Optimal estimates from below for biharmonic Green functions

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-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.Comment: 11 pages. To appear in "Proceedings of the AMS

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

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 and a large enough constant. 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. Our formula is more explicit than the result by T. Bayen, T. Lachand-Robert and \'E. Oudet, "Analytic parametrization of three-dimensional bodies of constant width" in Arch. Ration. Mech. Anal., 186 (2007), 225-249.Comment: 19 pages, 10 figure

A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

AbstractFourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u1 with u1,Δu1∈H˚1 and u2∈H2∩H˚1. We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains

When is the first eigenfunction for the clamped plate equation of fixed sign?

It is known that the first eigenfunction of the clamped plate equation, $Delta^2 varphi = lambda varphi$ in $Omega$ with $varphi=frac{partial}{partial n}varphi=0$ on $partialOmega$, is not necessarily of fixed sign. In this article, we survey the relations between domains $Omega$ and the sign of that first eigenfunction

POSITIVITY FOR THE NAVIER BILAPLACE, AN ANTI-EIGENVALUE AND AN EXPECTED LIFETIME

We address the question, for which lambda is an element of R is the boundary value problem {Delta(2)u + lambda u = f in Omega, u = Delta u = 0 on partial derivative Omega, positivity preserving, that is, f >= 0 implies u >= 0. Moreover, we consider what happens, when lambda passes the maximal value for which positivity is preserved

A Noncooperative Mixed Parabolic-Elliptic System And Positivity

Per quanto concerne la positività, i sistemi cooperativi ellittici e parabolici si comportano come le corrispondenti equazioni: una sorgente positiva implica che la soluzione è positiva. I sistemi con accoppiamento non cooperativo presentano invece un diverso comportamento. Per i sistemi ellittici non cooperativi sussiste un risultato limitato ma uniforme di positività mentre per i sistemi parabolici non cooperativi non esiste alcun risultato di positività. In questo lavoro si esaminano condizioni che assicurino la positività di un sistema intermedio di tipo misto parabolico-ellittico.Concerning positivity, cooperative elliptic and parabolic systems behave like the corresponding equations: a positive source implies that the solution is positive. Systems with a noncooperative coupling do not yield such type of behaviour. For noncoopemtive elliptic systems there is a restricted but uniform, positivity result and for the noncoopemtive parabolic system there is no positivity result at all. Here we address positivity presenting properties of an intermediate mixed parabolic-elliptic system