L p - L q estimates for Electromagnetic Helmholtz equation
LL p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZEQUATION.
ANDONI GARCIA
Abstract.
In space dimension n ≥
3, we consider the electromagnetic Schr¨odingerHamiltonian H = ( ∇− iA ( x )) − V and the corresponding Helmholtz equation( ∇ − iA ( x )) u + u − V ( x ) u = f in R n . We extend the well known L p - L q estimates for the solution of the free Helmholtzequation to the case when the electromagnetic hamiltonian H is considered. Introduction
This paper is devoted to the study of some estimates for the Helmholtz equationwith electromagnetic potential. A very natural question is to extend the knownresults for the Helmholtz equation with constant coefficients to the case when weconsider the perturbed Helmholtz equation by a potential. Our goal will be toextend the well known L p - L q estimates for the free Helmholtz equation given in[KRS], [CS], [Gut] and [Gut1] to the case when we perturb the equation withan electromagnetic potential. More precisely, conditions on the electric and themagnetic part of the potential will be given in order to ensure that the estimatesremain true. The L p - L q estimates for the free Helmholtz equation are the following:(1.1) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) , where u is a solution of(1.2) ∆ u + ( τ ± i(cid:15) ) u = − f τ, (cid:15) > . The exponents p and q in (1.1) have to verify some specific conditions that will bespecified later on. Here C can depend on τ , p , q and n and is independent of (cid:15) .The investigation of the estimates (1.1) started in [KRS], where the study ofuniform Sobolev estimates for constant coefficient second order differential opera-tors was accomplished. Later on, in [Gut], [Gut1] (See also [CS]) the range of theexponents p and q where the estimates (1.1) hold was determined.Moreover, in the purely electric case i.e., for Schr¨odinger Hamiltonians of thetype ∆ + V ( x ), where V : R n → R is the electric potential, some positive resultswere given in [RV].Therefore the aim of the paper is to prove the corresponding estimates (1.1) inthe case when the electromagnetic Schr¨odinger hamiltonian is considered .In the first part we will prove that the existing results for the free Helmholtzequation can be extended to the perturbed equation if we impose precise decayconditions at infinity for the electric and the magnetic potential. This can be donewithout assuming smallness, neither for the electric part, nor for the magnetic part.As it will be shown, for the electromagnetic case, the range for the exponents p and q where the estimates (1.1) are valid is not the same as the one for the free Date : November 20, 2018.2000
Mathematics Subject Classification.
Key words and phrases. dispersive equations, Helmholtz equation, magnetic potential.The author is supported by the grant BFI06.42 of the Basque Government. a r X i v : . [ m a t h . A P ] N ov ANDONI GARCIA case, hence in order to go further, we deal with the Helmholtz equation with purelyelectric potential, trying to obtain results in the same region of boundedness of thefree equation.Therefore, we consider the electromagnetic Schr¨odinger hamiltonian H of theform(1.3) H = ( ∇ − iA ( x )) − V ( x ) , and the corresponding Helmholtz equation in dimensions n ≥
3, namely,(1.4) ( ∇ − iA ( x )) u + u − V ( x ) u = f in R n . Here A : ( A , . . . , A n ) : R n → R n is the magnetic potential and V ( x ) : R n → R isthe electric potential. Since now on, we denote by ∇ A = ∇ − iA, ∆ A = ∇ A . The magnetic potential A is a mathematical construction which describes the in-teraction of particles with an external magnetic field. The magnetic field B , whichis the physically measurable quantity, is given by(1.5) B ∈ M n × n , B = DA − ( DA ) t , i.e. it is the anti-symmetric gradient of the vector field A (or, in geometrical terms,the differential dA of the 1-form which is standardly associated to A ). In dimension n = 3 the action of B on vectors is identified with the vector field curl A , namely(1.6) Bv = curl A × v n = 3 , where the cross denotes the vectorial