Sobolev inequalities Updated June 5, 2020 Plan 2 Outline: - - PowerPoint PPT Presentation

Sobolev inequalities

Updated June 5, 2020

Plan

2

Outline: Gagliardo-Nierenberg-Sobolev inequality L1-Sobolev and L2-Sobolev inequalities Sobolev embedding theorems Compact embedding and Rellich-Kondrachov’s theorem Boost in regularity via Moser iteration

Gagliardo-Nirenberg-Sobolev inequality

3

Morrey’s inequality: regularity for f P W1,ppRdq with p ą d Q: What happens for p ă d? Theorem For all d ě 2 and all p P r1, dq there is cpd, pq P p0, 8q such that @f P C8

c pRdq:

}f}p‹ ď cpd, pq}∇f}p where p‹ is the Sobolev conjugate of p defined by p‹ :“ pd d ´ p Proved independently by E. Gagliardo and L. Nirenberg for p “ 1 and by S.L. Sobolev for 1 ă p ă d

First some remarks

4

p‹ :“

pd d´p the only index for which this can hold. Indeed, take

ftpxq :“ fpx{tq. Then }ft}p‹ “ t

d p‹ }f}p‹ ^ }∇ft}p “ t d p´1}∇f}p

If }f}p‹ ď c}∇f}p is to hold for ft for all t ą 0, then we must have 1 p‹ “ 1 p ´ 1 d This forces p ď d. The case p “ d excluded in d ě 2 by fǫpxq :“ logpǫ ` |x|q gpxq with g P C8

c pRdq s.t. supppgq Ď Bp0, 1{2q and gp0q ‰ 0. Then

}∇fǫ}d ď cplogp1{ǫqq1{d yet }fǫ}8 ě |gp0q| logp1{ǫq so f ÞÑ }f}8{}∇f}d not bounded on C8

c pRdq.

L1-Sobolev inequality sufficient

5

For d “ 1 the inequality holds by @f P C8

c pRq:

}f}8 ď }f 1}1 Proved by writing fpxq “ ş

p´8,xs f 1dλ (as supppfq compact).

The case p “ 1 fundamental for all d ě 1: Lemma Let d ě 2 and suppose there is cpdq P p0, 8q such that @f P C8

c pRdq:

}f}

d d´1 ď cpdq}∇f}1

Then Sobolev inequality holds for all p P r1, dq (and p‹ as in (1)) with cpd, pq :“ cpdqd ´ 1 d ´ pp

Proof of Lemma

6

First extend L1-Sobolev inequality to f P C1

cpRdq.

Pick r ě 1 and, given f P C8

c pRdq, set ˜

f :“ f|f|r´1. Then ∇˜ f “ rf|f|r´2∇f (and so ˜ f P C1

cpRdq) which shows

}f}r

r

d d´1 “ }˜

f}

d d´1 ď c}∇˜

f}1 “ cr › ›f|f|r´2∇f › ›

1 “ cr

@ |f|r´1, |∇f| D H¨

• lder’s inequality with indices ˜

p, ˜ q P r1, 8s gives @ |f|r´1, |∇f| D ď › ›|f|r´1› ›

˜ p}∇f}˜ q “ }f}r´1 ˜ ppr´1q}∇f}˜ q

Now set set r :“ d´1

d´pp and observe that this implies

˜ ppr ´ 1q “ r d d ´ 1 “ p‹ ^ ˜ q “ p Putting these together we get the claim.

Proof of L1-Sobolev inequality

7

We thus focus on the proof of L1-Sobolev inequality. This will be derived from @f P C8

c pRdq:

}f}

d d´1 ď

d

ź

i“1

}Bif}1{d

1

from which we get L1-Sobolev via

d

ź

i“1

}Bif}1{d

1

ď 1 d › › ›

d

ÿ

i“1

|Bif| › › ›

1 ď

1 ? d › › › › ´ d ÿ

i“1

|Bif|2¯1{2› › › ›

1

“ 1 ? d }∇f}1 based on arithmetic-geometric/Cauchy-Schwarz inequalities. The above inequality holds for d “ 1 directly. We prove the

• ther cases by induction. Assume it holds for dimension d ´ 1

and let us prove it for d . . .

