arXiv subjects

Guido Sweers

Publications and source records attributed to Guido Sweers.

4 recordsLinked to original sources

arXiv2023

On a formula for all sets of constant width in 3d

In the recent paper "On a formula for sets of constant width in 2D", Comm. Pure Appl. Anal. 18 (2019), 2117-2131, we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the parametrization of the boundary of bodies of constant width in 3 dimensions, depending on one function defined on S^2. Each such function gives a minimal value r_0 and for all r \ge r_0 one finds a body of constant width 2r. Moreover, we show that all bodies of constant width in 3d have such a parametrization. The last result needs a tool that we describe as 'shadow domain' and that is explained in an appendix. The construction is explicit and and offers a parametrization different from the one given by T. Bayen, T. Lachand-Robert and É. Oudet, "Analytic parametrization of three-dimensional bodies of constant width" in Arch. Ration. Mech. Anal., 186 (2007), 225-249.

math.MG↗

arXiv2020

Note on a sign-dependent regularity for the polyharmonic Dirichlet problem

A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction between the influence on the solution of the positive and the negative part of the right-hand side.

math.AP↗

arXiv2019

Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators

We study fundamental solutions of elliptic operators of order $2m\geq4$ with constant coefficients in large dimensions $n\ge 2m$, where their singularities become unbounded. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as $n\geq3$, the polyharmonic operator $(-\Delta)^m$ may no longer serve as a prototype for the general elliptic operator. It is known from examples of [V.G. Maz'ya, S. A. Nazarov, Math. Notes 39 (1986); Transl. of Mat. Zametki 39 (1986)] and [E.B. Davies, Journal Differ. Equations 135 (1997)] that in dimensions $n\ge 2m+3$ fundamental solutions of specific operators of order $2m\geq4$ may change sign near their singularities: there are ``positive'' as well as ``negative'' directions along which the fundamental solution tends to $+\infty$ and $-\infty$ respectively, when approaching its pole. In order to understand this phenomenon systematically we first show that existence of a ``positive'' direction directly follows from the ellipticity of the operator. We establish an inductive argument by space dimension which shows that sign change in some dimension implies sign change in any larger dimension for suitably constructed operators. Moreover, we deduce for $n=2m$, $n=2m+2$ and for all odd dimensions an explicit closed expression for the fundamental solution in terms of its symbol. From such formulae it becomes clear that the sign of the fundamental solution for such operators depends on the dimension. Indeed, we show that we have even sign change for a suitable operator of order $2m$ in dimension $n=2m+2$. On the other hand we show that in the dimensions $n=2m$ and $n=2m+1$ the fundamental solution of any such elliptic operator is always positive around its singularity.

math.AP↗

arXiv2011

Optimal estimates from below for biharmonic Green functions

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,γ}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.

math.AP↗