product in R .One of the most interesting facts related to the L p - L q estimates for the electro-magnetic hamiltonian is that it seems that in order to conclude the boundedness ofthe solution, one should be able to bound the first order term that appears when thehamiltonian H is expanded. More concretely, when the term ( ∇ − iA ( x )) in (1.4)is expanded, a first orden term, namely A · ∇ , comes out and it is well known thatthere are no L p - L q estimates for the gradient of the solution of the free Helmholtzequation,(1.7) ∆ u + u = − f in R n . We will proceed in the following way. Let us consider the modified Helmholtzequation with electromagnetic potential and fixed frequency τ = 1. It reads asfollows(1.8) ( ∇ − iA ( x )) u + (1 ± i(cid:15) ) u − V ( x ) u = f in R n , (cid:15) (cid:54) = 0 . Remark . For convenience we will deal only with the case τ = 1, in contrastwith the case of general τ > L p - L q estimates, independent of (cid:15) , for the so-lution of (1.8). The independence of (cid:15) will imply that the result remains true forthe solution of (1.4). This can be seen in [IS].Our method is a mixture of an a priori estimate and perturbative arguments.This is what allows us to avoid smallness conditions in the potentials. Similar argu-ments have been used in the setting of the free Schr¨odinger equation, as can be seenin [BPST] and [DFVV]. Along the proof our basic tools will be the corresponding L p - L q estimates and a L -local estimate for the solution of the free Helmholtz equa-tion, together with an a priori estimate for the solution of the modified Helmholtzequation with electromagnetic potential (1.8). p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 3
Before we describe the results that we are going to use during our proof, let usintroduce some basic notation. For f : R n → C we define the Morrey-Campanatonorm as(1.9) ||| f ||| := sup R> R (cid:90) | x |≤ R | f | dx. Moreover, we denote, for j ∈ Z , the annulus C(j) by C ( j ) = { x ∈ R n : 2 j ≤ | x | ≤ j +1 } , (1.10) N ( f ) := (cid:88) j ∈ Z (cid:32) j +1 (cid:90) C ( j ) | f | dx (cid:33) / , and we easily see the duality relation (cid:90) f gdx ≤ ||| g ||| · N ( f ) . These norms were introduced by Agmon and H¨ormander in [AH].During the exposition, the truncated version of the norms appearing above will benecessary. We will denote them respectively by(1.11) ||| f ||| := sup R ≥ R (cid:90) | x |≤ R | f | dx, (1.12) N ( f ) := (cid:88) j ≥ (cid:32) j +1 (cid:90) C ( j ) | f | dx (cid:33) / . Let us also denote by L β ( R n ), for β ∈ R , the Hilbert space of all functions f suchthat (1 + | x | ) β f is square integrable over R n . The norm in this space is denoted by (cid:107) · (cid:107) β . Trivially we have that if β > / f ∈ L β ( R n ), then N ( f ) < + ∞ .As we have said, part of our method is perturbative, so in order to be able tostart, let us remind what is known for the free Helmholtz equation. Firstly, we aregoing to state the result concerning the L p - L q estimates which appears in [Gut]and [Gut1]. Let be A = (cid:18) n + 32 n , n − n (cid:19) , A (cid:48) = (cid:18) n + 12 n , n − n (cid:19) B = (cid:18) n + 4 n − n ( n + 1) , n − n (cid:19) , B (cid:48) = (cid:18) n + 12 n , n − n + 12 n ( n + 1) (cid:19) and ∆( n ), the set of points of [0 , × [0 ,
1] given by(1.13) ∆( n ) = (cid:26)(cid:18) p , q (cid:19) ∈ [0 , : 2 n + 1 ≤ p − q ≤ n , p > n + 12 n , q < n − n (cid:27) . The set ∆( n ) is the trapezium ABB (cid:48) A (cid:48) with the closed line segments AB and B (cid:48) A (cid:48) removed (see Figure 1). ANDONI GARCIA
Figure 1. ∆( n ), n ≥ Remark . In [Gut] (See also [Gut1]), estimates for the solution of the equationperturbed with generals τ > (cid:15) >
0, are given, namely,(1.14) ∆ u + ( τ + i(cid:15) ) u = − F, τ, (cid:15) > . The special case where the point (1 /p, /q ) lies on the open segment AA (cid:48) and on theduality line 1 /q = 1 − /p in Figure 1 was previously obtained in [[KRS], Theorem2.2 and 2.3 respectively].Recall that we will only deal with the case of fixed frequency τ = 1. The Theoremreads as follows. Theorem 1.1.
