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1 Sobolev s Iequality, Poicaré Iequality ad Compactess I. Sobolev iequality ad Sobolev Embeddig Theorems Theorem (Sobolev s embeddig theorem). Give the bouded, ope set R with 3 ad p<, the W,p 0 () L p p () ad W,p 0 () is cotiuously embedded i the space L p p (). This meas that the followig estimate () f L p p () C Df L p (), f W,p 0 (). holds true with a costat C = C(, p) (0, + ); here we deote the weak gradiet by Df =(D e f,,d e f) L p () L p (). Proof (L. Nireberg). (i) It suffices to prove the iequality () for all f C0 (). I this cotext we eed the geeralized Hölder iequality, amely, if f j L pj (), j =,,m, such that p + + p m =, the there holds (2) f (x) f m (x)dx f L p () f m L pm(), which ca be easily deduced from Hölder s iequality by iductio. (ii) At first, we deduce the estimate () i the case p =. Notig that f C0 (), we have the followig represetatio for all x R : This implies ad cosequetly f(x) = xi f(x) f(x) D ei f(x,,x i,t,x i+,,x )dt. xi D ei f dt ( i= D ei f dx i, D ei f dx i. We itegrate this iequality succeesively with respect to the variables x,,x, usig each time the geeralized Hölder iequality with p = = p m = ad m =. f(x) ( dx D e f dx i ( D e f dx i ( ( i=2 i=2 D ei f dx i dx D ei f dx i dx. Typeset by AMS-TEX
2 2 A similar itegratio over the variables x 2,,x yields ad fially ( f(x) dx D ei f dx i, R R i= (3) f ( D ei f dx i R ( ) D ei f dx i= i= Df dx = Df, f C 0 (). (iii) We ow cosider the case <p<. Here we iser f γ with γ> ito (3) ad obtai the followig relatio with the aid of Hölder s iequality ad the coditio p + q =: f γ = γ D f γ dx f γ Df dx γ f γ q Df p, ad cosequetly f γ γ γ f γ (γ )q Df p. Choosig we ifer Fially, we arrive at γ = ( )p p = p p p, γ =(γ )q = p p. with the costat C = f p p γ Df p, f C 0 (). p ( p). (iv) If ow u W,p 0 (), we approximate u i W,p -orm by C0 fuctios u m, ad apply () to the differece u l u m. It follows that {u m } is a Cauchy sequece i L p p. Thus u itself is cotaied i the same space ad satisfies ().
3 Corollary. If kp <, the Sobolev space W k,p 0 () is cotiuously embedded i L p kp (). That is, there exists a umber C depedig oly o k, p ad such that (4) u p C u,p L p () W k,p 0 (), f W0 (). 3 Proof. Suppose kp < ad u W k,p 0 (). Let p = p p The, sice Dα u L p () for all α k, the Sobolev iequality implies D β u L p () C u W k,p (), if β k, ad hece u W k,p (). - Similarly, we fid u W k 2,p (), where p = p = p 2, ad D γ u L p () C u W k,p (), if γ k 2, - Proceedig thusly, we fially obtai, after k steps, that (4) holds ad u W 0,q () = L q (), for q = p k. Equipped with the extesio operator E, we exted the embeddig theorem from the Sobolev spaces W k,p 0 () to the spaces W k,p (), if is a C k -domai. Namely, if u W k,p (), we cosider Eu W k,p 0 ( ), for some domai cotaiig, which the is cotaied i L p kp ( ), if kp<. Ad hece u L p kp (), by restrictio from to. Sice Eu = u o ad Eu W k,p ( ) c u W k,p () depedig o, we have thus proved the followig versio of the Sobolev embeddig theorem: Sobolev Embeddig Theorem*. Let R be a bouded C k -domai. If kp <, the Sobolev space W k,p () is cotiuously embedded i L p kp (). That is, there exists a umber C depedig oly o k, p, ad such that u L p p () C u W k,p (), u W,p ().
4 4 II. Poicaré Ieqiality. Poicaré Iequalities. Let R be a bouded domai. There exists a positive costat C p such that, for every u W k,p 0 (), (4) u Lp () C P u Lp (). Proof. First we prove the formula for u C0,p (); the, if u W0 (), select a sequece {u k } C0 () covergig to u i W,p -orm as k, i.e. v k v Lp () 0, v k v Lp () 0 I particular v k Lp () v Lp (), v k Lp () v Lp (). Sice (4) holds for every v k, we have v k L p () C P v k L p () Lettig k, we obtai (4) for u. Thus, it is eough to prove (4) for u C0 (). To this purpose, from the divergece theorem, we may write (5) div (v p x) dx =0, div (v p x) dx =0. {v>0} sice v =0o. Now {v<0} div (v p x)=pv v x + v p so that (5) yields v p dx = p v p v x dx, {v>0} {v>0} v p dx = p v p v x dx. {v<0} {v<0} Sice is bouded, we have max x = M< ; therefore, usig the Schwartz s x iequality, we get v p dx = p v p v x dx {v>0} {v>0} pm ( /q ( /p v p q dx v p {v>0} {v>0} = pm ( /q v dx) p v L p ( {v>0}). {v>0} Aalogous estimate ca be drived for the itegral over {v<0}. From this it follows (4) with C P = pm/. Iequality (4) implies that i W,p 0 (), the orm u W,p is equivalet to L p. Ideed, u W,p =( u p L + p u p L p)/p, ad from (4) u L p u W,p (C p P +)/p u L p.
