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Advances in Mathematics

Volume 371, 16 September 2020, 107246
Advances in Mathematics

Characterization of the traces on the boundary of functions in magnetic Sobolev spaces

https://doi.org/10.1016/j.aim.2020.107246Get rights and content

Abstract

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field A is differentiable and its exterior derivative corresponding to the magnetic field dA is bounded. In particular, we prove that, for d1 and p>1, the trace of the magnetic Sobolev space WA1,p(R+d+1) is exactly WA11/p,p(Rd) where A(x)=(A1,,Ad)(x,0) for xRd with the convention A=(A1,,Ad+1) when AC1(R+d+1,Rd+1). We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.

Introduction

The first-order magnetic Sobolev space WA1,p(Ω) on a given open set ΩRd+1 with d1 is defined, for a given exponent p[1,+), a vector field AC1(Ω,Rd+1), as [3], [4], [6], [11], [16], [19], [29], [31, §1.1]WA1,p(Ω){UWloc1,1(Ω,C);UWA1,p(Ω)pΩ|U|p+|AU|p<+}, where the weak covariant gradient AU associated with A of UWloc1,1(Ω,C) is defined as2AUU+iAU in Ω.

Magnetic Sobolev spaces arise naturally for p=2 and d=2 (corresponding to ΩR3) in quantum mechanics in the presence of a magnetic field described through its magnetic vector potential AC1(Ω,R3); the function U:ΩC is then a wave-function and the integral in (1.1) is the quadratic form associated to the quantum mechanical Hamiltonian of a particle in a magnetic field (see, e.g., [14, Chapter 16], [17, Chapter XV]). In physical models, the only observable quantities are the magnetic field B=×AdAC(Ω,2R3) and the probability density |U|2. Here and in what follows, dA denotes the exterior derivative of A; for this, we consider A as a differential form in C1(Ω,1Rd+1). The prevalent role of the magnetic field and of the probability density is reflected by the gauge invariance invariance of the model: the invariance of the Hamiltonian quadratic form defined by the right-hand side (1.1) under a change of variables AA+Φ and UeiΦU, for any phase shift ΦC1(Ω,R), see, e.g., [19, chapter 7]. Geometrically, the invariant quantity idA is the curvature of the associated U(1)–connection (see for example [32, Chapter 11]).

Magnetic Sobolev spaces WA1,p(Ω) generalize classical Sobolev spaces W1,p(Ω), in which A0, defined byW1,p(Ω){ULp(Ω);UW1,p(Ω)pΩ|U|p+|U|p<+}. For 0<s<1 and 1p<+, the fractional Sobolev (Sobolev–Slobodeckiĭ) space is defined asWs,p(Ω){uLp(Ω,C);uWs,p(Ω)puLp(Ω)p+|u|Ws,p(Ω)p<+}, where the Gagliardo seminorm |u|Ws,p(Ω) of the measurable function u:ΩC is given by|u|Ws,p(Ω)pΩ×Ω|u(y)u(x)|p|yx|d+spdxdy. When the set Ω is bounded and its boundary is of class C1, or Ω=R+d+1{(x,t)Rd×R;t>0}, and when p>1, the trace theory is well known since Gagliardo's pioneering work [13] (see also [9, §10.17–10.18 and Proposition 17.1], [22], [30], [36]). The trace linear operator Tr defined byTr:C1(Ω¯)C1(Ω)UTr=U|Ω, satisfies, for some positive constant Cp,Ω and for every UC1(Ω), the estimateTrUW11/p,p(Ω)Cp,ΩUW1,p(Ω), and therefore, the linear operator Tr extends to a bounded linear map from the Sobolev space W1,p(Ω) into fractional Sobolev space W11/p,p(Ω). Conversely, there exists a bounded linear operator Ext:W11/p,p(Ω)W1,p(Ω) such that for any uW11/p,p(Ω),Tr(Extu)=u on Ω and ExtuW1,p(Ω)Cp,ΩuW11/p,p(Ω), for some positive constant Cp,Ω independent of u. In particular, the map Tr is surjective. Consequently, the image under the trace operator of the space W1,p(Ω) is exactly the space W11/p,p(Ω). The space of traces can also be described as the real interpolations space (Lp(Rd),W1,p(Rd))11/p,p in the framework of interpolation of Banach spaces [20, Théorème VI.2.1].

