Van Schaftingen Research Articles

New estimates for a class of non-local approximations of the total variation

We consider a class of non-local functionals recently introduced by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung, which offers a novel way to characterize functions with bounded variation.We give a positive answer to an open question related to these functionals in the case of functions with bounded variation. Specifically, we prove that in this case the liminf of these functionals can be estimated from below by a linear combination in which the three terms that sum up to the total variation (namely the total variation of the absolutely continuous part, of the jump part and of the Cantor part) appear with different coefficients. We prove also that this estimate is optimal in the case where the Cantor part vanishes, and we compute the precise value of the limit in this specific scenario.In the proof we start by showing the results in dimension one by relying on some measure theoretic arguments in order to identify sufficiently many disjoint rectangles in which the difference quotient can be estimated, and then we extend them to higher dimension by a classical sectioning argument.

Brezis–Van Schaftingen–Yung formula in Orlicz spaces

In [13], Brezis–Van Schaftingen–Yung discovered new expression of the Sobolev semi-norm by replacing the Lp(R2N) norm in the Gagliardo Ws,p(RN) semi-norm with weak Lp(R2N) quasi-norm and setting s=1. In this paper we prove the same formula for functions in Orlicz–Sobolev spaces, in which the role of the power function t↦tp is performed by more general finite valued Young functions.

Asymptotic decay of solutions for sublinear fractional Choquard equations

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))f(u)onRNwhere s∈(0,1), N≥2, α∈(0,N), μ>0, Iα denotes the Riesz potential and F(t)=∫0tf(τ)dτ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than ∼1|x|N+2s. The result is new even for homogeneous functions f(u)=|u|r−2u, r∈[N+αN,2), and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case s=1 in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new “s-sublinear” threshold that we detect on the growth of f. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.

Brezis–Seeger–Van Schaftingen–Yung-type characterization of homogeneous ball Banach Sobolev spaces and its applications

Let [Formula: see text] and [Formula: see text] be a ball Banach function space satisfying some extra mild assumptions. Assume that [Formula: see text] or [Formula: see text] is an [Formula: see text]-domain for some [Formula: see text]. In this paper, the authors prove that a function [Formula: see text] belongs to the homogeneous ball Banach Sobolev space [Formula: see text] if and only if [Formula: see text] and [Formula: see text] where [Formula: see text] is related to [Formula: see text]. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when [Formula: see text] with [Formula: see text], while it is still new even when [Formula: see text] with [Formula: see text]. The novelty of this paper exists in that, to establish the characterization of [Formula: see text], the authors provide a machinery via using the generalized Brezis–Seeger–Van Schaftingen–Yung formula on [Formula: see text], the extension theorem on [Formula: see text], the Bourgain–Brezis–Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space [Formula: see text], and the method of extrapolation to overcome those difficulties caused by that [Formula: see text] might be neither the rotation invariance nor the translation invariance and that the norm of [Formula: see text] has no explicit expression.

Generalized Brezis–Seeger–Van Schaftingen–Yung formulae and their applications in ball Banach Sobolev spaces

Generalized Brezis–Seeger–Van Schaftingen–Yung formulae and their applications in ball Banach Sobolev spaces

Quantitative estimates for fractional Sobolev mappings in rational homotopy groups

Let N⊂RM be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of πN(N) is represented by a homomorphism deg:πN(N)→R.For maps f:SN→N we give a quantitative estimate of its rational homotopy group element deg([f])∈R in terms of its fractional Sobolev-norm. That is, we show that for all β∈(β0(deg),1], |deg([f])|≤C(deg)[f]Wβ,Nβ(SN)N+L(deg)β.Here C(deg)>0, L(deg)∈N, β0(deg)∈(0,1) are computable from the rational homotopy group represented by deg. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov’s integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière.

Semi-classical Analysis Around Local Maxima and Saddle Points for Degenerate Nonlinear Choquard Equations

