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Register a new userOn flows associated to Sobolev vector fields in Wiener spaces: an approach `a la DiPerna-Lions
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In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.
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In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions vanishing Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schr{o}dinger equations, taking into account unbounded or bounded potentials.
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Mazya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss work concerning the sharp constant in the Hardy-Sobolev-Mazya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Mazya inequality for bi-Laplacian in the upper half space of dimension five coincides with the Sobolev constant.
For an integer $rge0$, we prove the $r$th order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on $mathbb{R}^n$. Certain differential operators $A^{(m,r,l)} (0le lle r)$ on the sphere $mathbb{S}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.
This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $Omega$ with respect to the norm: $$|f|_{QH^{1,p}(v,mu;Omega)} = |f|_{L^p_v(Omega)} + | abla f|_{mathcal{L}^p_Q(mu;Omega)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $Omega$. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition of the form $$w(x)|xi|^p leq left(xicdot Q(x)xiright)^{p/2}leq tau(x)|xi|^p$$ for a pair of $p$-admissible weights $wleq tau$ in $Omega$. We also give explicit examples demonstrating the sharpness of our hypotheses.
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.
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