Proof of L1-Sobolev inequality continued ...

8

Write points of Rd as px, zq where x P R and z P Rd´1. Set θ :“ d´2

d´1. Then d d´1 “ θ d´1 d´2 ` p1 ´ θq and so by H¨

• lder:

@x P Rd : ż ˇ ˇfpx, zq ˇ ˇ

d d´1 dz ď

´ż ˇ ˇfpx, zq ˇ ˇ

d´1 d´2 dz

¯ d´2

d´1 ´ż ˇ

ˇfpx, zq ˇ ˇdz ¯

1 d´1

Now @x P Rd : ż ˇ ˇfpx, zq ˇ ˇdz ď }B1f}1 Induction assumption and multivariate H¨

• lder in turn yield

ż ´ż ˇ ˇfpx, zq ˇ ˇ

d´1 d´2 dz

¯ d´2

d´1 dx ď

ż

d

ź

i“2

› ›Bifpx, ¨q › ›

1 d´1

1

dx ď

d

ź

i“2

}Bif}

1 d´1

1

Putting together we get ż |f|

d d´1 dλ ď

d

ź

i“1

}Bif}

1 d´1

1

which readily gives the claim.

Sobolev embedding

9

Corollary Let d ě 2. Then @p P r1, dq: W1,ppRdq Ď č

pďp1ď dp

d´p

Lp1pRdq. Proof: Sobolev inequality gives W1,ppRdq Ď Lp‹. By definition W1,ppRdq Ď Lp. Now apply interpolation of Lp norms.

L2-Sobolev inequality

10

The case p “ 2 frequently used, but excluded in d “ 2. This is by including Lp-norm on r.h.s. as in: Theorem Let d ě 2. For each q P r2, 2d

d´2q (where 2d d´2 is interpreted as 8

when d “ 2) there is ˜ cpd, qq P p0, 8q such that @f P C8

c pRdq:

}f}q ď ˜ cpd, qq ` }f}2 ` }∇f}2 ˘ ˜ cpd, qq is bounded in d ě 3, the bound extends to q “

2d d´2.

L2-Sobolev in d ě 3

11

Let q P r2, 2d

d´2q “ p‹ for p “ 2. As p‹ ą p, find θ P r0, 1s such that 1 q “ θ p‹ ` 1´θ 2 . Interpolation of Lp-norms:

}f}q ď }f}θ

p‹ }f}1´θ 2

ď cpd, 2q}∇f}θ

2 }f}1´θ 2

ď cpd, 2q ´ θ}∇f}2 ` p1 ´ θq}f}2 ¯ where arithmetic-geometric inequality used in the last step.

L2-Sobolev in d ě 2

12

Need some Fourier transform facts: Lemma For each f P C8

c pRdq,

ż p1 ` 4π2|k|2q ˇ ˇp fpkq ˇ ˇ2dk “ }f}2

2 ` }∇f}2 2

Moreover, for all f P C8

c pRdq there is c P p0, 8q s.t.

@k P Rd : ˇ ˇp fpkq ˇ ˇ ď c p1 ` 4π2|k|2qd . In particular, p f P L1. Proof: Write x ∇fpkq “ ´2πikp fpkq and use that Fourier transform is an isometry in L2. For second part, take ˜ f :“ p1 ´ ∆qdf to get above with c :“ }p f}8 ď }p f}1.

Proof of L2-Sobolev in d ě 2

13

Take q P p2, 2d

d´2q and let p P r1, 2q be the H¨

• lder dual and let

f P C8

c pRdq. Then x ÞÑ fp´xq is thus the Fourier transform of p

f. Hausdorff-Young inequality gives }f}q ď }p f}p “ ´ż p1 ` 4π|k|2qp{2ˇ ˇp fpkq ˇ ˇp p1 ` 4π|k|2q´p{2dk ¯1{p ď ´ż p1 ` 4π2|k|2q ˇ ˇp fpkq ˇ ˇ2dk ¯1{2´ ż p1 ` 4π2|k|2q´

p 2´p dk

¯ 1

p´ 1 2 .

by H¨

• lder’s inequality with parameters 2{p and

2 2´p. The

integral converges when

2p 2´p ą d which is equivalent

to q ă

2d d´2. The claim follows from above lemma.

