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We consider functions , where is a smooth bounded domain, and is an integer. For all , such that , we prove that with , where is a smooth positive function which coincides with dist near , and denotes any partial differential operator of order .
Nonlinear Schrödinger equations (NLS) with strongly singular potential on a bounded domain are considered. If and , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here is excluded because is not equal to , where is nonnegative and selfadjoint in . On the other hand, if is a smooth and bounded domain with , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....
We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided . To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.
The estimate is shown to hold if and only if is elliptic and canceling. Here is a homogeneous linear differential operator of order on from a vector space to a vector space . The operator is defined to be canceling if . 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...
This paper introduces two novel nonlinear anisotropic Picone identities with variable exponents that expand upon the traditional identity used for the ordinary Laplace equation. Additionally, the research explores potential applications of these findings in anisotropic Sobolev spaces featuring variable exponents.
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.
We consider the 3D quantum BBGKY hierarchy which corresponds to the -particle Schrödinger equation. We assume the pair interaction is . For the interaction parameter , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the 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 in [28]. This allows, in the case , for the application of the Klainerman–Machedon uniqueness theorem...
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...
Given a measurable set of positive measure, it is not difficult to show that if and only if is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If is small, is close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between and its convex hull in terms of .
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 in a smooth bounded domain . In particular, we obtain new and bounds for the extremal solution when the domain is strictly convex. More precisely, we prove that if and if .
A family of linear homogeneous 4th order elliptic differential operators with real constant coefficients, and bounded nonsmooth convex domains are constructed in so that the have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces .
Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to . 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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