Fractional Korn's inequality on subsets of the Euclidean space
aa r X i v : . [ m a t h . F A ] F e b FRACTIONAL KORN’S INEQUALITY ON SUBSETS OF THE EUCLIDEAN SPACE
ARTUR RUTKOWSKI
Abstract.
We give a new, simpler proof of the fractional Korn’s inequality for subsets of R d . We also show aframework for obtaining Korn’s inequality directly from the appropriate Hardy-type inequality. Introduction
Let D be a bounded open subset of R d , d >
1, and let p ∈ (1 , ∞ ). For x ∈ R d we will use the notation x = ( x ′ , x d ) with x ′ ∈ R d − , x d ∈ R . Whenever we mention a vector field on D ⊆ R d , we mean a measurablemapping from D into R d . The space L p ( D ) consists of all the vector fields u for which the norm k u k L p ( D ) :=( R D | u ( x ) | p dx ) /p is finite.We define the fractional Sobolev space of the vector fields as follows: W s,p ( D ) = (cid:26) u ∈ L p ( D ) : | u | pW s,p ( D ) := Z D Z D (cid:12)(cid:12) u ( x ) − u ( y ) (cid:12)(cid:12) p | x − y | d + sp dx dy < ∞ (cid:27) .W s,p ( D ) is endowed with the norm given by the formula k u k W s,p ( D ) := ( k u k pL p ( D ) + | u | pW s,p ( D ) ) /p . We alsointroduce the Sobolev space with the projected difference quotient: X s,p ( D ) = (cid:26) u ∈ L p ( D ) : | u | p X s,p ( D ) := Z D Z D (cid:12)(cid:12) ( u ( x ) − u ( y )) ( x − y ) | x − y | (cid:12)(cid:12) p | x − y | d + sp dx dy < ∞ (cid:27) . We equip X s,p ( D ) with the norm k u k X s,p ( D ) := ( k u k pL p ( D ) + | u | p X s,p ( D ) ) /p . Furthermore, we define the spaces W s,p ( D ) and X s,p ( D ) as the closures of ( C c ( D )) d in the norms k · k W s,p ( D ) and k · k X s,p ( D ) , respectively.Obviously, k u k X s,p ( D ) ≤ k u k W s,p ( D ) . Our main goal here is to establish a reverse inequality with a multi-plicative constant on the left-hand side, which is known as the fractional Korn’s inequality. Theorem 1.1.
Assume that p > , s ∈ (0 , , and sp = 1 . If D is a bounded C open set or a bounded Lipschitzset with sufficiently small Lipschitz constant, then there exists C ≥ such that for every u ∈ X s,p ( D ) we have C k u k X s,p ( D ) ≥ k u k W s,p ( D ) . In particular, X s,p ( D ) = W s,p ( D ) . This result was obtained very recently by Mengesha and Scott [3] with the use of a complicated extensionoperator intrinsic to studying projected seminorms. We present a significantly shorter proof which omits theextension: we first obtain the inequality for the epigraphs in Theorem 3.1 by using the result of Mengesha and
Mathematics Subject Classification . 26D10, 46E35, 46E40.
Key words and phrases . Korn’s inequality, fractional Sobolev spaces.
Scott [6, Theorem 1.1] for the half-space together with an appropriate change of variables, and then we applyan argument via the partition of unity.In Section 4 we show that the Korn’s inequality for the vector fields of the class ( C c ( D )) d can be obtaineddirectly from the appropriate Hardy-type inequality for D with the use of an operator which extends the vectorfield by 0 to the whole space. This in particular yields a simpler proof of the Korn’s inequality for the half-spacethan the original one due to Mengesha [2]. It may also facilitate the proofs for more general sets D in thefuture.For applications, open problems, and a wider context concerning the fractional Korn’s inequality we referto the aforementioned works of Mengesha and Scott. We remark that the arguments below were obtainedindependently of the ones in [3]. Acknowledgements.
I thank Bart lomiej Dyda for helpful discussions and remarks to the manuscript. Re-search was partially supported by the Faculty of Pure and Applied Mathematics, Wroc law University of Scienceand Technology, 049U/0052/19. 2.
Preliminaries and auxiliary results
Let f : R d − → R be a continuous function. The open set { ( x ′ , x d ) ∈ R d : f ( x ′ ) < x d } will be called theepigraph of f . The epigraph of a Lipschitz (resp. C ) function will be called a Lipschitz (resp. C ) epigraph. Definition 2.1.
We say that an open set D ⊆ R d is Lipschitz with constant L > B , . . . B n with centers belonging to ∂D , epigraphs U , . . . , U n of functions f , . . . , f n with Lipschitz constant L or better,and rigid motions R , . . . R n , such that the following conditions are satisfied • ∂D ⊂ n S i =1 B i , • R i ( B i ∩ D ) ⊂ U i and R i ( B i ∩ ∂D ) = R i ( B i ) ∩ ∂U i .We say that D is a C set if the above conditions are satisfied with the difference that the epigraphs U , . . . , U n and the functions f , . . . , f n are C instead of Lipschitz.We also consider an additional open set (not necessarily a ball) B n +1 , relatively compact in D , such that D ⊂ n +1 S i =1 B i . Note that if D is C , then the balls and the epigraphs may be so chosen that D is Lipschitz withan arbitrarily small constant L .Throughout the paper we will use certain Lipschitz maps as substitutions in the integration process. Belowwe establish some basic facts about these transformations. Lemma 2.2.