Let u be a solution of (1.15) ∆ u + (1 + i(cid:15) ) u = − F, (cid:15) > . Then, there exists a constant C, independent of (cid:15) , such that (1.16) (cid:107) u (cid:107) L q ( R n ) = (cid:107) (∆ + (1 + i(cid:15) )) − F (cid:107) L q ( R n ) ≤ C (cid:107) F (cid:107) L p ( R n ) when ( p , q ) ∈ ∆( n ) , n ≥ . As we mentioned, another tool that will be crucial in the proof is an L -localestimate, which bounds the truncated Morrey-Campanato norm of the solution ofthe free equation, defined in (1.11), in terms of the L p norm of the RHS data. ThisTheorem also appears in [RV], [Gut] and [Gut1]. The statement is the following. Theorem 1.2.
Let u be a solution of (1.17) ∆ u + (1 + i(cid:15) ) u = − F, (cid:15) > . If (i) n = 3 or and n +1 ≤ p − < , or(ii) n ≥ and n +1 ≤ p − ≤ n , p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 5 then, there exists a constant C , independent of (cid:15) , such that (1.18) sup R ≥ (cid:18) R (cid:90) B R | u ( x ) | dx (cid:19) / ≤ C (cid:107) F (cid:107) L p ( R n ) . The last result concerns an a priori estimate for the solution of the perturbedequation. It states that, given precise conditions, without assuming smallness, onthe decay at infinity for the the electric potential, the magnetic potential and theradial derivative of the electric potential, we have a precise control for (cid:107)∇ A u (cid:107) − (1+ δ )2 and (cid:107) u (cid:107) − (1+ δ )2 . This result appears in [IS] .The Theorem is the following one. Theorem 1.3.
Let n ≥ , and u ∈ C ∞ be a solution of ( ∇ − iA ( x )) u + (1 ± i(cid:15) ) u − V ( x ) u = f, (cid:15) (cid:54) = 0 . Let us assume that: (V) V ( x ) can be decomposed as V ( x ) = V ( x ) + V ( x ) , and there exist strictlypositive constants C and µ such that ( V ) | V ( x ) | ≤ C | x | − µ , ( ∂ r V )( x ) ≤ C | x | − − µ , | x | ≥ , ( V ) | V ( x ) | ≤ C | x | − − µ , | x | ≥ , (B) | B ( x ) | ≤ C | x | − − µ , | x | ≥ . Choose δ > sufficiently small (so that δ ≤ µ/ , δ < ). Then, there exists aconstant C = C ( δ ) , which depends uniformly in (cid:15) , such that the following a prioriestimate holds (1.19) (cid:107)∇ A u (cid:107) − (1+ δ )2 + (cid:107) u (cid:107) − (1+ δ )2 ≤ C (cid:107) f (cid:107) δ . Remark . Observe that, since the electric potential V and magnetic potential A must satisfy the conditions of the theorem, singularities at the origin are notallowed. Remark . Notice that the unique continuation property holds for the differentialoperator H = ( ∇− iA ( x )) − V ( x ), as can be seen in [R]. The assumptions ( V ), ( V ),( V ) and ( B ), together with this observation implies that the limiting absorptionprinciple holds. Remark . The conditions on the decay for the electric potential V and themagnetic field B given by ( V ), ( V ) and ( B ) respectively, are sufficient for us, duewe have fixed the frequency τ = 1. It can be seen in [IS], that the result is trueprovided τ and (cid:15) belong to the following set denoted by K (1.20) K = { k = τ + i(cid:15) ∈ C /τ ∈ ( τ , τ ) , (cid:15) ∈ (0 , (cid:15) ) } , where 0 < τ < τ < ∞ and 0 < (cid:15) < ∞ .Hence, the critical case τ = 0 is excluded. This situation requires more decayfor both potentials (typically (cid:104) x (cid:105) − (2+ (cid:15) ) , (cid:15) >
0, for B and V if n = 3 and (cid:104) x (cid:105) − for n ≥
4, where (cid:104) x (cid:105) = (1 + | x | ) / ), in order to obtain a priori estimates for thesolution of the perturbed equation, as can be seen in [F], where Morrey -Campanatotype estimates, uniform in (cid:15) , are obtained for τ ≥ B τ := x | x | B . ANDONI GARCIA
Once we have described all the tools which are going to be used, it is necessaryto introduce the region where we are able to extend the known results for the freeHelmholtz equation to the case when electromagnetic perturbations are considered.During the discussion, it will appear a subregion of ∆( n ), for n ≥
3, which will bedenoted by ∆ ( n ), given by(1.21) ∆ ( n ) = (cid:26)(cid:18) p , q (cid:19) ∈ ∆( n ) : 1 n + 1 ≤ p − , n + 1 ≤ − q (cid:27) . The set ∆ ( n ) is the solid triangle determined by the points Q , Q (cid:48) and Q (cid:48)(cid:48) (seeFigure 2).This will be the region of boundedness for the perturbed Helmholtz equation. Figure 2. ∆ ( n ), n ≥ Remark . For the case of the perturbed electromagnetic equation we are notable to obtain a positive result of boundedness for the whole region ∆( n ), since wehave not control for the gradient term, namely A ·∇ , outside ∆ ( n ). However, whenwe set A ≡
0, and consider the electric hamiltonian, the results can be extendedoutside ∆( n ) by imposing more decay on V .2. Electromagnetic Helmholtz Equation.
