Weighted Spectral Gap and a Unique Continuation Result for the Magnetic Differential Elliptic Operator

In this paper we study the second order differential elliptic operator generalizing the magnetic Schrödinger operator namely the magnetic elliptic differential operator denoted DA = −∇A(M∇A), where ∇A = ∇ + iA, A being a 1-form called the magnetic potential, B = CurlA being the magnetic field yielding from the potential A and M = (g jk)1≤ j,k≤N a positive definite matrix on a domain Ω subset of RN . It is important to note that positivity of general differential operator of second order have been extensively studying over past decades, see for example (Agmon (1982), Pinchover, (1988), Pinsky (1995), Pinchover & Tintarev (2006)) and more recently the work of (Abbas & Ragusa (2021)), and when a magnetic field is added see (Melgaard(1996).


Introduction
In this paper we study the second order differential elliptic operator generalizing the magnetic Schrödinger operator namely the magnetic elliptic differential operator denoted D A = −∇ A (M∇ A ), where ∇ A = ∇ + iA, A being a 1-form called the magnetic potential, B = CurlA being the magnetic field yielding from the potential A and M = (g jk ) 1≤ j,k≤N a positive definite matrix on a domain Ω subset of R N . It is important to note that positivity of general differential operator of second order have been extensively studying over past decades, see for example (Agmon (1982), Pinchover, (1988), Pinsky (1995), Pinchover & Tintarev (2006)) and more recently the work of (Abbas & Ragusa (2021)), and when a magnetic field is added see (Melgaard(1996).
The present paper generalize a results of (Ensted & Tintarev (2009)) which deal with the magnetic Schrödinger operator ∆ A . We prove that if q V ≥ 0, A satisfies a local condition of integrability and B 0 in the sense of distribution q A,V has a weighted spectral gap that is there exists W > 0 such that q A,V [u] ≥ Ω W(x)|u| 2 dx, u ∈ C ∞ 0 (Ω). The use of ellipticity condition lead us to set the problem in weighted spaces with weight function γ. Let 0 ≤ γ ∈ L 1 loc (Ω), where Ω is a sub-domain of R N . Then, we will need some weighted embedding theorems see (Bourgain (2000), Kilpelinen (1997), Leonardi (1994)).

Preliminaries
Let us define a new measure denoted by dµ = γ dx. Then, we have the following weighted Laplace and Sobolev spaces respectively are the closure of C ∞ 0 (Ω) with respect to the norms ||u|| µ = ( Ω |u| 2 + |∇u| 2 dµ) 1 2 and ||u|| µ,A = ( Ω |u| 2 + |∇ A u| 2 dµ) 1 2 . We denote K ⊂⊂ Ω, if K is relatively compact in Ω. To ensure the ellipticity condition of the operator D = ∇(M∇), we suppose that for every K ⊂⊂ Ω there exists Λ K > 1 such that respectively associated with the Schrodinger operator D and D A .
We will suppose throughout the paper that the weight function γ satisfies the following two assumptions: There exists a real s ≥ 1 p−1 such as γ −1 ∈ L s loc (Ω). The assumption (A1) guarantees that the weighted space W m,p (γ, Ω) is well defined and contains C ∞ 0 (Ω) as a subset and (A2) will allow us to pass from weighted Sobolev space to non-weighted Sobolev space. So we announce the following lemma see (Drabek & Als (1996)).

Picone type equality
We give here a Picone type equality for the elliptic differential operator where M is a positive definite matrix with elements in C.
Lemma 2.2. (Picone type equality) Let v a positive smooth function. And let u ∈ C ∞ 0 (Ω), then we have the following Picone type identity

Existence of Spectral Gap
The main result of this section reads as follow.
Theorem 3.1. Suppose that A ∈ L ∞ loc (Ω) and CurlA 0 in the sense of distributions on Ω then the quadratic form q A,V admit a weighted gap in Ω.
Before proving this result, we need some auxiliary results. i.e. for any u ∈ C ∞ 0 (Ω) And using the Picone type identity we obtain the desired result.
Let v be a positive solution of Eq (3.1). Then we have Proof: Let u ∈ C ∞ 0 (Ω) and v a positive function. We have Let consider the last term in the integral, we have And then Proof: This proof follows the same pattern as (Arioli & Szulkin (2003)).

