- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 16:20:06
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=7&iss=4&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2005
4
On the Newton partially flat minimal resistance body type problems
M.
Comte
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Jesús Ildefonso
Díaz
Universidad Complutense de Madrid, MADRID, SPAIN
stationary points, minimal resistance body problems, Newton problem
We study the flat region of stationary points of the functional $\int_{\Omega }F(\left\vert \nabla u(x)\right\vert )dx$ under the constraint $u\leq M,$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^{2}$. Here $F(s)$ is a function which is concave for $s$ small and convex for $s$ large, and $M>0$ is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when $\Omega$ is a ball. We analyze also some other qualitative properties. Moreover, we show the uniqueness of a radial solution minimizing the above mentioned functional. Finally, we consider nonsymmetric domains $\Omega$ and provide sufficient conditions which insure that a stationary solution has a flat part.
Calculus of variations and optimal control; optimization
Partial differential equations
General
395
411
10.4171/JEMS/33
http://www.ems-ph.org/doi/10.4171/JEMS/33
Moser-Trudinger and logarithmic HLS inequalities for systems
Itai
Shafrir
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Gershon
Wolansky
Technion - Israel Institute of Technology, HAIFA, ISRAEL
We prove several optimal Moser-Trudinger and logarithmic Hardy-Littlewood-Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S2$, on a bounded domain $\Omega\subset\R2$ and on all of $\R2$. In some cases we also address the question of existence of minimizers.
Partial differential equations
General
413
448
10.4171/JEMS/34
http://www.ems-ph.org/doi/10.4171/JEMS/34
Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity
Olivier
Rey
École Polytechnique, PALAISEAU CEDEX, FRANCE
Juncheng
Wei
University of British Columbia, VANCOUVER, CANADA
semilinear elliptic Neumann problems, critical Sobolev exponent, blow-up
We show that the critical nonlinear elliptic Neumann problem \[ \Delta u -\mu u + u^{7/3} = 0 \ \ \mbox{in} \ \Om, \ \ u >0 \ \mbox{in} \ \Om \ \mbox{and} \ \frac{ \partial u}{\partial \nu} = 0 \ \ \mbox{on} \ \partial \Om\] where $\Om$ is a bounded and smooth domain in $\R^5$, has arbitrarily many solutions, provided that $\mu>0$ is small enough. More precisely, for any positive integer $K$, there exists $\mu_K >0$ such that for $0
Partial differential equations
General
449
476
10.4171/JEMS/35
http://www.ems-ph.org/doi/10.4171/JEMS/35
On polynomials and surfaces of variously positive links
Alexander
Stoimenow
University of Tokyo, TOKYO, JAPAN
positive link, quasipositive link, almost positive link, almost alternating link, Alexander polynomial, Jones polynomial, fiber surface, ribbon genus
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is $1$, with a similar relation for links. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property of canonical Seifert surfaces associated to almost positive (and almost alternating) link diagrams.
Manifolds and cell complexes
General
477
509
10.4171/JEMS/36
http://www.ems-ph.org/doi/10.4171/JEMS/36