In this section we will give the precise statement and the proof of the theorem,where we extend the known result for the free Helmholtz equation to the case whenelectromagnetic perturbations are considered. The basic theorems which will beused along the proof were given in the section 1, as well as the basic notation. Firstwe will announce the result for the general electromagnetic case and afterwards, bysetting A ≡
0, the electric case will be treated by extending our previous result.Let us start by considering the solution of the Helmholtz equation with elec-tromagnetic potential that satisfies either the ingoing or the outgoing Sommerfeldradiation condition. For n ≥
3, it reads,(2.1) ( ∇ − iA ( x )) u + u − V ( x ) u = f in R n , where A : ( A , . . . , A n ) : R n → R n is the magnetic potential and V ( x ) : R n → R isthe electric potential. p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 7
We will assume that the magnetic potential A satisfies the Coulomb gauge con-dition(2.2) ∇ · A = 0 . We will prove L p - L q estimates for the solution of the equation (2.1). In orderto do that we will consider the solution of (2.1) as the solution of the modifiedHelmholtz electromagnetic equation,(2.3) ( ∇ − iA ( x )) u + (1 ± i(cid:15) ) u − V ( x ) u = f in R n , (cid:15) (cid:54) = 0 , via limiting absorption principle, by taking the limit of the solution of (2.3) when (cid:15) goes to 0. We will obtain the corresponding L p - L q estimates, independent of (cid:15) ,for the solution of (2.3), so these will remain true for the solution of (2.1). This isguaranteed by the results appearing in [IS].The goal is to determine the region of p and q where the solution of (2.3) satisfies L p - L q estimates, namely,(2.4) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) . with C independent of (cid:15) .The main result of this section is the following. Theorem 2.1.
Let u be a solution of (2.5) ( ∇ − iA ( x )) u + (1 ± i(cid:15) ) u − V ( x ) u = f in R n , n ≥ , (cid:15) (cid:54) = 0 . Let V and A satisfy ( V ) , ( V ) , ( V ) and ( B ) in Theorem 1.3, and suppose that thereexist constants C, µ > such that (2.6) | A ( x ) | ≤ C (1 + | x | ) µ , | V ( x ) | ≤ C (1 + | x | ) µ . Then, there exists a constant C, independent of (cid:15) , such that (2.7) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) , when (cid:16) p , q (cid:17) ∈ ∆ ( n ) .Proof.Remark . Notice that there are no smallness assumption neither for the electricpotential V nor for the magnetic potential A . Also we bound the solution onlyassuming short-range decay for V and A . As we said in Remark 1.3, singularitiesat the origin for V and A are not considered. Step 1.
It will be proved that, whenever (cid:16) p , q (cid:17) ∈ ∆ ( n ) then we get the following(2.8) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) δ . Let u be a solution of (2.5). Since ∇ · A ≡
0, we can expand the term ( ∇ − iA ) inthe following form(2.9) ( ∇ − iA ) u = ∆ u − iA · ∇ A u + | A | u. This is the key point in order to consider the electromagnetic case as a perturbationof the free equation. As can be seen, there appear terms of order zero and orderone. So by passing terms to the RHS, we have that u is solution of the followingequation(2.10) ∆ u + (1 ± i(cid:15) ) u = f + 2 iA · ∇ A u − | A | u + V u.
