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A remarkably interesting, simple P.D.E.

Daniel W. Stroock (2006)

Banach Center Publications

In this note, I will summarize and make a couple of small additions to some results which I obtained earlier with David Williams in [1]. Williams and I hope to expand and refine these additions in a future paper based on work that is still in process.

A strong maximum principle for the Paneitz operator and a non-local flow for the Q -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

In this paper we consider Riemannian manifolds ( M n , g ) of dimension n ≥ 5 , with semi-positive Q -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive Q -curvature. Modifying the test function construction of Esposito-Robert, we show...

A survey of results on nonlinear Venttsel problems

Darya E. Apushkinskaya, Alexander I. Nazarov (2000)

Applications of Mathematics

We review the recent results for boundary value problems with boundary conditions given by second-order integral-differential operators. Particular attention has been paid to nonlinear problems (without integral terms in the boundary conditions) for elliptic and parabolic equations. For these problems we formulate some statements concerning a priori estimates and the existence theorems in Sobolev and Hölder spaces.

Absence de principe du maximum pour certaines équations paraboliques complexes

Pascal Auscher, Thierry Coulhon, Philippe Tchamitchian (1996)

Colloquium Mathematicae

Le but de cette note est de montrer que le principe du maximum, même dans une version affaiblie, n’est pas vérifıé pour la classe des opérateurs paraboliques du type d / d t + L , où L est un opérateur différentiel elliptique d’ordre 2 sous forme divergence à coefficients complexes mesurables et bornés en dimension supérieure ou égale à 5. Le principe de démonstration repose sur un résultat abstrait de la théorie des semi-groupes permettant d’utiliser le contre-exemple présenté dans [MNP] à la régularité des...

An existence result for an interior electromagnetic casting problem

Mohammed Barkatou, Diaraf Seck, Idrissa Ly (2006)

Open Mathematics

This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C 2. Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem.

An Hadamard maximum principle for the biplacian on hyperbolic manifolds

Håkan Hedenmalm (1999)

Journées équations aux dérivées partielles

We prove the existence of a maximum principle for operators of the type Δ ω - 1 Δ , for weights ω with log ω subharmonic. It is associated with certain simply connected subdomains that are produced by a Hele-Shaw flow emanating from a given point in the domain. For constant weight, these are the circular disks in the domain. The principle is equivalent to the following statement. THEOREM. Suppose ω is logarithmically subharmonic on the unit disk, and that the weight times area measure is a reproducing measure...

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