sharp inequalities and related problems · sharp inequalities and related problems 85th annual...

Sharp Inequalities and related problems85th Annual Meeting of the Indian Academy of Sciences

University of Hyderabad

K. Sandeep

[email protected]

Tata Institute of Fundamental Research

Center for Applicable Mathematics, Bangalore

November 8-10, 2019

K Sandeep 85th Annual Meeting

Inequalities : Discrete

2xy ≤ x2 + y2, for x , y ∈ R



2xy ≤ x2 + y2, for x , y ∈ R

xy ≤ xp

p+

yq

qwhere

1

p+

1

q= 1, x , y ∈ R



2xy ≤ x2 + y2, for x , y ∈ R

xy ≤ xp

p+

yq

qwhere

1

p+

1

q= 1, x , y ∈ R

For xi , yi ∈ R, i = 1, ..., n

n∑i=1

|xiyi | ≤ (n∑

i=1

|xi |p)1p (

n∑i=1

|yi |q)1q


Poincare Inequality

Let −∞ < a < b < ∞ and u ∈ C 1([a, b]), u(a) = u(b) = 0


Poincare Inequality


|u(x)|2 = |∫ x

au′(t) dt|2 ≤ (b − a)

∫ b

a|u′|2dt


Poincare Inequality


|u(x)|2 = |∫ x

au′(t) dt|2 ≤ (b − a)

∫ b

a|u′|2dt

1

(b − a)2

∫ b

a|u|2dt ≤

∫ b

a|u′|2dt


Poincare Inequality

There exists an optimal constant λ1 > 0 such that

λ1

∫ b

a|u|2dt ≤

∫ b

a|u′|2dt

holds for all functions u ∈ C 1([a, b]), u(a) = u(b) = 0.


Poincare Inequality

Let Ω be a bounded domain in Rn, then there exists an oplimal

constant λ1(Ω) > 0 such that

λ1(Ω)

∫Ω|u(x)|2 dx ≤

∫Ω|∇u(x)|2 dx

holds for all u ∈ C 1c (Ω).


Poincare Inequality

Let Ω be a bounded domain in Rn, then there exists an oplimal

constant λ1(Ω) > 0 such that

λ1(Ω)

∫Ω|u(x)|2 dx ≤

∫Ω|∇u(x)|2 dx

holds for all u ∈ C 1c (Ω).

λ1(Ω) is the first eigen value of −Δ


Inequalities in Rn

Can we have this inequality in Rn ?:

C

∫Rn

|u(x)|2 dx ≤∫Rn

|∇u(x)|2 dx , u ∈ C 1c (R

n)


Inequalities in Rn


C

∫Rn

|u(x)|2 dx ≤∫Rn

|∇u(x)|2 dx , u ∈ C 1c (R

n)

Answer : NO


Inequalities in Rn


C

∫Rn

|u(x)|2 dx ≤∫Rn

|∇u(x)|2 dx , u ∈ C 1c (R

n)

Answer : NO

More generally can we have

C

[∫Rn

|u(x)|q dx

] pq

≤∫Rn

|∇u(x)|p dx , u ∈ C 1c (R

n)


Inequalities in Rn

Suppose we have such an inequality for some p, q then

q =np

n − p:= p∗


Sobolev Inequality

Sobolev Inequality : Let 1 ≤ p < n, there exists an optimal

constant Sp,n > 0 such that

Sp,n

[∫Rn

|u(x)|p∗ dx

] pp∗

≤∫Rn

|∇u(x)|p dx , u ∈ C 1c (R

n)


Sobolev Inequality

Sobolev Inequality : Let 1 ≤ p < n, there exists an optimal

constant Sp,n > 0 such that

Sp,n

[∫Rn

|u(x)|p∗ dx

] pp∗

≤∫Rn

|∇u(x)|p dx , u ∈ C 1c (R

n)

Morrey’s Inequality : Let p > n, there exists an optimal constant

Sp,n > 0 such that

supx �=y

u(x)− u(y)

|x − y |1− np

≤ Sn,p

[∫Rn

|∇u(x)|p dx

] 1p


Best Constants and Extremals

First consider the case p = 1. i.e we have the inequality

S1,n

[∫Rn

|u(x)| nn−1 dx

] n−1n

≤∫Rn

|∇u(x)| dx , u ∈ C 1c (R

n)




S1,n

[∫Rn

|u(x)| nn−1 dx

] n−1n

≤∫Rn

|∇u(x)| dx , u ∈ C 1c (R

n)

S1,n = n1−1n [ωn−1]

1n

where ωn−1 = Hn−1(Sn−1), the surface measure of the boundary

of unit ball in Rn.