It remains to show that
Proceed by absurd, we suppose now for every > 0, then we can find a sequence u n in H 1 (γ, Ω) with ||u|| H 1 (γ,Ω) = 1 such that Since u n is bounded in H 1 (γ, Ω) we can extract a subsequence still denoted u n converging weakly to u in H 1 (γ, Ω), and since A ∈ L N (Ω) then Ω γ|Au n ||∇u n | dx −→ Ω γ|Au||∇u| dx.
Hence passing to the limit in (1.7) we get If u 0 this is a contradiction, and if u = 0 again a contradiction since (1.6) can be write and passing to the limit we get 1 ≤ 0.
Lemma 3.5. Let B ⊂ Ω and B is compact in Ω, V ∈ L ∞ loc (Ω) and A ∈ L N loc (Ω). If CurlA 0 as a distribution on B, then Proof: Assume that C B = 0. Then, there exists a sequence ( By the uniform ellipticity condition we get So, (u k ) k∈N is a bounded sequence in the separable weighted Hilbert space H 1 A (γ, B). Therefore by the Banach-Alaoglu theorem we can extract from (u k | B ) a subsequence still denoted by u k | B converging weakly to w in H 1 A (γ, B). Taking into account (1.11) and the weakly lower semi-continuity of the form Now using the diamagnetic inequality we get . We may therefore conclude from (1.9) that C > 0. Moreover applying successively Lemma 1.4 and Lemma 1.1 we conclude that w belongs to a non-weighted Sobolev space say W 1,p (B). By the lifting theorem by ( Bethuel & Zheng (1988)) there exists a function φ ∈ W 1,p (B) such that w v = C exp{iφ}. Hence applying (1.12) we get And thus ∇φ + A = 0 as an element of L 2 (γ, B). Since curl∇φ = 0(Also in the sense of weak derivatives) we conclude that curlA = 0 in the sense of distributions on B, which is a contradiction. We can now prove Theorem 1.1.
Proof of theorem 1.1: Let (B k ) k be a partition of the domain Ω such that any B k is i.e. a relatively compact subset of Ω with Lipschitz boundary and B k ⊂ B k+1 .
Assume without loss of generality that curlA 0 as a distribution on B 1 , then curl A 0 as a distribution on each B k . Let We can always find a function u ∈ C ∞ 0 (Ω) with B k |u| 2 = 1 such that q A,V [u] ≤ 1 this fact with Lemma 1.5 imply 0 < c k ≤ 1. Now, choosing 0 ≤ ψ k ≤ 1 in C ∞ 0 (Ω), with ψ k = 1 on B k and supp ψ k ⊂ B k+1 we get

An UCP Result
In this part, we prove a weak ucp property of the magnetic differential elliptic operator D A . First of all, let us give the definitions of different notion of UCP. For more details see (Regbaoui (2001)).
Definition 3.6. A function u ∈ L 2 (Ω) has a zero of infinite order at x 0 ∈ Ω if for each n ∈ N, there exists a constant C > 0 such that If u = 0 on a set of positive measure E, then u has a zero of infinite order.
Lemma 3.11. Let the magnetic potential A be such that A ∈ L N loc (Ω). Then for any u ∈ H 1 (Ω) we have |A| 2 |u| 2 ≤ C u 2 + 2 ∇u 2 . (3.14) Proof: Let A ∈ L N loc (Ω) and u ∈ H 1 (Ω). For any M > 0, let decompose A = A 1 + A 2 where here 1 E denote the characteristic function of the set E.
Clearly, A 1 ∈ L ∞ . As for A 2 , by the dominated convergence theorem it converges to zero in L N . Indeed, |A 2 | ≤ |A| and A 2 converges to 0 when M → +∞. Therefore, choosing M sufficiently large we can make the L N -norm of A 2 arbitrarily small.

Now by Hölder and Sobolev inequalities
Now, let give the following inverse Poincaré's Inequality.
Lemma 3.12. Let B r and B 2r be two concentric balls contains in Ω. If u is a solution of 3.1 then, we have
Taking r 0 smaller if necessary we can assume B r 0 (x 0 ) ⊂ Ω. Since u = 0 on E, the Hölder inequality and inequality (3.23) yield and fix n ∈ N, choose > 0 such that C 2 N = 2 −n . Since r 0 depends on then r 0 will also depends on n. (3.25) can be written as f (r) ≤ 2 −n f (2r), for r ≤ r 0 .