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Born-Infeld equations in the electrostatic case

Pietro dAvenia

Dipartimento di Meccanica, Matematica e ManagementPolitecnico di Bari

Workshop in Nonlinear PDEs,Bruxelles, September 8, 2015

joint work with Denis Bonheure and Alessio Pomponio

September 8, 2015 1 / 33

Introduction The infinity problem

Let us consider the Poisson equation

= in R3. (1)

In the classical Maxwell theory, is the electrostatic potentialgenerated by the charge density .

If = 0, we get the

infinity problem associated with a point charge source:

the solution of (1) is (x) = 1/(4|x|), but its energy is

H = 12

R3|E|2 dx = 1

2

R3||2 dx = +.

September 8, 2015 2 / 33

Introduction The infinity problem

When L1(R3), which is another relevant physical case, we cannotsay, in general, that

= (1)

admits a solution with finite energy.Indeed

(i) by Gagliardo-Nirenberg-Sobolev inequality it is easy to see that if L6/5(R3), then (1) has a unique and finite energy solution;

(ii) if, e.g.

(x) =1

|x|5/2 + |x|7/2( L1(R3) \ L6/5(R3))

then (1) has no radial solutions with H < +.

September 8, 2015 3 / 33

Introduction Born solution

To avoid the violation of the principle of finiteness, Max Born in

M. Born, Modified field equations with a finite radius of theelectron, Nature 132 (1933), 282.

M. Born, On the quantum theory of the electromagnetic field, Proc.Roy. Soc. London Ser. A 143 (1934), 410437.

proposed a nonlinear theory starting from a modification of MaxwellsLagrangian density.

September 8, 2015 4 / 33

Introduction Born solution

Newtons mechanics Einsteins mechanicsLN = 12mv

2 LE = mc2(1

1 v2/c2)

(i) one of the simplest which is real only when v2 < c2;(ii) for small velocities LN LE.

September 8, 2015 5 / 33

Introduction Born solution

By analogy, starting from Maxwells Lagrangian density in the vacuum

LM = FF

4,

whereF = A A;(A0, A1, A2, A3) = (,A) is the electromagnetic potential;(x0, x1, x2, x3) = (t, x);j denotes the partial derivative with respect to xj ;

and Born introduced the new Lagrangian density

LB = b2(

1

1 +FF

2b2

)det(g),

whereb is a constant having the dimensions of e/r20 (e and r0 beingrespectively the charge and the radius of the electron);g is the Minkowski metric tensor with signature (+).

September 8, 2015 6 / 33

Introduction Born-Infeld action

Borns action, as well as Maxwells action, is invariant only for theLorentz group of transformations (orthogonal transformations).

Some months later, Born and Infeld in

M. Born, L. Infeld, Foundations of the new field theory, Nature 132(1933), 1004.

M. Born, L. Infeld, Foundations of the new field theory, Proc. Roy.Soc. London Ser. A 144 (1934), 425451.

introduced a modified version of the Lagrangian density

LBI = b2(det(g)

det

(g +

Fb

)),

whose integral is now invariant for general transformation.

September 8, 2015 7 / 33

Introduction Born-Infeld action

Since the electromagnetic field (E,B) is given by

B = A and E = tA,

we get

LM =|E|2 |B|2

2, LB = b2

(1

1 |E|

2 |B|2b2

)

and

LBI = b2(

1

1 |E|2 |B|2b2

(E B)2

b4

).

September 8, 2015 8 / 33

Introduction The electrostatic case

In the electrostatic case we infer that

LB = LBI = b2(

1

1 |E|2

b2

)= b2

(1

1 ||

2

b2

).

In presence of a charge density , we formally get the equation

div

(

1 ||2/b2

)= ,

which replaces the Poisson equation.

September 8, 2015 9 / 33

Introduction Remarks

RemarkWhen = 0, one can easily explicitly compute the solution.

M.H.L. Pryce, On a Uniqueness Theorem, Math. Proc. CambridgePhilos. Soc. 31 (1935), 625628.

(r) = 1

1 + r2N2.

RemarkThe operator

Q() = div

(

1 ||2

),

also naturally appears in string theory and in classical relativity, whereQ represents the mean curvature operator in Lorentz-Minkowskispace.

September 8, 2015 10 / 33

The problem Our equation

We consider the problemdiv

(

1 ||2

)= , x RN ,

lim|x|

(x) = 0,

(BI)

for general non-trivial charge distributions .