5 III. Compactess Theorem of Rellich ad Kodrachov We call the Baach space (B, )iscompactly embedded ito the Baach space (B 2, 2 ) if the ijective mappig I : B B 2 is compact; this meas that bouded sets i B are mapped oto precompact sets i B 2. Compactess Theorem of Rellich ad Kodrachov. Let deote a bouded, domai i R. (i.) Let p<. The for all q< p p, W,p 0 () is compactly embedded ito L q (). This meas for each sequece {f k } k=,2, W,p 0 () with f W,p () s [0, ) we ca select a subsequece {f kl } l=,2, ad elemet f L q () satisfyig lim l f kl f =0. (i.2) If is Lipschize, the W,p () is compactly embedded ito L q (). (ii.) W,2 0 () is compactly embedded ito L 2 (). (ii.2) If is Lipschize, the W,2 () is compactly embedded ito L q (). To prove (i.) ad (i.2), we shall use the followig result. Iterpolatio Iequality. If the expoets p q r fulfill 5 q = λ p + λ r with λ [0, ], the f q f λ p f λ r, f L r (). Proof of Iterpolatio Iequality. Notig = λq p + ( λ)q r = ( ) ( p r + λq ( λ)q ) we obtai ( f q = ( f λq f ( λ)q dx f p dx ) λ p ( /q λ f r r dx = f λ p f λ r. Proof of (i.). Step. We start with a arbitrary sequece {f k } with f W,p () s [0, ), ad make the trasitio to a sequece {g k } k=,2, C0 () with the property g k f k W,p () k. The latter satisfies the restrictio g k W,p () +s, k N.
6 6 If we maage to select a subsequece {g kl } coverget i L () from the sequece {g k }, the the sequece {f kl } is coverget i L () as well; here we observe g k f k L () c g k f k W,p () c k. Step 2. I order to show that the sequece {f kl } coverges eve i the space L q () with <q< p p, we apply the iterpolatio iequality by choosig λ (0, ) with the property p = λ +( λ) q p. The iterpolatio iequality ad the Sobolev iequality yield the estimate f q f λ f λ p/( p) f λ (C Df p λ, f W,p 0 (). Therefore, we have f kl f km q C f kl f km λ 0, as l, m Step 3. It still remais to select a subsequece i L () from the sequece {g k } k=,2, C 0 (). Therefore, we take a arbitrary ε (0, ) ad cosider the sequece of fuctios g k,ε (x) := ( ) x y ε ρ g k (y)dy = ρ(z)g k (x εz) C0 (Θ), R ε R where Θ={x R : dist(x, ) < }. For each fixed ε (0, ], the sequece of fuctios {g k,ε } is uiformly bouded ad equicotiuous, sice we have the followig estimates for all x Θ: g k,ε (x) ( ) x y ε g k (y) dy C 0 ε ε sup ρ(x) ε R ρ ad Dg k,ε (x) ε + R ( ) x y Dρ g k (y) dy ε g k (y) dy R ε (+) sup Dρ(z) C 0 ε + sup Dρ(z). Step 4. For each ε>0, the Arzelá-Ascoli theorem thus yields a subsequece {g kl,ε} of the sequece {g k,ε } covergig uiformly i the set. - We ow set ε m = m with m N. Usig the Cator s diagoal procedure we select a subsequece {g kl } of the sequece {g k } such that, for each fixed m N, the sequece {g kl,ε m } coverges uiformly i the set.