We study existence of semi-classical states for the nonlinear Choquard equation: -ε2Δv+V(x)v=1εα(Iα∗F(v))f(v)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\varepsilon ^2\\Delta v+ V(x)v = {1\\over \\varepsilon ^\\alpha }(I_\\alpha *F(v))f(v) \\quad \ ext {in}\\ {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 3, alpha in (0,N), I_alpha (x)=A_alpha /|{x}|^{N-alpha } is the Riesz potential, Fin C^1({mathbb {R}},{mathbb {R}}), F'(s)=f(s) and varepsilon >0 is a small parameter. We develop a new variational approach, in which our deformation flow is generated through a flow in an augmented space to get a suitable compactness property and to reflect the properties of the potential. Furthermore our flow keeps the size of the tails of the function small and it enables us to find a critical point without introducing a penalization term. We show the existence of a family of solutions concentrating to a local maximum or a saddle point of V(x)in C^N({mathbb {R}}^N,{mathbb {R}}) under general conditions on F(s). Our results extend the results by Moroz and Van Schaftingen (Calc Var Partial Differ Equ 52:199–235, 2015) for local minima (see also Cingolani and Tanaka (Rev Mat Iberoam 35(6):1885–1924, 2019)) and Wei and Winter (J Math Phys 50:012905, 2009) for non-degenerate critical points of the potential.

Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki–Lions type conditions

We prove the existence and asymptotic behavior of solutions to the following problem: − Δ u + V ( x ) u − g ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where g ( x ) : = μ | x | is called the Coulomb potential, g ( x ) : = β | x | 2 is called the Hardy potential (the inverse-square potential). μ , β > 0 are parameters, I α : R N ⟶ R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ ∈ ( 0 , α ( N − 2 ) / 2 ( α + 1 ) ) and β ∈ ( 0 , α ( N − 2 ) 2 / 4 ( 2 + α ) ), under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at s = 1 and s = 0

In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo s-seminorm of a function f as s→1− and s→0+, which extend the results due to Bourgain-Brezis-Mironescu (BBM) and Maz'ya-Shaposhnikova (MS) respectively. Recently, Brezis, Van Schaftingen and Yung provided a different approach by replacing the strong Lp norm in the Gagliardo s-seminorm by the weak Lp quasinorm. They characterized the case for s=1 that complements the BBM formula. The corresponding MS formula for s=0 was later established by Yung and the first author. In this paper, we follow the approach of Brezis-Van Schaftingen-Yung and show the anisotropic versions of s=1 and s=0. Our result generalizes the work by Brezis, Van Schaftingen, Yung and the first author and complements the work by Ludwig.

Stein-Weiss inequality in 𝐿¹ norm for vector fields

In this work, we investigate the limit case p = 1 p=1 of the classical Stein-Weiss inequality for the Riesz potential. Our main result is a characterization of this inequality for a special class of vector fields associated to cocanceling operators introduced by Van Schaftingen [J. Eur. Math. Soc. (JEMS) 15 (2013), pp. 877–921]. As application, we recover and improve some fractional inequalities associated to vector fields.

Brezis–Van Schaftingen–Yung formulae in ball Banach function spaces with applications to fractional Sobolev and Gagliardo–Nirenberg inequalities

Let X be a ball Banach function space on $${\mathbb R}^n$$ . In this article, under some mild assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any locally integrable function f with $$\Vert \,|\nabla f|\,\Vert _{X}<\infty $$ , $$\begin{aligned} \sup _{\lambda \in (0,\infty )}\lambda \left\| \left| \left\{ y\in {{\mathbb {R}}}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\right\} \right| ^{\frac{1}{q}} \right\| _X\sim \Vert \,|\nabla f|\,\Vert _X \end{aligned}$$ with the positive equivalence constants independent of f, where the index $$q\in (0,\infty )$$ is related to X and $$|\{y\in {\mathbb R}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\}|$$ is the Lebesgue measure of the set under consideration. In particular, when $$X:=L^p({\mathbb R}^n)$$ with $$p\in [1,\infty )$$ , the above formulae hold true for any given $$q\in (0,\infty )$$ with $$n(\frac{1}{p}-\frac{1}{q})<1$$ , which when $$q=p$$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo–Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, Orlicz-slice (generalized amalgam) spaces, and weak Morrey spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on $$X'$$ of the Hardy–Littlewood maximal operator, and the exquisite geometry of $${\mathbb {R}}^n.$$

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space W˙1,p. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodeckiĭ spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best n-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces

Let (X,ρ,μ) be a metric measure space of homogeneous type, and let Cc⁎(X) denote the set of all Lipschitz functions f on X such that limr→0+⁡supy∈B(⋅,r)⁡|f(⋅)−f(y)|/r converges uniformly to a compactly supported continuous function Lip f on X. Let p∈[1,∞) and let f∈Cc⁎(X) be such that the pair (f,Lipf) satisfies a certain Poincaré inequality. Assume that α∈R∖{0} if p∈(1,∞), and α∈(0,∞) if p=1. Under these assumptions, the authors prove the following result which extends a recent formula of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung on finite dimensional Euclidean spaces:supλ∈(0,∞)⁡λp∬Dλ[U(x,y)]α−1dμ(x)dμ(y)∼∫X[Lipf(x)]pdμ(x), whereDλ:={(x,y)∈X×X:|f(x)−f(y)|>λρ(x,y)[U(x,y)]αp} for any λ∈(0,∞), V(x,y):=μ(B(x,ρ(x,y))) and U(x,y):=min⁡{V(x,y),V(y,x)} for any x,y∈X, and the positive constants of equivalence are independent of f. A similar result remains true if p∈(1,∞) and f lies in a wider class of functions in the Hajłasz–Sobolev space. As an application, the authors establish new fractional Sobolev and Gagliardo–Nirenberg inequalities on X. These results are applicable to many classical examples, such as Euclidean spaces, with weighted Lebesgue measure, and complete finite dimensional Riemannian manifolds, with non-negative Ricci curvature.

Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities

We prove existence of infinitely many solutions u in H^1_r({mathbb {R}}^N) for the nonlinear Choquard equation -Δu+μu=(Iα∗F(u))f(u)inRN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} - {\\varDelta } u + \\mu u =(I_\\alpha *F(u)) f(u) \\quad \\hbox {in}\\ {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 3, alpha in (0,N), I_alpha (x) := frac{{varGamma }(frac{N-alpha }{2})}{{varGamma }(frac{alpha }{2}) pi ^{N/2} 2^alpha } frac{1}{|x|^{N- alpha }}, x in {mathbb {R}}^N setminus {0} is the Riesz potential, and F is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: mu is a fixed positive constant or mu is unknown and the L^2-norm of the solution is prescribed, i.e. int _{{mathbb {R}}^N} |u|^2 =m>0. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions (Arch Ration Mech Anal 82(4):347–375, 1983), we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results due to Moroz and Van Schaftingen (Trans Am Math Soc 367(9):6557–6579, 2015).

Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung

We provide answers to some questions raised in a recent work by H. Brezis, J. Van Schaftingen and P.-L. Yung [2,3] concerning the Gagliardo semi-norm |u|Ws,q computed at s=1, when the strong Lq is replaced by weak Lq. In particular, we address generalization of the results in [2,3] for a general domain and non-smooth functions.

Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

AbstractWe are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent−Δu+V(x)u=Iα∗[Q(x)|u|N+αN]Q(x)|u|αN−1u,x∈RN.$$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N. \end{array}$$By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent,Proceedings of the Royal Society of Edinburgh, Section A Mathematics,150(2020), 1377–1400].

A new formula for the Lp norm

Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the W˙1,p norm of a function in Cc∞(RN). The formula was obtained by replacing the Lp(R2N) norm in the Gagliardo semi-norm for W˙s,p(RN) with a weak-Lp(R2N) quasi-norm and setting s=1. This provides a characterization of such W˙1,p norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula [1]. In this paper, we obtain an analog for the case s=0. In particular, we present a new formula for the Lp norm of any function in Lp(RN), which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on Lp(RN), which complements a formula by Maz′ya and Shaposhnikova [12]. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space F2s,p(RN) (i.e. the Bessel potential space (I−Δ)−s/2Lp(RN)), as well as its homogeneous counterpart F˙2s,p(RN), for s∈(0,1), p∈(1,∞).

On the asymptotic decay of the Schrödinger–Newton ground state

The asymptotics of the ground state u(r) of the Schrödinger–Newton equation in R3 was determined by V. Moroz and J. van Schaftingen to be u(r)∼Ae−r/r1−‖u‖22/8π for some A>0, in units in which the ground state energy is −1. They left open the value of ‖u‖22, the squared L2 norm of u. Here it is rigorously shown that 21/33π2⩽‖u‖22⩽27/2π3/2. It is reported that numerically ‖u‖22≈14.03π, revealing that the monomial prefactor of e−r increases with r in a concave manner. Asymptotic results are proposed for the Schrödinger–Newton equation with external ∼−K/r potential, and for the related Hartree equation of a bosonic atom or ion.

Sobolev martingales

We introduce new spaces of martingales that behave like L_1 -based Sobolev spaces on \mathbb R^d . We prove a martingale analog of Van Schaftingen’s theorem and give sharp estimates on the lower Hausdorff dimension of the terminal distributions of these martingales. We also provide martingale analogs of trace theorems for Sobolev functions.