Characterization of the traces on the boundary of functions in magnetic Sobolev spaces
Introduction
The first-order magnetic Sobolev space on a given open set with is defined, for a given exponent , a vector field , as [3], [4], [6], [11], [16], [19], [29], [31, §1.1] where the weak covariant gradient associated with A of is defined as2
Magnetic Sobolev spaces arise naturally for and (corresponding to ) in quantum mechanics in the presence of a magnetic field described through its magnetic vector potential ; the function 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 and the probability density . Here and in what follows, dA denotes the exterior derivative of A; for this, we consider A as a differential form in . 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 and , for any phase shift , see, e.g., [19, chapter 7]. Geometrically, the invariant quantity is the curvature of the associated –connection (see for example [32, Chapter 11]).
Magnetic Sobolev spaces generalize classical Sobolev spaces , in which , defined by For and , the fractional Sobolev (Sobolev–Slobodeckiĭ) space is defined as where the Gagliardo seminorm of the measurable function is given by When the set Ω is bounded and its boundary is of class , or , and when , 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 by satisfies, for some positive constant and for every , the estimate and therefore, the linear operator Tr extends to a bounded linear map from the Sobolev space into fractional Sobolev space . Conversely, there exists a bounded linear operator such that for any , for some positive constant independent of u. In particular, the map Tr is surjective. Consequently, the image under the trace operator of the space is exactly the space . The space of traces can also be described as the real interpolations space in the framework of interpolation of Banach spaces [20, Théorème VI.2.1].
The trace theory for can be easily derived from the one of when the magnetic potential A is bounded. In fact, by the triangle inequality, it follows that in this case. Hence the trace space of is the space as well. The situation becomes more delicate when 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 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 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 , , and , we define, for any measurable function , the magnetic Gagliardo semi-norm where the potential is defined for each by Here and in what follows ⋅ denotes the usual complex scalar product defined for and by . For a vector field , we will consider A's parallel component on the boundary defined for each by
Our first main result is
Theorem 1.1 Let and . There exists a positive constant depending only on d and p such that if and , then for each , for each , there exists depending linearly on u and depending on β such that in and
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 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 :
Theorem 1.2 Let and . Assume that and that . The trace mapping is linear and continuous. There also exists a linear continuous mapping such that is the identity on . Moreover, the corresponding estimates of Theorem 1.1 with and are valid.
In the case where the magnetic field dA is constant, we obtain the following improvements:
Theorem 1.3 Let and . Assume that and that dA is constant. We have, with , Moreover, for every , there exists such that and
We later show that the space with and is the trace space of the space 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 and a magnetic potential . It is worth noting that the trace theory in this setting is known as in the case . 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 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 defined in by (1.5) encodes the information the trace space of , 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 , , and . There exists a positive constant depending only on d, s and p such that if , if and if , then and
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 , , and . There exists a positive constant depending only on d, s, and p such that for every with and for any with compact support, one can find depending linearly on u and depending on β such that in , and
Characterizations of trace spaces
For , we define the weighted space where It is standard to check that the space is complete. We also have the following density result:
Lemma 4.1 Let , and . If , then the space is dense in .
Proof The proof of the completeness is standard. For the density of smooth maps, we first observe that if , , and on
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 , , and . There exists a constant such that if with constant dA, and , then
The first ingredient of the proof Proposition 5.1 is the following lemma:
Lemma 5.2 Let , , and . For every , there exists a positive constant such that, for every
Trace and extension on domains
In this section, we consider the trace problem on a domain of class 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 be an open set, , and . The pull-back of A by ψ is defined for each by where is the adjoint of . We first recall the following elementary result whose proof follows from the chain rule
Interpolation of magnetic spaces
We define for every and , the functional space [34, Definition 1.8.1/1] where
For every and , one has [34, Lemma 1.8.1]. In particular, the corresponding trace space can be defined by [34, Definition 1.8.1/2] 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 , where
Theorem 8.1 Let , and . There exists a constant such that for every such that dA is bounded and every , there exists such that on . Moreover, if ,
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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.