ANDONI GARCIA
Now we apply the result coming from Theorem 1.2. By considering the dual esti-mate of (1.18), we get that, if (cid:16) p , q (cid:17) ∈ ∆ ( n ) it holds (cid:107) u (cid:107) L q ( R n ) = (cid:107) (∆ + (1 ± i(cid:15) )) − ( f + 2 iA · ∇ A u − | A | u + V u ) (cid:107) L q ( R n ) (2.11) ≤ C ( N ( f ) + N (2 iA · ∇ A u ) + N ( | A | u ) + N ( V u )) . with C independent of (cid:15) and N defined in (1.12).Now we continue by treating the terms appearing on the RHS of (2.11). Firstwe deal with the term N (2 iA · ∇ A u ). From (2.6), we get that this term can bebounded as N (2 iA · ∇ A u ) = C (cid:88) j ≥ j +1 (cid:90) C ( j ) | A · ∇ A u | dx (2.12) ≤ C (cid:88) j ≥ j (cid:90) C ( j ) | A | |∇ A u | dx ≤ C (cid:88) j ≥ j ( δ − µ ) (cid:90) R n |∇ A u | (1 + | x | ) δ dx. Therefore, we finally have(2.13) N (2 iA · ∇ A u ) ≤ C (cid:107)∇ A u (cid:107) − (1+ δ )2 . Let us continue with the term N ( | A | u ). As before, from (2.6), we can treatthis term as follows N ( | A | u ) = (cid:88) j ≥ j +1 (cid:90) C ( j ) || A | u | dx (2.14) ≤ C (cid:88) j ≥ j (cid:90) C ( j ) | A | | u | dx ≤ C (cid:88) j ≥ j ( − δ − µ ) (cid:90) R n | u | (1 + | x | ) δ dx. Hence, we get(2.15) N ( | A | u ) ≤ C (cid:107) u (cid:107) − (1+ δ )2 . The last term is N ( V u ). Similarly, we obtain from (2.6) N ( V u ) = (cid:88) j ≥ j +1 (cid:90) C ( j ) | V u | dx (2.16) ≤ C (cid:88) j ≥ j (cid:90) C ( j ) | V | | u | dx ≤ C (cid:88) j ≥ j ( δ − µ ) (cid:90) R n | u | (1 + | x | ) δ .dx So, it verifies(2.17) N ( V u ) ≤ C (cid:107) u (cid:107) − (1+ δ )2 . p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 9
From (2.11), (2.13), (2.15) and (2.17), we get that, whenever (cid:16) p , q (cid:17) ∈ ∆ ( n ), the L q norm of u can be bounded as (cid:107) u (cid:107) L q ( R n ) = (cid:107) (∆ + (1 ± i(cid:15) )) − ( f + 2 iA · ∇ A u − | A | u + V u ) (cid:107) L q ( R n ) (2.18) ≤ C ( N ( f ) + N (2 iA · ∇ A u ) + N ( | A | u ) + N ( V u )) . ≤ CN ( f ) + C (cid:107)∇ A u (cid:107) − (1+ δ )2 + C (cid:107) u (cid:107) − (1+ δ )2 . Now we remind the a priori estimate given by Theorem 1.3, which ensures that,under the assumptions ( V ), ( V ), ( V ) and ( B ) for V and A respectively, thereexists a constant C , such that the following holds(2.19) (cid:107)∇ A u (cid:107) − (1+ δ )2 + (cid:107) u (cid:107) − (1+ δ )2 ≤ C (cid:107) f (cid:107) δ . Remark . The constant C which appears here depends uniformly in (cid:15) .So, from (2.18) and (2.19), we get(2.20) (cid:107) u (cid:107) L q ( R n ) ≤ CN ( f ) + C (cid:107) f (cid:107) δ . Moreover, it holds that N ( f ) can be bounded as(2.21) N ( f ) ≤ C (cid:107) f (cid:107) δ . This, together with (2.20) concludes that, if u is a solution of (2.5), it verifies(2.22) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) δ . Therefore, we get the desired estimate.
Step 2.
By applying duality to the last estimate, we get that if (cid:16) p , q (cid:17) ∈ ∆ ( n ),(2.23) (cid:107) u (cid:107) − (1+ δ )2 ≤ C (cid:107) f (cid:107) L p ( R n ) Remark . The adjoint operator is the one corresponding to ∓ (cid:15) . Since we can dothe same argument for both signs, all the computations are justified. Step 3.
