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A Hardy type inequality for W 0 m , 1 ( Ω ) functions

Hernán Castro, Juan Dávila, Hui Wang (2013)

Journal of the European Mathematical Society

We consider functions u W 0 m , 1 ( Ω ) , where Ω N is a smooth bounded domain, and m 2 is an integer. For all j 0 , 1 k m - 1 , such that 1 j + k m , we prove that i u ( x ) d ( x ) m - j - k W 0 k , 1 ( Ω ) with k ( i u ( x ) d ( x ) m - j - k ) L 1 ( Ω ) C u W m , 1 ( Ω ) , where d is a smooth positive function which coincides with dist ( x , Ω ) near Ω , and l denotes any partial differential operator of order l .

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

Nonlinear Schrödinger equations (NLS) a with strongly singular potential a | x | - 2 on a bounded domain Ω are considered. If Ω = N and a > - ( N - 2 ) 2 / 4 , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here a = - ( N - 2 ) 2 / 4 is excluded because D ( P a ( N ) 1 / 2 ) is not equal to H 1 ( N ) , where P a ( N ) : = - Δ - ( N - 2 ) 2 / ( 4 | x | 2 ) is nonnegative and selfadjoint in L 2 ( N ) . On the other hand, if Ω is a smooth and bounded domain with 0 Ω , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

The estimate D k - 1 u L n / ( n - 1 ) A ( D ) u L 1 is shown to hold if and only if A ( D ) is elliptic and canceling. Here A ( D ) is a homogeneous linear differential operator A ( D ) of order k on n from a vector space V to a vector space E . The operator A ( D ) is defined to be canceling if ξ n { 0 } A ( ξ ) [ V ] = { 0 } . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...

Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds

Emmanuel Russ, Yannick Sire (2011)

Studia Mathematica

Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

We consider the 3D quantum BBGKY hierarchy which corresponds to the N -particle Schrödinger equation. We assume the pair interaction is N 3 β 1 V ( B β ) . For the interaction parameter β ( 0 , 2 / 3 ) , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the N limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for β ( 0 , 3 / 5 ) in [28]. This allows, in the case β ( 0 , 3 / 5 ) , for the application of the Klainerman–Machedon uniqueness theorem...

On the local Cauchy problem for first order partial differential functional equations

Elżbieta Puźniakowska-Gałuch (2010)

Annales Polonici Mathematici

A theorem on the existence of weak solutions of the Cauchy problem for first order functional differential equations defined on the Haar pyramid is proved. The initial problem is transformed into a system of functional integral equations for the unknown function and for its partial derivatives with respect to spatial variables. The method of bicharacteristics and integral inequalities are applied. Differential equations with deviated variables and differential integral equations can be obtained...

Quantitative stability for sumsets in n

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

Given a measurable set A n of positive measure, it is not difficult to show that | A + A | = | 2 A | if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If ( | A + A | - | 2 A | ) / | A | is small, is A close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between A and its convex hull in terms of ( | A + A | - | 2 A | ) / | A | .

Regularity of stable solutions of p -Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchón (2015)

Journal of the European Mathematical Society

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of Δ p u = g ( u ) in a smooth bounded domain Ω n . In particular, we obtain new L r and W 1 , r bounds for the extremal solution u when the domain is strictly convex. More precisely, we prove that u L ( Ω ) if n p + 2 and u L n p n - p - 2 ( Ω ) W 0 1 , p ( Ω ) if n > p + 2 .

Trudinger–Moser inequality on the whole plane with the exact growth condition

Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi (2015)

Journal of the European Mathematical Society

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function....

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