September 8, 2015 11 / 33

The problem References

This problem has motivated several publications in the past years.

R. Bartnik and L. Simon, Comm. Math. Phys. 87 (1982).(its ideas are fundamental in our arguments)

Moreover, the operator Q has been studied in other situations bymany authors in the recent years (Azzollini, Bereanu, Bonheure,Brezis, Coelho, Corsato, Derlet, De Coster, Fortunato, Jebelean,Kiessling, Mawhin, Mugnai, Obersnel, Omari, Orsina, Pisani, Rivetti,Torres, Wang, Yu, ...).

September 8, 2015 12 / 33

Functional setting The space

Assuming N > 3, we work on

X = D1,2(RN ) { C0,1(RN ) | 6 1},

equipped with the norm defined by

X :=(

RN||2 dx

)1/2.

September 8, 2015 13 / 33

Functional setting Properties of X

Lemma

(i) X is continuously embedded in W 1,p(RN ), for allp > 2 = 2N/(N 2);

(ii) X is continuously embedded in L(RN );

(iii) if X , then lim|x| (x) = 0;

(iv) X is weakly closed.

September 8, 2015 14 / 33

Functional setting Weak solutions

For a X , weak solutions are understood in the following sense.

Definition

A weak solution of (BI) is a function X such that for all X , wehave

RN

1 ||2

dx = , , (2)

where , denotes the duality pairing between X and X .

RemarkIf is a distribution, the weak formulation of (2) extends to any testfunction Cc (RN ).

September 8, 2015 15 / 33

Functional setting The functional

As Born-Infeld equation is formally the Euler equation of the actionfunctional

I() =

RN

(1

1 ||2

)dx , ,

we expect that one can derive existence and uniqueness of thesolution from a variational principle.

Lemma

The functional I is bounded from below, coercive, continuous, strictlyconvex, weakly lower semi-continuous.

Thus one can look for the solution as the minimizer of I in X by thedirect methods of the Calculus of Variations.However, one needs to pay attention to the lack of regularity of thefunctional when = 1.Hence we use the following classical definitions

September 8, 2015 16 / 33

Critical point in weak sense Definition

DefinitionLet X be a real Banach space and : X (,+] be a convexlower semicontinuous function. Let D() = {u X | (u) < +} bethe effective domain of . For u D(), the set

(u) = {u X | (v) (u) > u, v u, v X}

is called the subdifferential of at u. If, moreover, we consider afunctional I = + , with as above and C1(X,R), thenu D() is said to be critical in weak sense if (u) (u), that is

(u), v u+ (v) (u) > 0, v X.

A. Szulkin, Ann. Inst. H. Poincar Anal. Non Linaire 3 (1986),77109.

September 8, 2015 17 / 33

Critical point in weak sense CP and minimum

Remark

Observe that, according to the previous definition, is a critical pointin weak sense for the functional I if and only if, for any X we getRN

(1

1 ||2

)dx

RN

(1

1 ||2

)dx > , , ,

which is simply equivalent to require that is a minimum for I.

September 8, 2015 18 / 33

Critical point in weak sense Existence and uniqueness

...and

Proposition

The infimum m = infX I() is achieved by a unique X \ {0}.

easily follows from the properties of I.Thus we can conclude with

Theorem

For any X , there exists a unique critical point in weak sense ofI.

September 8, 2015 19 / 33

Critical point in weak sense Further properties

Proposition

Assume X . If X is a weak solution of (BI), then = .

QuestionIs it true that the unique minimizer is always a weak solution of(BI)?

We are not able to answer this question in its full generality but weconjecture a positive answer and the following statement goes in thatdirection.

September 8, 2015 20 / 33

Critical point in weak sense Further properties

Proposition

Assume X and let be the unique minimizer of I in X . Then

E = {x RN | || = 1}

is a null set (with respect to Lebesgue measure) and the function satisfies

RN

||21 ||2

dx 6 , .

Moreover, for all X , we have the variational inequalityRN

||21 ||2

dxRN

1 ||2

dx 6 , , .

September 8, 2015 21 / 33

Critical point in weak sense Further properties

Remark

If satisfies furtherRN

||21 ||2

dx = , ,

then it is easy to see that is a weak solution of (BI).

September 8, 2015 22 / 33

Radially distributed charge densities Definition and Theorem

For O(N), X and X , we define X as (x) = (x),for all x RN , and X as , = , , for all X .

DefinitionWe say that X is radially distributed if = , for any O(N).

We next define

Xrad = { X |