This is the final step in the proof. As we said in the introduction the maindifficulty will be to handle the first order term given by A · ∇ A u , since there areno L p - L q estimates for the gradient of the solution of the free Helmhotz equation.Instead of considering this norm our argument will end up by treating (cid:107)∇ A u (cid:107) − (1+ δ )2 ,and this norm will be under control in the region ∆ ( n ).Consequently, we have that if (cid:16) p , q (cid:17) ∈ ∆ ( n ), from the L p - L q estimates for thesolution of the free equation, namely (1.16), given in Theorem 1.1, and proceedingas in the step 1 for the terms 2 iA · ∇ A u , | A | u and V u , we get (cid:107) u (cid:107) L q ( R n ) = (cid:107) (∆ + (1 ± i(cid:15) )) − ( f + 2 iA · ∇ A u − | A | u + V u ) (cid:107) L q ( R n ) (2.24) ≤ C (cid:107) f (cid:107) L p ( R n ) + C ( N (2 iA · ∇ A u ) + N ( | A | u ) + N ( V u )) . where C and C do not depend on (cid:15) .Let us remind that, from (2.6) it holds N (2 iA · ∇ A u ) ≤ C (cid:107)∇ A u (cid:107) − (1+ δ )2 , (2.25) N ( | A | u ) ≤ C (cid:107) u (cid:107) − (1+ δ )2 ,N ( V u ) ≤ C (cid:107) u (cid:107) − (1+ δ )2 . Therefore, we get(2.26) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) + C (cid:107)∇ A u (cid:107) − (1+ δ )2 + C (cid:107) u (cid:107) − (1+ δ )2 . Finally we conclude that if u is solution of (2.5), by applying (2.23) we can boundits L q norm as follows(2.27) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) + C (cid:107)∇ A u (cid:107) − (1+ δ )2 . It remains to bound (cid:107)∇ A u (cid:107) − (1+ δ )2 . This can be done in the following way. Let usconsider a radial function ϕ = ϕ ( | x | ) ∈ C ∞ . By multiplying Helmholtz equation(2.5) by ϕ ¯ u in the L -sense and taking the resulting real parts it gives the identity − (cid:90) R n ϕ |∇ A u | dx + 12 (cid:90) R n ∆ ϕ | u | dx − (cid:90) R n ϕV | u | dx + (cid:90) R n ϕ | u | dx (2.28) = (cid:60) (cid:90) R n f ϕ ¯ udx. From this, passing some terms to the RHS and taking modulus, it holds (cid:90) R n ϕ |∇ A u | dx ≤ (cid:90) R n | ∆ ϕ || u | dx + (cid:90) R n | ϕ || V || u | dx + (cid:90) R n | ϕ || u | dx (2.29) + (cid:90) R n | f ϕ ¯ u | dx. Assume for simplicity that f has compact support and therefore from Step 1 we have that(2.30) (cid:107) u (cid:107) L q ( R n ) < + ∞ . Then we proceed by density.Now we pass to choose the appropriate function ϕ . Let us consider ϕ = ϕ ( | x | ) = | x | ) δ .Now, we start to bound some of the terms appearing in the RHS of (2.29). Wetrivially have(2.31) | ∆ ϕ | ≤ C | ϕ | , so it holds(2.32) (cid:90) R n | ∆ ϕ || u | dx ≤ C (cid:90) R n | ϕ || u | dx, and from (2.6), we obtain(2.33) (cid:90) R n | ϕ || V || u | dx ≤ C (cid:90) R n | ϕ || u | dx. Then, we conclude(2.34) (cid:90) R n |∇ A u | (1 + | x | ) δ dx ≤ C (cid:90) R n | u | (1 + | x | ) δ dx + (cid:90) R n | f ϕ ¯ u | dx For the last term, by applying H¨older inequality with 1 = p + q + r , we can boundit as(2.35) (cid:90) R n | f ϕ ¯ u | dx ≤ (cid:107) f (cid:107) L p (cid:107) u (cid:107) L q (cid:107) ϕ (cid:107) L r , where r is to be determined. From (2.34) and (2.35) we obtain that(2.36) (cid:90) R n |∇ A u | (1 + | x | ) δ dx ≤ C (cid:90) R n | u | (1 + | x | ) δ dx + C (cid:107) f (cid:107) L p (cid:107) u (cid:107) L q . This leads to the crucial estimate(2.37) (cid:107)∇ A u (cid:107) − (1+ δ )2 ≤ C (cid:107) u (cid:107) − (1+ δ )2 + α (cid:107) u (cid:107) L q + C ( α ) (cid:107) f (cid:107) L p , for α, C ( α ) > p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 11
