An isoperimetric inequality for a nonlinear eigenvalue problem - - PowerPoint PPT Presentation

An isoperimetric inequality for a nonlinear eigenvalue problem

Gisella Croce joint work with A. Henrot and G. Pisante

PICOF ’12 - Ecole Polytechnique, Palaiseau

Let Ω be an open bounded subset of RN. λp,q(Ω) := inf    ∇vLp(Ω) vLq(Ω) : v = 0, v ∈ W 1,p (Ω),

• Ω

|v|q−2v = 0   

Let Ω be an open bounded subset of RN. λp,q(Ω) := inf    ∇vLp(Ω) vLq(Ω) : v = 0, v ∈ W 1,p (Ω),

• Ω

|v|q−2v = 0    Among the sets of given volume, which one minimizes λp,q(Ω)?

Let Ω be an open bounded subset of RN. λp,q(Ω) := inf    ∇vLp(Ω) vLq(Ω) : v = 0, v ∈ W 1,p (Ω),

• Ω

|v|q−2v = 0    Among the sets of given volume, which one minimizes λp,q(Ω)?

THEOREM (C., Henrot and Pisante, Annales de l’IHP)

Assume 1 < p < ∞ and

• 1 < q < p∗ =

Np N−p ,

if 1 < p < N 1 < q < ∞ , if p ≥ N . λp,q(Ω) ≥ λp,q(B1 ∪ B2), where B1 and B2 are two disjoint balls of volume |Ω|/2.

Freitas-Henrot, On the first twisted Dirichlet eigenvalue: λ2,2(Ω) Motivation: Barbosa-B´ erard, Eigenvalue and twisted eigenvalue problems, applications to cmc surfaces: second variation of constant mean curvature immersions.

Freitas-Henrot, On the first twisted Dirichlet eigenvalue: λ2,2(Ω) Motivation: Barbosa-B´ erard, Eigenvalue and twisted eigenvalue problems, applications to cmc surfaces: second variation of constant mean curvature immersions. Dacorogna-Gangbo-Sub´ ıa, Sur une g´ en´ eralisation de l’in´ egalit´ e de Wirtinger: the minimizers of λp,q((−1, 1)) are odd.

Freitas-Henrot, On the first twisted Dirichlet eigenvalue: λ2,2(Ω) Motivation: Barbosa-B´ erard, Eigenvalue and twisted eigenvalue problems, applications to cmc surfaces: second variation of constant mean curvature immersions. Dacorogna-Gangbo-Sub´ ıa, Sur une g´ en´ eralisation de l’in´ egalit´ e de Wirtinger: the minimizers of λp,q((−1, 1)) are odd. λp,q

per((−1, 1)):=inf

   v′Lp((−1,1)) vLq((−1,1)) : v ∈ W 1,p

per ((−1, 1)), 1

• −1

|v|q−2v = 0    = λp,q((−1, 1))

Freitas-Henrot, On the first twisted Dirichlet eigenvalue: λ2,2(Ω) Motivation: Barbosa-B´ erard, Eigenvalue and twisted eigenvalue problems, applications to cmc surfaces: second variation of constant mean curvature immersions. Dacorogna-Gangbo-Sub´ ıa, Sur une g´ en´ eralisation de l’in´ egalit´ e de Wirtinger: the minimizers of λp,q((−1, 1)) are odd. λp,q

per((−1, 1)):=inf

   v′Lp((−1,1)) vLq((−1,1)) : v ∈ W 1,p

per ((−1, 1)), 1

• −1

|v|q−2v = 0    = λp,q((−1, 1)) [L(∂A)]2 ≥ 4λp,p′

per ((−1, 1))|A|

∀ A ⊆ R2 L(∂A) =

1

• −1

(x′(t), y′(t))lp, if ∂A = {(x(t), y(t)) : t ∈ [−1, 1]}

The constraint

• Ω

|u|q−2u = 0 makes the problem difficult. Why?

The constraint

• Ω

|u|q−2u = 0 makes the problem difficult. Why?

A classical example: the first eigenvalue of the laplacian

Among the sets of given volume, which one minimizes λ(Ω) = inf

• ∇vL2(Ω)

vL2(Ω) : v = 0, v ∈ H1

0(Ω)

• ?

The constraint

• Ω

|u|q−2u = 0 makes the problem difficult. Why?

A classical example: the first eigenvalue of the laplacian

Among the sets of given volume, which one minimizes λ(Ω) = inf

• ∇vL2(Ω)

vL2(Ω) : v = 0, v ∈ H1

0(Ω)

• ?

The ball!

Since u is positive, one has λ(Ω) = ∇uL2(Ω) uL2(Ω) ≥ ∇u∗L2(B) u∗L2(B) ≥ λ(B) where u∗ is the Schwarz rearrangement of u (rearrangement of the level sets of u in balls of same volume) and B is a ball with |B| = |Ω|.

Since u is positive, one has λ(Ω) = ∇uL2(Ω) uL2(Ω) ≥ ∇u∗L2(B) u∗L2(B) ≥ λ(B) where u∗ is the Schwarz rearrangement of u (rearrangement of the level sets of u in balls of same volume) and B is a ball with |B| = |Ω|. For λp,q(Ω) = inf    ∇vLp(Ω) vLq(Ω) : v = 0, v ∈ W 1,p (Ω),

• Ω

|v|q−2v = 0    the minimizers are forced to change sign, because of the constraint!

IDEA OF THE PROOF OF OUR RESULT:

• 1. Let u be a minimizer and Ω± = {u ≷ 0}.

Let B± be two balls such that |B±| = |Ω±|. Then λp,q(Ω) ≥ λp,q(B+ ∪ B−) We reduce to Ω = B1 ∪ B2, |B1 ∪ B2| fixed; u radial on B1 and B2

IDEA OF THE PROOF OF OUR RESULT:

• 1. Let u be a minimizer and Ω± = {u ≷ 0}.