L2-Sobolev embedding

14

Corollary For all d ě 2, W1,2pRdq Ď č

2ďqă 2d

d´2

LqpRdq.

Compact embedding

15

Theorem (Rellich-Kondrachov) Let d ě 2 and let O Ď Rd be bounded and open. Given p P r1, dq let q P r1, p‹q, where p‹ :“

dp d´p. Then every non-empty bounded set

C Ď W1,p

0 pOq is has a compact closure in Lq.

Criterion of compactness in Lp

16

Lemma (Kolmogorov-Riesz compactness theorem) Let p P r1, 8q and let C Ď LppRdq be non-empty. Then C is totally bounded in LppRdq if and only if it is bounded, sup

fPC

}f}p ă 8 tight lim

rÑ8 sup fPC

}f1Bp0,rqc}p “ 0 and, denoting fhpxq :“ fpx ` hq for h P Rd, obeys lim

ǫÓ0 sup |h|ăǫ

sup

fPC

}fh ´ f}p “ 0. Note: Called Riesz-Tamarkin, Kolmogorov-Riesz, or Fr´ echet-Kolmogorov theorem A.N. Sudakov showed that boundedness redundant.

Proof of Lemma

17

Suffices to show that C contains a finite δ-net for each δ ą 0. Let χǫ by our standard mollifier and set Cǫ :“

• pf1Bp0,1{ǫqq ‹ χǫ : f P C

( . As }χǫ}1 “ 1, Minkowski’s integral inequality implies › ›pf1Bp0,1{ǫqq ‹ χǫ ´ f › ›

p ď

› ›pf1Bp0,1{ǫqq ´ f › ›

p ` sup |h|ăǫ

}fh ´ f}p So suffices to find a δ-net in Cǫ. But H¨

• lder gives

› ›∇pf1Bp0,1{ǫq ‹ χǫq › ›

8 ď }f}p}∇χǫ}q

and so, since }pf1Bp0,1{ǫqq ‹ χǫ}8 ď }f1Bp0,1{ǫq}1 ď c}f}p, we get Cǫ is an equicontinuous family As suppppf1Bp0,1{ǫqq ‹ χǫq Ď Bp0, ǫ ` 1{ǫq, the claim follows from Arzel` a-Ascoli in sup-norm and then also Lp-norm by compact

• support. For converse, see the notes.

Proof of Rellich-Kondrachov’s Theorem

18

Let C Ď W1,2

0 pOq. Tightness trivial as O bounded. Sobolev

implies }f}q ď λpOq

1 q ´ 1 p‹ }f}p‹ ď cλpOq 1 q ´ 1 p‹ }∇f}p

so C bounded also in Lq. For the last condition find θ s.t. θ P r0, 1s by 1

q “ 1 ´ θ ` θ p‹ . Then

}fh ´ f}q ď }fh ´ f}1´θ

1

}fh ´ f}θ

p‹ ď p2cqθ }fh ´ f}1´θ 1

}∇f}θ

p.

As 1 ´ θ ą 0 because q ă p‹, it suffices to show lim

hÑ0 sup fPC

}fh ´ f}1 “ 0 Writing ˇ ˇfhpxq ´ fpxq ˇ ˇ ď |h| ż ˇ ˇ∇fpx ` thq ˇ ˇdt we get }fh ´ f}1 ď |h| diampOq}∇f}1 ď |h| diampOqλpOq1´ 1

p }∇f}p

Since supfPC }∇f}p ă 8, we are done.

Boost in regularity via example

19

Let d ě 2, pick O Ď Rd bounded open and consider inf !