Therefore, we have to find the region for p and q , for ( p , q ) ∈ ∆ ( n ), where thefollowing conditions are satisfied(2.38) nr ≤ , p + 1 q + 1 r . We have that (2.38) holds if ( p , q ) ∈ ∆ − ( n ), where ∆ − ( n ) is given by(2.39) ∆ − ( n ) = (cid:26)(cid:18) p , q (cid:19) ∈ ∆ ( n ) : 1 q ≤ − p (cid:27) . The region ∆ − ( n ) is determined by the points (cid:16) p , q (cid:17) ∈ ∆ ( n ) under the diagonal1 /q = 1 − /p , including the points (cid:16) p , q (cid:17) in the duality line (see Figure 3). Figure 3. ∆ − ( n ), n ≥ p , q ) ∈ ∆ − ( n ), from (2.23) and (2.37) it holds(2.40) (cid:107)∇ A u (cid:107) − (1+ δ )2 ≤ C (cid:107) f (cid:107) L p + α (cid:107) u (cid:107) L q + C ( α ) (cid:107) f (cid:107) L p , for α, C ( α ) >
0. This, together with (2.27) leads to(2.41) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) + α (cid:107) u (cid:107) L q + C ( α ) (cid:107) f (cid:107) L p , for ( p , q ) ∈ ∆ − ( n ). Now we choose α sufficiently small and conclude for ( p , q ) ∈ ∆ − ( n ) the desired bound(2.42) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) . Finally, by applying duality we have that the result is true for ( p , q ) ∈ ∆ ( n ). Theproof is complete. (cid:3) Notice that the region of boundedness for the solution of the Helmholtz equationwith electromagnetic potential is smaller than the region obtained in the free case.This is due to the presence of the first order term. Therefore it is natural toconsider if, whenever A ≡ n ). This will be our nextaim. This idea is resumed in the following result, where we extend the boundednessof the solution to the whole ∆( n ). We have been able to prove the next Theorem. Theorem 2.2.
Let u be a solution of (2.43) − ∆ u + (1 ± i(cid:15) ) u + V ( x ) u = f, in R n , n ≥ , (cid:15) (cid:54) = 0 . If V satisfies ( V ) , ( V ) and ( V ) in Theorem 1.3, and suppose that there existconstants C, γ > such that (2.44) | V ( x ) | ≤ C (1 + | x | ) γ + µ , and γ satisfies for (cid:16) p , q (cid:17) ∈ ∆( n ) \ ∆ ( n ) , (2.45) γ > δ − µ + n (cid:110) n +1 − ( − q ) (cid:111) , n < − q < n +1 , δ − µ + n (cid:110) n +1 − ( p − ) (cid:111) , n < p − < n +1 , then, there exists a constant C, independent of (cid:15) , such that (2.46) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) . Proof.
As for the proof of Theorem 2.1, this will be divided in three steps. Thefirst two steps are the same, so we will skip them. The main difference appears inStep 3.
Remark . As we have seen in Theorem 2.1, whenever (cid:16) p , q (cid:17) ∈ ∆ ( n ), the decayassumption (2.6) for the electric potential V is sufficient ir order to prove the L p - L q estimates. As we will see, outside this region, more decay for the electric potential V potential is needed. Step 3.