Let B± be two balls such that |B±| = |Ω±|. Then λp,q(Ω) ≥ λp,q(B+ ∪ B−) We reduce to Ω = B1 ∪ B2, |B1 ∪ B2| fixed; u radial on B1 and B2

• 2. u solves

−div(|∇u|p−2∇u) = [λp,q(Ω)]p up−q

Lq(Ω) |u|q−2u

= ⇒

• ∂u1

∂ν1

• p−1

|∂B1| =

• ∂u2

∂ν2

• p−1

|∂B2|

IDEA OF THE PROOF OF OUR RESULT:

• 1. Let u be a minimizer and Ω± = {u ≷ 0}.

Let B± be two balls such that |B±| = |Ω±|. Then λp,q(Ω) ≥ λp,q(B+ ∪ B−) We reduce to Ω = B1 ∪ B2, |B1 ∪ B2| fixed; u radial on B1 and B2

• 2. u solves

−div(|∇u|p−2∇u) = [λp,q(Ω)]p up−q

Lq(Ω) |u|q−2u

= ⇒

• ∂u1

∂ν1

• p−1

|∂B1| =

• ∂u2

∂ν2

• p−1

|∂B2|

• 3. ˙

λp,q(B1 ∪ B2) = 0 ⇔

• ∂u1

∂ν1

• =
• ∂u2

∂ν2

• .

⇓ |B1| = |B2|

CONCLUSION:

• 1. λp,q(Ω) ≥ λp,q(B+ ∪ B−)

CONCLUSION:

• 1. λp,q(Ω) ≥ λp,q(B+ ∪ B−)
• 2. λp,q(B+ ∪ B−) ≥ λp,q(B′

1 ∪ B′ 2) for |B′ 1| = |B′ 2| = |B+∪B−| 2

CONCLUSION:

• 1. λp,q(Ω) ≥ λp,q(B+ ∪ B−)
• 2. λp,q(B+ ∪ B−) ≥ λp,q(B′

1 ∪ B′ 2) for |B′ 1| = |B′ 2| = |B+∪B−| 2

• 3. λp,q is decreasing: if Ω1 ⊂ Ω2, then λp,q(Ω1) ≥ λp,q(Ω2)

CONCLUSION:

• 1. λp,q(Ω) ≥ λp,q(B+ ∪ B−)
• 2. λp,q(B+ ∪ B−) ≥ λp,q(B′

1 ∪ B′ 2) for |B′ 1| = |B′ 2| = |B+∪B−| 2

• 3. λp,q is decreasing: if Ω1 ⊂ Ω2, then λp,q(Ω1) ≥ λp,q(Ω2)

= ⇒ λp,q(B′

1 ∪ B′ 2) ≥ λp,q(B1 ∪ B2) for |B1| = |B2| = |Ω| 2

CONCLUSION:

• 1. λp,q(Ω) ≥ λp,q(B+ ∪ B−)
• 2. λp,q(B+ ∪ B−) ≥ λp,q(B′

1 ∪ B′ 2) for |B′ 1| = |B′ 2| = |B+∪B−| 2

• 3. λp,q is decreasing: if Ω1 ⊂ Ω2, then λp,q(Ω1) ≥ λp,q(Ω2)

= ⇒ λp,q(B′

1 ∪ B′ 2) ≥ λp,q(B1 ∪ B2) for |B1| = |B2| = |Ω| 2

Summarizing we get λp,q(Ω) ≥ λp,q(B1 ∪ B2) , |B1| = |B2| = |Ω| 2

Technique of Freitas-Henrot (p = q = 2):

• 1. Reduction to pairs of balls

Technique of Freitas-Henrot (p = q = 2):

• 1. Reduction to pairs of balls
• 2. A minimizer u = u(r) solves

u′′ + N − 1 r u′ + [λ2,2(Ω)]2u = µ0

• n [0, R1] and [0, R2].

N.B.: explicit expression of the solution (Bessel functions)

Current projects:

• 1. lim

p→1 λp,q(Ω):

Current projects:

• 1. lim

p→1 λp,q(Ω): we think that at least lim p→1 λp,p(Ω) equals either

inf

• max

|∂E1| |E1| , |∂E2| |E2|

• , |E1| = |E2|, E1 ∩ E2 = ∅, E1, E2 ⊂ Ω
• r

Current projects:

• 1. lim

p→1 λp,q(Ω): we think that at least lim p→1 λp,p(Ω) equals either

inf

• max

|∂E1| |E1| , |∂E2| |E2|

• , |E1| = |E2|, E1 ∩ E2 = ∅, E1, E2 ⊂ Ω
• r

inf {max {h(E1), h(E2)} , |E ∗

1 | = |E ∗ 2 |, E1 ∩ E2 = ∅, E1, E2 ⊂ Ω}

where, for any set E, h denotes its Cheeger constant: h(E) = inf

D⊂E

|∂D| |D| = h(E ∗)

Current projects:

• 1. lim

p→1 λp,q(Ω): we think that at least lim p→1 λp,p(Ω) equals either

inf

• max

|∂E1| |E1| , |∂E2| |E2|

• , |E1| = |E2|, E1 ∩ E2 = ∅, E1, E2 ⊂ Ω
• r

inf {max {h(E1), h(E2)} , |E ∗

1 | = |E ∗ 2 |, E1 ∩ E2 = ∅, E1, E2 ⊂ Ω}

where, for any set E, h denotes its Cheeger constant: h(E) = inf

D⊂E

|∂D| |D| = h(E ∗) 2. lim

p→∞ λp,q(Ω)

Thank you for your attention!