1 2}∇f}2 2 : f P C8 c pOq ^ }f}r “ 1

) We will take r ą 1 in what follows even though r “ 1 very interesting too. For r “ 2, this describes base frequency of a drum shaped as O.] Q: Is there a minimizing f (with f P Lr and ∇f P L2)? Q: And if so, how regular is it?

Existence of minimizer

20

Lemma For r P p1, 8q, there exists f P W1,2

0 pOq with }f}r “ 1 such that

1 2}∇f}2

2 “ inf

!

1 2}∇h}2 2 : h P C8 c pOq ^ }h}r “ 1

) Moreover, there is β ě 0 such that f satisfies ∆f “ ´β f|f|r´2 where the Laplacian is taken in the sense of weak derivatives; i.e., @φ P C8

c pOq:

ż p´∆φqfdλ “ β ż φ f|f|r´2dλ.

Proof of Lemma

21

Let tfnuně1 Ď C8

c pOq be minimizing sequence. Then

supně1 }∇fn}2 ă 8 so Sobolev inequality gives @q P “ 1, 2d

d´2

˘ : sup

ně1

}fn}q ă 8 This includes q “ 2 and so tfnuně1 is bounded in W1,2

0 pOq and

thus also in W1,p

0 pOq for all p P r1, 2q.

Rellich-Kondrachov theorem shows (for a subsequence if needed) that @q1 P “ 1, 2d

d´2

˘ : fn Ñ f in Lq1 (Limit same for all q1.) This includes q1 “ 2 and so fn Ñ f in L2. Integrate w.r.t. φ P C8

c pOq to get

ż φp∇fnqdλ “ ż p∇φqfndλ Ñ ż p∇φqfdλ As C8

c pOq is dense in L2 and t∇fnuně1 bounded in L2, f is

weakly differentiable and ∇fn Ñ ∇f weakly in L2. Now . . .

Proof of Lemma continued ...

22

. . . lower semicontinuity of Lp norms under weak limits shows }∇f}2 ď lim inf

nÑ8 }∇fn}2.

So 1

2}∇f}2 2 ď inft. . . u. Can have “ă” because f can be

approximated by C8

c pOq-functions. So f is a minimizer.

To show that f solves PDE, take ˜ f :“ pf ` tφq{}f ` tφ}1 for φ P C8

c pOq to get

}∇f ` t∇φ}2

2 ě }∇f}2 2 ` }∇f}2 2

` }f ` tφ}2

r ´ }f}2 r

˘ Now expand in lowest order in t to get ż p∇fq ¨ p∇φqdλ “ 2r}∇f}2

2

ż f|f|r´2φ dλ Finally, set β :“ 2r}∇f}2

2.

Remarks

23

β is closely related to the value of the infimum for r “ 2 we get an eigenvalue problem for the Laplacian

• perator on W1,2

0 pOq,

´∆f “ βf May have multiple solutions, even with same β. r “ 1 excluded because formal derivative of f ÞÑ |f| does not capture everything; instead we get ´∆f “ β sgnpfq ` β1δfp0q Needs theory of distributions

Boost in regularity via Moser iteration

24

Lemma Let r P p1, 2d

d´2q and β ě 0. Then there is cpd, r, β, Oq P p0, 8q such

that any f P W1,2

0 pOq that solves

´∆f “ βf|f|r´2 weakly obeys }f}8 ď cpd, r, β, Oq}f}2 In particular, f P L8.

Proof of Moser’s iterative estimate

25

Pick q P p2, 2d

d´2q with q ą r and let f P W1,2pOq solves the PDE.

Assume f P L2s for some s ě 1. Then fs :“ f|f|s´1 obeys fs P L2. By approximation via smooth functions, ∇fs “ ∇p f|f|s´1q “ s|f|s´1∇f so L2-Sobolev gives }f}s

qs “ }f|f|s´1}q ď ˜

c ´ }f}s

2s `

› ›∇pf|f|s´1q › ›

2

¯ Now › ›∇pf|f|s´1q › ›2

2 “ s2

ż |f|2s´2|∇f|2dλ and integration by parts shows ż |f|2s´2|∇f|2dλ “ ´p2s ´ 2q ż |f|2s´2|∇f|2dλ ´ ż |f|2s´1∆fdλ Use PDE ∆f “ ´βf|f|r´2 to wrap this into . . .