Now let us consider (cid:16) p , q (cid:17) ∈ ∆( n ) such that n < − q < n +1 . Thenfrom Theorem 1.1 and observing that, since the dual estimate of (1.18) in Theorem1.2 can not be applied for the perturbative term V u because we are outside theallowed range for q , we conclude that the L q norm of the solution of the equation(2.43) can be bounded as follows (cid:107) u (cid:107) L q ( R n ) = (cid:107) ( − ∆ + (1 ± i(cid:15) )) − ( f − V u ) (cid:107) L q ( R n ) (2.47) ≤ C ( (cid:107) f (cid:107) L p ( R n ) + (cid:107) V u (cid:107) L p ( R n ) ) . where C does not depend on (cid:15) .Here p is given by(2.48) 1 p − q = 2 n + 1 , n < − q < n + 1 . We have taken the point p being in the line 1 /p − /q = 2 / ( n + 1) in order therequire the smallest decay for V .Now, we can bound the term V u in (2.47) as follows. By applying H¨olderinequality we trivially get (cid:107)
V u (cid:107) L p ( R n ) ≤ C (cid:107) (1 + | x | ) − ( γ + µ )+ δ u (1 + | x | ) − δ (cid:107) L p ( R n ) (2.49) ≤ C (cid:107) (1 + | x | ) − ( γ + µ )+ δ (cid:107) L r ( R n ) (cid:107) u (1 + | x | ) − δ (cid:107) L ( R n ) . where r is given by(2.50) 1 p = 2 n + 1 + 1 q = 2 n + 1 − (cid:18) − q (cid:19) + 12 = 1 r + 12 . We have that(2.51) (cid:107) (1 + | x | ) − ( γ + µ )+ δ (cid:107) L r ( R n ) < ∞ , p - L q ESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 13 provided that(2.52) γ > δ − µ + n (cid:26) n + 1 − ( 12 − q ) (cid:27) . Finally, from (2.47), (2.49) and reminding that (cid:107) u (cid:107) − (1+ δ )2 is still bounded for (cid:16) p , q (cid:17) ∈ ∆( n ) such that n < − q < n +1 , we can conclude the final estimate(2.53) (cid:107) u (cid:107) L q ( R n ) ≤ C (cid:107) f (cid:107) L p ( R n ) . Now, by applying duality we have that the result is true whenever (cid:16) p , q (cid:17) ∈ ∆( n )such that n < p − < n +1 . This ends the proof. (cid:3) Remark . Notice that, from (2.45), the necessary decay γ for V in order to obtainthe result grows as we approach the upper frontier of ∆( n ) (See line segment AB Figure 3). However, we can always take γ < | V ( x ) | ≤ C/ (1 + | x | ) γ , where C is not necessarily small. References [AH]
Agmon, S., H¨ormander, L. , Asymptotic properties of solutions of differential equationswith simple characteristics,
J. Anal. Math. , , 1–38, (1976).[BPST] Burq, N., Planchon, F., Stalker, J.G., Tahvildar-Zadeh, A.S. , Strichartz estimatesfor the wave and schr¨odinger equations with the inverse-square potential,
J. Funct. Anal , , 519–549, (2003).[CS] Carbery, A., Soria, F. , Almost-everywhere convergence of Fourier integrals for func-tions in Sobolev spaces, and an L -localisation principle, Rev. Mat. Iberoamericana , ,319–337, (1988) (2).[DFVV] D’Ancona, P., Fanelli, L., Vega, L., Visciglia, N. , Endpoint Strichartz estimates forthe magnetic Schr¨odinger equation,
J. Funct. Anal. , , 3227–3240, (2010).[F] Fanelli, L. , Non-trapping magnetic fields and Morrey-Campanato estimates forSchr¨odinger operators,
J. Math. Anal. Appl. , , 1–14, (2009).[Gut] Guti´errez, S. , Un problema de contorno para la ecuaci´on de Ginzburg-Landau. PhD.Thesis,
Basque Country University , (2000).[Gut1]
Guti´errez, S. , Non trivial L q solutions to the Ginzburg-Landau equation, Math. Ann. , , 1–25, (2004).[IS] Ikebe, T., Saito, Y. , Limiting absorption method and absolute continuity for theSchr¨odinger operator,
J. Math. Kyoto Univ. , , 513–542, (1972).[KRS] Kenig, C.E., Ruiz, A., Sogge, D. , Uniform Sobolev inequalities and unique continuationfor second order constant coefficient differential operators,
Duke Math. J. , , 329–347,(1987).[R] Regbaoui, R. , Strong Uniqueness for Second Order Differential Operators,
J. Differen-tial Equations , , 201–217, (1997).[RV] Ruiz, A., Vega, L. , On local regularity of Schr¨odinger equations,
Internat. Math. Res.Notices , 13–27, (1993).
Andoni Garcia: Universidad del Pais Vasco, Departamento de Matem ´ aticas, Apartado644, 48080, Bilbao, Spain E-mail address ::