction

Bounds are obtained for the $L^p$ norm of the torsion function $v_Ω$, i.e. the solution of $-Δv=1,\, v\in H_0^1(Ω),$ in terms of the Lebesgue measure of $Ω$ and the principal eigenvalue $λ_1(Ω)$ of the Dirichlet Laplacian acting in $L^2(Ω)$. We show that these bounds are sharp for $1\le p\le 2$.

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arXiv2018

Heat flow in Riemannian manifolds with non-negative Ricci curvature

Let $Ω$ be an open set in a geodesically complete, non-compact, $m$-dimen-sional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary. We study the heat flow from $Ω$ into $M-Ω$ if the initial temperature distribution is the characteristic function of $Ω$. We obtain a necessary and sufficient condition which ensures that an open set $Ω$ with infinite measure has finite heat content for all $t>0$. We also obtain upper and lower bounds for the heat content of $Ω$ in $M$. Two-sided bounds are obtained for the heat loss of $Ω$ in $M$ if the measure of $Ω$ is finite.

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arXiv2018

Heat content in non-compact Riemannian manifolds

Let $Ω$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $Ω$ has infinite measure, and if $Ω$ has finite heat content $H_Ω(T)$ for some $T>0$, then $H_Ω(t)<\infty$ for all $t>0$. Comparable two-sided bounds for $H_Ω(t)$ are obtained for such $Ω$.

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