7 7 Step 5. We have the iequality g k (z) g k,ε (x) ρ(z) g k (x) g k (x εz) dz ρ(z) ε 0 Dg k (x tz) dtdz, for all x, which implies the estimate (5) g k (z) g k,ε (x) dx ε Dg k (x tz) dx C ε, k N. Choosig a arbitrary umber ε>0, we obtai the relatio (6) g kl g kl2 L () g kl g kl,εm L () + g g kl,εm k l2,εm L () + g g kl2,εm k l2 L () (2C + )ε, l,l 2 some costat l 0 (ε). Cosequetly, {g kl } represets a Cauchy sequece i L () ad hece possesses a limit i L (). Proof of (ii.). (a) We agai start with a sequece {f k } with f k W,2 () s [0, ), ad make the trasitio to a sequece {g k } k=,2, C0 () with the property g k f k W,2 () k. Thus {g k} satisfies the restrictio g k W,2 () +s, k N. To prove Propositio 2, it suffices to select a subsequece {g kl } coverget i L 2 () from the sequece {g k }. (b) For this purpose, we agai take a arbitrary ε (0, ) ad cosider the sequece of fuctios where g k,ε (x) := ε R ρ ( x y ε ) g k (y)dy = ρ(z)g k (x εz) C0 (Θ), R Θ={x R : dist(x, ) < }. As show i Step 3 of the proof of Propositio, the sequece of fuctios {g k,ε } is uiformly bouded ad equicotiuous, for each fixed ε (0, ]. The, as i Step 4 of the proof of Propositio, the Arzelá-Ascoli theorem ad Cator s digoal process yield a s subsequece {g kl } of the sequece {g k } such that, for each fixed m N, the sequece {g kl,/m} coverges uiformly i the set. (c) To show that {g kl } coverget i L 2 (), the crucial step is to establish, aalogously to (5), (7) g k (z) g k,ε (x) L2 () ε Dg k L2 (), k N.
8 8 I fact, we have ( g k (z) g k,ε (x) 2 [ ρ(z) g k (x) g k (x εz) dz ( ε ρ(z) for all x, which implies the estimate [ ( ε g k (z) g k,ε (x) 2 dx [ = ( ε 2 ρ(z) ρ(z)dz 0 0 ) 2 ) 2 Dg k (x tz) dt dz], ) 2 Dg k (x tz) dt dz] dx (ρ(z)/2 (ρ(z)/2 ( ε )( Dg k (x) dx, k N. 0 ) 2 Dg k (x tz) dt dz] dx ) ρ(z)ε 2 Dg k (x) 2 dxdz Aalogously to (6), we use the triagle iequality for L 2 -orm ad (7) to coclude that {g kl } is a Cauchy sequece i L 2 () ad hece possesses a limit i L 2 (). Proof of (i.2) ad (ii.2). If <p< ad {f k } is a bouded sequece i W,p (), we cosider Ef k W,p 0 ( ), for some domai cotaiig, i.e. Eu = u o ad (8) Ef k W,p ( ) c f k W,p (), for some costat c depedig o. By (8), the sequece {Ef k } is also bouded i W,p (). Hece, if p<, the (i) implies {Ef k } cotais a subsequece {Ef kl } which coverges i L p/( p) ( ). Sice Ef k = f k i, the sequece f kl coverges i L p/( p) (). By (ii.), {Ef k } cotais a subsequece {Ef kl } which coverges i L 2 ( ). Sice Ef k = f k i, the sequece f kl coverges i L 2 ().
9 IV. Poicaré Ieqiality: revisted. We have already proved that there exists a positive costat C p such that, for every u W k,p 0 (), (9) u L p () C P u L p (). O the other had, (9) caot hold if u = costat. Roughly speakig, the hypotheses that guaratee the validity of (9) require that u vaishes i some otrivial set. For istace, uder each of the followig coditios, (9) holds: (i) u W,p 0,Γ 0 (); i.e. u has zero trace o a oempty relativly ope subset Γ 0 ; (ii) u W,2 () ad u = 0 o a set E Γ with positive measure E = α>0; (iii) u W,2 () ad u = 0, i.e. u has mea value zero i. Poicaré Iequality: revisted. Let be a bouded, Lipschitz domai. Assume that u satisfies oe of the hypotheses (i), (ii), (iii) above. The, there exists C P such that (8) holds. Proof. Assume that oe of the hypotheses (i), (ii), (iii) holds. By cotradictio suppose (9) is ot true. This meas that j N, u j satisfies the same hypothesis such that (0) u j L p () >j u j L p (). Normalize u j i L p () by settig The, from (0), u j w j =. u j L p () w j Lp () = ad w j Lp () < j. Thus {w j } is bouded i W,p () ad by Rellich s theorem there exists a sequece {w jk } ad w also satisfies the same hypothesis such that { wj w strogly i L p (), w j w weakly i L p (). The cotiuity of the orm gives w Lp () = lim w j Lp () =. j O the other had, the weak semicotiuity of the orm yields w Lp () lim if w j Lp () =0 j so that w = 0. Sice is coected, w is costat ad sice w satisfies oe of the hypotheses (i), (ii), (iii), we ifer w = 0, i cotradictio to w Lp () =. Corollary. If u W,p (), let The udx= u. u u L p () C P u L p (). 9
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