The trace theory for WA1,p(Ω) can be easily derived from the one of W1,p(Ω) when the magnetic potential A is bounded. In fact, by the triangle inequality,|AuLp(Ω)uLp(Ω)|AL(Ω)uLp(Ω), it follows that WA1,p(Ω)=W1,p(Ω) in this case. Hence the trace space of WA1,p(Ω) is the space W11/p,p(Ω) as well. The situation becomes more delicate when Ω=R+d+1 and A is not assumed to be bounded but its total derivative DA or, even more physically, its exterior derivative dA is bounded. This type of assumption on A appears naturally in many problems in physics for which A is linear in simple settings. Moreover, even when A is bounded, the quantitative bounds resulting (1.3) depend on the uniform norm AL(Ω) which is not gauge-invariant; it would be desirable to have estimates depending rather on dA. To our knowledge, a characterization of the trace of WA1,p(R+d+1) is not known under such assumption on A. The goal of this work is to give a complete answer to this question. Besides its own interest concerning boundary values in problems of calculus of variations and partial differential equations, this is closely related to classes of fractional magnetic problems motivated by relativistic magnetic quantum physical models [15] that have been studied recently [1], [2], [5], [8], [12], [18], [25], [26], [27], [28], [33], [37].

Given 0<s<1, 1p<+, and AC(Rd,Rd), we define, for any measurable function u:RdC, the magnetic Gagliardo semi-norm|u|WAs,p(Rd,C)pRd×Rd|eiIA(x,y)u(y)u(x)|p|yx|d+spdxdy, where the potential IA:Rd×RdR is defined for each x,yRd byIA(x,y)01A((1t)x+ty)(yx)dt. Here and in what follows ⋅ denotes the usual complex scalar product defined for v=(v1,,vd) and w=(w1,,wd)Cd by vw=v1w¯1++vdw¯d. For a vector field A=(A1,,Ad+1)C(R+d+1,Rd+1), we will consider A's parallel component on the boundary A:RdRd defined for each xRd byA(x)(A1,,Ad)(x,0).

Our first main result is

Theorem 1.1

Let d1 and 1<p<+. There exists a positive constant Cd,p depending only on d and p such that if AC1(R+d+1,Rd+1) and dAL(R+d+1)β, then

  • (i)

    for each UCc1(R+d+1,C),|U(,0)|WA11/p,p(Rd)+β1212pU(,0)Lp(Rd)Cd,p(AULp(R+d+1)+β12ULp(R+d+1)),

  • (ii)

    for each uCc1(Rd,C), there exists UCc1(R+d+1) depending linearly on u and depending on β such that U(x,0)=u(x) in Rd andAULp(R+d+1)+β12ULp(R+d+1)Cd,p(|u|WA11/p,p(Rd)+β1212puLp(Rd)).

The conclusions of Theorem 1.1 are gauge-invariant: all the functional norms are gauge-invariant and the constants only depend through β which is an upper bound of the norm dAL(R+d+1) of the magnetic field on the half-space.

As a consequence of Theorem 1.1, by a standard density argument (see Section 4), we obtain the following characterization of the trace of the space WA1,p(R+d+1):

Theorem 1.2

Let d1 and 1<p<+. Assume that AC1(R+d+1,Rd+1) and that dAL(R+d+1,2Rd+1). The trace mappingTr:WA1,p(R+d+1,C)WA11/p,p(Rd,C)U(x,xd+1)U(x,0) is linear and continuous. There also exists a linear continuous mappingExt:WA11/p,p(Rd,C)WA1,p(R+d+1,C) such that TrExtR+d+1 is the identity on WA11/p,p(Rd). Moreover, the corresponding estimates of Theorem 1.1 with u=TrU and U=Extu are valid.