Proof of Moser’s iterative estimate continued ...

26

› ›∇pf|f|s´1q › ›2

2 “

βs2 2s ´ 1}f}2s`r´2

2s`r´2

Summarizing: }f}s

qs ď ˜

c ˆ }f}s

2s `

´ βs2 2s ´ 1 ¯1{2 }f}s`pr´2q{2

2s`r´2

˙ . Assuming r ď 2 for simplicity, H¨

• lder gives

}f}s`pr´2q{2

2s`r´2

ď λpOq

s 2s`r´2 ´ s 2s }f}s

2s

From here we get the basic iterative estimate @s ě 1: f P L2s ñ }f}qs ď ´ ˜ c ` 1 ` ˜ c1?s ˘¯1{s }f}2s, where ˜ c1 :“ a β λpOq1{2.

Proof of Moser’s iterative estimate continued ...

27

We now iterate: Set sn :“ pq{2qn. Then f P L2s0 by assumption and so so f P L2sn for each n ě 0 with }f}2sn ď ˆ n´1 ź

k“0

´ ˜ c ` 1 ` ˜ c1?sk ˘¯1{sk˙ }f}2. The product converges as n Ñ 8 so }f}8 “ lim

nÑ8 }f}2sn ď

ˆ 8 ź

k“0

´ ˜ c ` 1 ` ˜ c1?sk ˘¯1{sk˙ }f}2 The argument for r ě 2 similar; just observe qs ą 2s ` r ´ 2 as implied by q ą r.

Remarks

28

Key ingredients for Moser iteration: a higher norm of f related to a lower norm of f and ∇f a PDE that converts the norm of ∇f to a (lower) norm of f. Applies to elliptic PDEs (eigenvalue problems, etc), parabolic PDEs (heat equation etc) including non-linear ones.

What’s next?

29

To appreciate above derivation, note that any solution f P W1,2pOq of the Poisson equation ´∆f “ g

• n O

admits integral representation f “ ż

O

Kpx, yqgpyqdy, where K is the Green function in O. Smooth away from diagonal tpx, xq: x P Ou with power law singularity Kpx, yq „

yÑx

c |x ´ y|d´2 when d ě 3 and logarithmic singularity in d “ 2. Key point: convergence requires g P Ld{2`ǫ for some ǫ ą 0!

Back to our variational problem

30

For us g :“ βf|f|r´2 and, by Moser iteration, g P L8 (for r P r2, 2d

d´2q).

Continuity+integrability of K ñ f is continuous. Since y ÞÑ ∇yKpx, yq remains integrable, we even get ∇f P C1,α for all α P p0, 1q. Further iteration possible: ∇yKpx, yq “ ∇xKpx, yq ` Op1q with normal derivative of Kpx, ¨q vanishing on BO. So for r “ 2, integrating by parts shows that from ∇f P CpOq we get ∇2f P CpOq etc. This ultimately gives f P C8pOq.

19th Hilbert problem

31

The above set of ideas was developed in the solution of 19th Hilbert problem that asked: Are all the minimizers of Φpfq :“ ż

O

ϕp∇fqdλ with ϕ strictly convex and smooth, necessarily smooth? Solved by E. De Giorgi in 1957, similar ideas used also by

• J. Nash for parabolic equations. J. Moser in 1960 came up with

“Moser iteration” technique used above.

Other important inequalities

32

The Sobolev inequality comes along with three other types of inequalities Poincar´ e inequality Nash inequality Log-Sobolev inequality which we will now discuss or at least review.