S1,n

[∫Rn

|u(x)| nn−1 dx

] n−1n

≤∫Rn

|∇u(x)| dx , u ∈ C 1c (R

n)

S1,n = n1−1n [ωn−1]

1n

where ωn−1 = Hn−1(Sn−1), the surface measure of the boundary

of unit ball in Rn.

S1,n is not achieved


Isoperimetric Inequality

Isoperimetric problem :Determine the shape of the closed plane

curve having a given length and enclosing the maximum area


Isoperimetric Inequality

Let Ω be a bounded open set in Rn with smooth boundary then

Hn−1(∂Ω) ≥ n1−1n [ωn−1]

1n |Vol Ω| n

n−1

Iquality holds iff Ω is a ball


Equivalence

Federer-Fleming : Sobolev inequality with p = 1 is equivalent to

the Isoperimetric Inequality


Cartan-Hadamard Manifold

Conjecture : Let (M, g) be a Cartan-Hadamard manifold then the

isoperimetric inequality :

Hn−1(∂Ω) ≥ n1−1n [ωn−1]

1n |Vol Ω| n

n−1

holds in (M, g).

• n=2, Weil, 1926

• n=3, Kleiner, 1992

• n=4, Croke,1984

• n ≥ 5, Joel Spruck,et al...2019 (???)


The case p = 2

S2,n

[∫Rn

|u(x)| 2nn−2 dx

] n−2n

≤∫Rn

|∇u(x)|2 dx , u ∈ C 1c (R

n)


The case p = 2

S2,n

[∫Rn

|u(x)| 2nn−2 dx

] n−2n

≤∫Rn

|∇u(x)|2 dx , u ∈ C 1c (R

n)

Extremal functions exist


The case p = 2

S2,n

[∫Rn

|u(x)| 2nn−2 dx

] n−2n

≤∫Rn

|∇u(x)|2 dx , u ∈ C 1c (R

n)

Extremal functions exist

If u is an Extremal then u solves

−Δu = un+2n−2 , u > 0 ,

∫Rn

|∇u|2 < ∞.


Extremals

• Rotational Symmetry


Extremals

• Rotational Symmetry

• Conformal invariance

Let K : Rn → Rn be a conformal map and u is a solution of the

PDE, the

u := J(K )n−22 u(K (x))

is also a solution


Extremals

• All solutions are of the from

U(x) =

[√N(N − 2)ε

ε2 + |x − x0|2]N−2

2

for some ε > 0 and x0 ∈ RN


Equivalent Geometric problem

Obata : Let g be a metric on the unit sphere Sn conformal to the

standard metric on Sn, then g is of constant scalar curvature 1 iff

g is of constant sectional curvature 1.


Yamabe Problem

Let (M, g) be a compact Riemannian manifold of dimension N and

scalar curvature Kg , can we find a metric g conformal to g such

that g has constant scalar curvature.?

If g = u4

N−2 g , then this is equivalent to solving the PDE

−4(N − 1)

N − 2Δgu + Kgu = ku

N+2N−2

where k is a constant.


Hardy-Sobolev -Maz’ya Inequality.

Let 2 ≤ k < N, RN = R

k × Rh. Denote a point x ∈ R

N by

x = (y , z) ∈ Rk × R

h, then for t ∈ [0, 2) there exists an optimal

constant S = St,N,k > 0 such that

S

⎛⎝∫RN

|u|2∗(t)|y |t dx

⎞⎠

22∗(t)

≤∫RN

|∇u|2dx − λ

∫RN

|u|2|y |2 dx

holds for all u ∈ D1,2(RN) where 2∗(t) = 2(N−t)N−2 and

0 ≤ λ ≤ (k−2)2

4 .


Euler Lagrange Equation.

−Δu − λu

|y |2 =up(t)−1

|y |t , u > 0, u ∈ D1,2(RN)

where x = (y , z) ∈ Rk × R

h = RN


Classification of Solution

Question : Uniqueness of Solution

Difficulty : Lack of rotational symmetry


Classification of Solution.

• (G Mancini , Sandeep) The problem has Hyperbolic symmetry

and showed that the Solution space is of N − k + 1 dimensional


Classification of solutions when t = 1.

Theorem

(Fabri, Mancini, S) Let u0 be the function given by

u0(x) = u0(y , z) = cN,k

((1 + |y |)2 + |z |2)−N−2

2

where cN,k = {(N − 2)(k − 1)}N−22 . Then u is a solution of

−Δu =u

NN−2

|y | in RN , u > 0, u ∈ D1,2(RN)

if and only if u(y , z) = λN−22 u0(λy , λz + z0) for some λ > 0 and

z0 ∈ RN−k


Thank You


sharp inequalities and related problems · sharp inequalities and related problems 85th annual...

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