In the case where the magnetic field dA is constant, we obtain the following improvements:

Theorem 1.3

Let d1 and 1<p<+. Assume that AC1(R+d+1,Rd+1) and that dA is constant. We have, with uTrU,|u|WA11/p,p(Rd)Cd,pAULp(R+d+1). Moreover, for every uWA1,p(Rd), there exists UWA1,p(R+d+1) such that TrU=u andAULp(R+d+1)+dA12ULp(R+d+1)Cd,p|u|WA11/p,p(Rd).

We later show that the space WAs,p(Rd) with 0<s<1 and p1 is the trace space of the space WA,1(1s)p1,p(Rd) whose definition is given in (4.1); moreover, the corresponding estimates hold (see Theorem 4.3, Theorem 5.4).

We establish similar estimates for a smooth bounded domain ΩRd+1 and a magnetic potential AC1(Ω¯,Rd+1). It is worth noting that the trace theory in this setting is known as in the case A0. Nevertheless, our estimates (Proposition 6.4 and Proposition 6.5) are gauge-invariant, and sharpen estimates in the semi-classical limit (Proposition 6.6).

As a consequence of the trace theorems, we derive a characterization of the space WAs,p(Rd,C) as an interpolation space (Theorem 7.2). We also observe that the characterization of traces is also independent on the side of the hyperplane from which the trace is taken or to which the extension is made (this fact is not completely trivial, see Remark 8.2). Consequently, the trace theorem provides an extension theorem from a half-space to the whole space (Theorem 8.1).

In an appendix, we show that under the assumption that some derivatives of the vector field A are bounded, our magnetic fractional spaces have equivalent norms to other families of fractional spaces defined in the literature (Proposition A.1).

We now describe briefly the idea of the proof of the trace theory. The proofs of the trace estimates and of the construction of the extension are based on standard strategies that go back to Gagliardo's seminal work [13]. Concerning Theorem 1.1 and its variants (Proposition 2.1, Proposition 3.1), the key point of our analysis lies on the observation that A defined in Rd by (1.5) encodes the information the trace space of WA1,p(R+d+1), and on an appropriate extension formula given in (3.1). The proof of the trace estimates also involves Stokes theorems (Lemma 2.2) and a simple, useful, observation given in Lemma 2.3. Concerning Theorem 1.3 and its variants (Theorem 5.4), the new part is the trace estimates (see, e.g., (1.6)). To this end, the Stokes formula and an averaging argument are used while taking into account the fact dA is constant. The proof for a domain Ω uses the results in the half space via local charts.

Section snippets

Trace estimate for bounded magnetic fields

In this section, we prove the following trace estimate on the boundary of the half-space with a bounded magnetic field, which covers (i) in Theorem 1.1.

Proposition 2.1

Let d1, 0<s<1, and 1p<+. There exists a positive constant Cd,s,p depending only on d, s and p such that if AC1(R+d+1,Rd+1), if dAL(R+d+1)β and if UCc1(R+d+1,C), then|U(,0)|WAs,p(Rd)pCd,s,pRd×(0,+)|AU(z,t)|p+βp2|U(z,t)|pt1(1s)pdzdt andU(,0)Lp(Rd)pCd,s,p(Rd×(0,+)|AU(z,t)|pt1(1s)pdtdz)1s(Rd×(0,+)|U(z,t)|pt1(1s)pdtdz)

Extension to the half-space

In this section, we prove the following extension result which implies (ii) of Theorem 1.1.

