Michiel van Den Berg
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Michiel van Den Berg has a total of 38 co-authors. Here are the top co-authors by number of shared publications:
- Dorin Bucur
Université Savoie Mont Blanc, Le-Bourget-du-Lac, France
9 shared publications - Thomas Kappeler
Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich Switzerland
6 shared publications - Giuseppe Buttazzo
Dipartimento di Matematica, Università di Pisa, Italy
5 shared publications - Peter Gilkey
Mathematics Department, University of Oregon, Eugene, USA
5 shared publications Katie Gittins
Department of Mathematical Sciences Durham University Durham UK
4 shared publications- Katie Gittins
Department of Mathematical Sciences Mathematical Sciences and Computer Science Building Durham University Upper Mountjoy Campus Stockton Road
4 shared publications
Latest Publications
- Open Access
- 25 July 2026
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 36 (9) , 298
On the Gradient of the Torsion Function for Elongated Cylinders
Let $$(0,L)\times E_a\subset \mathbb {R}^{d+1}$$ denote the cylinder of length L , and base $$E_a\subset \mathbb {R}^{d},$$ 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 $$E_a$$ 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.- 25 July 2026
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 36 (9) , 298
Open AccessOn the Gradient of the Torsion Function for Elongated Cylinders
Let $$(0,L)\times E_a\subset \mathbb {R}^{d+1}$$ denote the cylinder of length L , and base $$E_a\subset \mathbb {R}^{d},$$ 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 $$E_a$$ 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. - 8 September 2025
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 64 (8) , 1-21
On functionals involving the p-capacity and the q-torsional rigidity
- 8 September 2025
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 64 (8) , 1-21
On functionals involving the p-capacity and the q-torsional rigidity
- Open Access
- 12 May 2025
- Published by Wiley in Bulletin of the London Mathematical Society
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.
- 12 May 2025
- Published by Wiley in Bulletin of the London Mathematical Society
Open AccessQualitative 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.
- 27 January 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in EMS Surveys in Mathematical Sciences
- Vol. 12 (1) , 1-25
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 \kappa -localise in L^{2} .
- 27 January 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in EMS Surveys in Mathematical Sciences
- Vol. 12 (1) , 1-25
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 \kappa -localise in L^{2} .
- Open Access
- 23 November 2024
- Published by Springer Nature in Potential Analysis
- Vol. 63 (1) , 413-429
On the Torsion Function for Simply Connected, Open Sets in $\mathbb {R}^2$For an open set $$\Omega \subset \mathbb {R}^2$$ let $$\lambda (\Omega )$$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $$L^2(\Omega )$$ . Let $$w_\Omega $$ be the torsion function for $$\Omega $$ , and let $$\Vert .\Vert _p$$ denote the $$L^p$$ norm. It is shown that there exist $$\eta _1>0,\eta _2>0$$ such that (i) $$\Vert w_{\Omega }\Vert _{\infty } \lambda (\Omega )\ge 1+\eta _1$$ for any non-empty, open, simply connected set $$\Omega \subset \mathbb {R}^2$$ with $$\lambda (\Omega ) >0$$ , (ii) $$\Vert w_{\Omega }\Vert _1\lambda (\Omega )\le {(1-\eta _2)}|\Omega |$$ for any non-empty, open, simply connected set $$\Omega \subset \mathbb {R}^2$$ with finite measure $$|\Omega |$$ .- 23 November 2024
- Published by Springer Nature in Potential Analysis
- Vol. 63 (1) , 413-429
Open AccessOn the Torsion Function for Simply Connected, Open Sets in $\mathbb {R}^2$For an open set $$\Omega \subset \mathbb {R}^2$$ let $$\lambda (\Omega )$$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $$L^2(\Omega )$$ . Let $$w_\Omega $$ be the torsion function for $$\Omega $$ , and let $$\Vert .\Vert _p$$ denote the $$L^p$$ norm. It is shown that there exist $$\eta _1>0,\eta _2>0$$ such that (i) $$\Vert w_{\Omega }\Vert _{\infty } \lambda (\Omega )\ge 1+\eta _1$$ for any non-empty, open, simply connected set $$\Omega \subset \mathbb {R}^2$$ with $$\lambda (\Omega ) >0$$ , (ii) $$\Vert w_{\Omega }\Vert _1\lambda (\Omega )\le {(1-\eta _2)}|\Omega |$$ for any non-empty, open, simply connected set $$\Omega \subset \mathbb {R}^2$$ with finite measure $$|\Omega |$$ . - 19 June 2024
- Published by World Scientific Pub Co Pte Ltd in Communications in Contemporary Mathematics
- Vol. 27 (05)
On some isoperimetric inequalities for the Newtonian capacity
Upper bounds are obtained for the Newtonian capacity of compact sets in [ ] in terms of the perimeter of the [ ]-parallel neighborhood of [ ]. For compact, convex sets in [ ] with a [ ] boundary the Newtonian capacity is bounded from above by [ ], where [ ] is the integral of the mean curvature over the boundary of [ ] with equality if [ ] is a ball. For compact, convex sets in [ ] with non-empty interior the Newtonian capacity is bounded from above by [ ] with equality if [ ] is a ball. Here, [ ] is the perimeter of [ ] and [ ] 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 [ ] with radius [ ] and time length [ ].
- 19 June 2024
- Published by World Scientific Pub Co Pte Ltd in Communications in Contemporary Mathematics
- Vol. 27 (05)
On some isoperimetric inequalities for the Newtonian capacity
Upper bounds are obtained for the Newtonian capacity of compact sets in [ ] in terms of the perimeter of the [ ]-parallel neighborhood of [ ]. For compact, convex sets in [ ] with a [ ] boundary the Newtonian capacity is bounded from above by [ ], where [ ] is the integral of the mean curvature over the boundary of [ ] with equality if [ ] is a ball. For compact, convex sets in [ ] with non-empty interior the Newtonian capacity is bounded from above by [ ] with equality if [ ] is a ball. Here, [ ] is the perimeter of [ ] and [ ] 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 [ ] with radius [ ] and time length [ ].
- Open Access
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 $\kappa$-localise in $L^2$.Open AccessOn 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 $\kappa$-localise in $L^2$. - Open Access
On efficiency and localisation for the torsion function
We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$ of finite Lebesgue measure $0<|\Omega|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.Open AccessOn efficiency and localisation for the torsion function
We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$ of finite Lebesgue measure $0<|\Omega|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction. - Open Access
- 1 December 2022
- Published by Springer Nature in Potential Analysis
- Vol. 57 (4) , 571-600
On Efficiency and Localisation for the Torsion Function
- 1 December 2022
- Published by Springer Nature in Potential Analysis
- Vol. 57 (4) , 571-600
Open AccessOn Efficiency and Localisation for the Torsion Function
- Open Access
On some variational problems involving capacity, torsional rigidity, perimeter and measure
We investigate the existence of a maximiser among open, bounded, convex sets in $\R^d,\,d\ge 3$ for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if $d=2$), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.Open AccessOn some variational problems involving capacity, torsional rigidity, perimeter and measure
We investigate the existence of a maximiser among open, bounded, convex sets in $\R^d,\,d\ge 3$ for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if $d=2$), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.