L1-Poincar´ e inequality

33

Lemma (L1-Poincar´ e inequality) For each d ě 2, @f P C8

c pRdq:

}f}1 ď cpdqλpsupppfqq1{d}∇f}1, where cpdq is the constant from L1-Sobolev inequality. Proof: H¨

• lder’s inequality with parameters d and

d d´1 shows

}f}1 “ ż 1supppfq|f|dλ ď λpsupppfqq1{d}f}

d d´1 .

Now apply L1-Sobolev inequality.

Lp-Poincar´ e inequality

34

Just as for Sobolev, applying this to power function and taking extensions we get: Proposition For all d ě 2 and let O Ď Rd be non-empty and open with λpOq ă 8. Then for all p P r1, 8q, @f P W1,p

0 pOq:

}f}p ď p cpdqλpOq1{d}∇f}p where cpdq is the constant from L1-Sobolev inequality. Note: Fails for f P W1,ppOq in general! E.g., f :“ 1O has ∇f “ 0!

Proof of Proposition

35

Let f P C8

c pOq, set g :“ f|f|p´1 and note that ∇g “ pf|f|p´2∇f.

Then }∇g}1 “ p ż |f|p´1|∇f|dλ ď p}f}p´1

p

}∇f}p by H¨

• lder’s inequality. Applying the L1-Poincar´

e inequality along with supppfq Ď O, }f}p

p “ }g}1 ď cpdqλpOq1{d}f}p´1 p

}∇f}p Now cancel }f}p´1

p

and apply density of C8

c pOq in W1,p 0 pOq.

Marriage of Poincar´ e, Morrey and Sobolev

36

Corollary Let d ě 2 and let O Ď Rd be non-empty open with λpOq ă 8. For p P r1, 8q denote p‹‹ :“

dp d´p if p ă d and p‹‹ :“ 8 if p ě d. Then

for each q P p0, p‹‹q there is cpd, p, q, Oq P p0, 8q such that @f P W1,p

0 pOq:

}f}q ď cpd, p, q, Oq}∇f}p. Proof: For p ą d this is by Morrey’s inequality. For p ă d and q P rp, p‹q, we interpolate q-norm into p-norm and p‹-norm and then apply Sobolev to p‹ norm and Poincar´ e to p-norm. The case p “ d handled by a perturbation argument.

Poincar´ e-Wirtinger inequality

37

Extension to W1,ppOq possible assuming O has a nice boundary. The key underlying observation is: Lemma Let d ě 2 and let O Ď Rd be non-empty, open with λpOq ă 8. Then for all p P r1, 8q and all f P C8

c pOq,

ˆż

O

ˇ ˇ ˇ f ´ 1 λpOq ż

O

fdλ ˇ ˇ ˇ

p

dλ ˙1{p ď 2p cpdqλpOq1{d}∇f}p. Proof: Denote ¯ f :“ 1O

1 λpOq

ş

O fdλ. Then LHS bounded by

}f}p ` }¯ f}p ď 2}f}p where second ď comes from Jensen’s inequality. Now apply Lp-Poincar´ e.

Nash inequality

38

Lemma (Nash inequality) For each d ě 2 there is c P p0, 8q such that @f P W1,2pRdq X L1pRdq: }f}1`2{d

2

ď c}∇f}2}f}2{d

1

Key in analyzing the heat flow which preserves the L1-norm!

Log-Sobolev inequality

39

Problem with Sobolev:

dp d´p Ñ p as d Ñ 8, so no estimate

uniform in d ě 2 possible. For analysis in infinite dimensions, we instead use: Lemma (Log-Sobolev inequality) For all d ě 1 there is c P p0, 8q such that @f P W1,1pRdq: ż |f| log ´ |f| }f}1 ¯ ď c}∇f}1 Proved by A.J. Stam for Gaussian measures (1959).

• P. Federbush (1969) extended this to Rd based on

hypercontractivity estimates of Nelson. Full power realized by

• L. Gross.

What do I owe you?

40

Classical treatment of Fourier analysis Analysis and functional analysis on CpXq — Arzel` a-Ascoli, Stone-Weierstrass and Riesz-Markov representation theorem Theory of distributions Abstract functional analysis. Well, some other time ...

41

THANK YOU

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