Proposition 3.1

Let d1, 0<s<1, and 1p<+. There exists a positive constant Cd,s,p depending only on d, s, and p such that for every AC1(R+d+1,Rd+1) with dAL(R+d+1)β and for any uCc1(Rd,C) with compact support, one can find UCc1(R+d+1) depending linearly on u and depending on β such that U(x,0)=u(x) in Rd,Rd×(0,+)|AU(x,t)|pt1(1s)pdxdtCd,s,p(|u|WAs,p(Rd)p+βsp2uLp(Rd)p) andRd×(0,+)|U(z,t)|pt1(1s)pdxdt

Characterizations of trace spaces

For γR, we define the weighted spaceWA,γ1,p(R+d+1){uWloc1,1(R+d+1,C);UWA,γ1,p(Ω)<+}, whereUWA,γ1,p(Ω)pRd×(0,+)(|U(x,t)|p+|AU(x,t)|p)tγdxdt It is standard to check that the space WA,γ1,p(R+d+1) is complete. We also have the following density result:

Lemma 4.1

Let 1p<+, γR and AC(R+d+1,Rd+1). If 1<γ<p1, then the space Cc(R+d+1) is dense in WA,γ1,p(R+d+1).

Proof

The proof of the completeness is standard. For the density of smooth maps, we first observe that if χCc(Rd+1), 0χ1, and χ=1 on B1

Constant magnetic field on the half-space

We begin with an improvement of Proposition 2.1 in the case where the magnetic field dA is constant.

Proposition 5.1

Let d1, 0<s<1, and 1p<+. There exists a constant Cd,s,p>0 such that if AC1(R+d+1,Rd+1) with constant dA, UCc1(R+d+1) and u=U(,0), then|u|WAs,p(Rd)p+dAsp2Rd|u|pCd,s,pRd×(0,+)|AU(z,t)|pt1(1s)pdzdt.

The first ingredient of the proof Proposition 5.1 is the following lemma:

Lemma 5.2

Let d1, 0<s<1, and 1p<+. For every λ>0, there exists a positive constant Cd,s,p,λ such that, for every AC1(R+d

Trace and extension on domains

In this section, we consider the trace problem on a domain ΩRd+1 of class C1 with estimates depending only on the magnetic field dA. We first develop the tools to work with a magnetic derivative on the boundary ∂Ω via local charts. Let WRd+1 be an open set, ψC1(W,Rd+1), and A:ψ(W)Rd+1. The pull-back ψA of A by ψ is defined for each xW by(ψA)(x)Dψ(x)A(ψ(x)), where Dψ(x) is the adjoint of Dψ(x). We first recall the following elementary result whose proof follows from the chain rule

Interpolation of magnetic spaces

We define for every p[1,+) and γ(0,+), the functional space [34, Definition 1.8.1/1]WA,γ1,p(Rd)={U:(0,+)(WA1,p(Rd)+Lp(Rd));U is weakly differentiable and UWA,γ1,p(Rd)<+}, whereUWA,γ1,p(Rd)(0+(U(t)WA1,p(Rd)p+U(t)Lp(Rd)p)tγdt).

For every T(0,+) and UWA,γ1,p(Rd), one has UC([0,T],WA1,p(Rd)+Lp(Rd)) [34, Lemma 1.8.1]. In particular, the corresponding trace space can be defined by [34, Definition 1.8.1/2]TA,γ1,p{U(0);UWA,γ1,p(Rd)}. By a classical result in interpolation theory

Extension from a half-space

Finally, we obtain a result about the extension from half-space to the whole space of functions in magnetic Sobolev spaces. Set, for γR,WA,γ1,p(Rd+1){UWloc1,1(Rd+1);UWA,γ1,p(Rd+1)<+}, whereUWA,γ1,p(Rd+1)(R+d+1(|AU(x,t)|p+|U(x,t)|p)|t|γdxdt)1p.

Theorem 8.1

Let d1, 1<γ<p1 and 1p<+. There exists a constant C>0 such that for every AC1(Rd+1,1Rd+1) such that dA is bounded and every UWA,γ1,p(R+d+1), there exists U¯WA,γ1,p(Rd+1) such that U¯=U on R+d+1. Moreover, if βdAL(Rd+1),Rd+1(|AU¯(x,t

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    J. Van Schaftingen was partially supported by the Projet de Recherche (Fonds de la Recherche Scientifique - FNRS) No. T.1110.14 “Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations”. J. Van Schaftingen acknowledges the hospitality of the EPFL where a substantial part of this work was performed.

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