Let U and V be open subsets of R d and assume that T : U → V is a bijection such that T and T − are Lipschitz with constant K ≥ . Then for every non-negative measurable function u : V → R we have (1 /K ) d Z V u ( x ) dx ≤ Z U u ( T x ) dx ≤ K d Z V u ( x ) dx. Proof.
This fact follows conveniently from the result of Haj lasz [1, Appendix], see also Rado and Reichelderfer[5, V.2.3]. To verify the validity of the constants we first claim that J T — the Jacobian of T satisfies | J T | ≤ K d RACTIONAL KORN’S INEQUALITY 3 almost everywhere in U . Indeed, let x ∈ U and r > B ( x , r ) ⊂ U . If we take f = T and u = T B ( x ,r ) in [1], then we get that Z B ( x ,r ) | J T ( x ) | dx = Z T B ( x ,r ) dy. Thus,(2.1) 1 | B ( x , r ) | Z B ( x ,r ) | J T ( x ) | dx = | T B ( x , r ) || B ( x , r ) | . The limit r → + on the left-hand side of (2.1) exists and equals J T ( x ) for almost every x ∈ U by theLebesgue differentiation theorem. Furthermore, we have T B ( x , r ) ⊆ B ( T ( x ) , Kr ), hence the right-hand sideof (2.1) is bounded from above by K d . This proves the claim that | J T ( x ) | ≤ K d for almost every x ∈ U .Similarly we show that | J T − | ≤ K d almost everywhere in V . Thus, the lemma follows from [1] and the formula | J T − ( T x ) | = | J T ( x ) | − . (cid:3) We will commonly map the epigraph of a Lipschitz function f : R d − → R to the half-space R d + as follows: T ( x ′ , x d ) = ( x ′ , x d − f ( x ′ )) , x ′ ∈ R d − , x d > f ( x ′ ) . Clearly this is a bijection with the inverse T − ( x ′ , x d ) = ( x ′ , x d + f ( x ′ )) , x ′ ∈ R d − , x d > . Lemma 2.3. If f is Lipschitz with constant L , then for every x and y in the epigraph of f we have (2.2) C ( L ) − | x − y | ≤ | T x − T y | ≤ C ( L ) | x − y | , where C ( L ) = p L ( L + 1) . In particular, C ( L ) → as L → + .Proof. Let x and y belong to the epigraph of f . We may reduce the problem to the planar geometry. Let a = | x ′ − y ′ | , b = | x d − y d | , c = | x − y | , c ′ = | T x − T y | , d = | f ( x ′ ) − f ( y ′ ) | . Note that d ≤ La . x ya bc T x T ya b + dc ′ Figure 1.
The picture presents the pessimistic variant with ( x d − y d )( f ( x ′ ) − f ( y ′ )) > x = y , we have( c ′ ) c = 1 + 2 bd + d a + b ≤ Lba + L a a + b = 1 + L La + 2 aba + b ≤ L ( L + 1) a + b a + b ≤ L ( L + 1) . (2.3)This gives the right-hand side part of (2.2). A similar argument may be used with T − in place of T , givingthe left-hand side of (2.2). (cid:3) ARTUR RUTKOWSKI
Lemma 2.4.
Assume that D is bounded and that ψ ∈ C ∞ b ( D ) . Then | uψ | X s,p ( D ) . k u k X s,p ( D ) . An analogousresult holds with W s,p in place of X s,p .Proof. We have Z D Z D (cid:12)(cid:12) ( uψ ( x ) − uψ ( y )) ( x − y ) | x − y | (cid:12)(cid:12) p | x − y | d + sp dx dy . Z D Z D (cid:12)(cid:12) ( ψ ( x ) − ψ ( y )) u ( x ) ( x − y ) | x − y | (cid:12)(cid:12) p | x − y | d + sp dx dy + Z D Z D (cid:12)(cid:12) ψ ( y )( u ( x ) − u ( y )) ( x − y ) | x − y | (cid:12)(cid:12) p | x − y | d + sp dx dy. The latter integral is smaller than k ψ k pL ∞ ( D ) | u | p X s,p ( D ) . For the former we use the fact that | ψ ( x ) − ψ ( y ) | . | x − y | to get that it does not exceed c k u k pL p ( D ) . The proof for W s,p is identical. (cid:3) Let B δ = { x ∈ B : d ( x, ∂B ) > δ } . In the subsequent section we will apply an argument using a partitionof unity subordinate to B , . . . , B n , B n +1 from Definition 2.1. This in particular will require extending vectorfields given on B i ∩ D and supported in ( B i ) δ ∩ D for some fixed δ >
0, to a rotated epigraph (1 ≤ i ≤ n ) or tothe whole of R d ( i = n + 1). The following result enables us to perform such operations. Lemma 2.5.
Let the open sets
B, U ⊆ R d satisfy U ∩ B δ = ∅ and U \ B = ∅ . Assume that u ∈ X s,p ( U ∩ B ) has support contained in U ∩ B δ for fixed δ > . If we let e u = u in U ∩ B and e u = 0 in U \ B , then k e u k X s,p ( U ) . k u k X s,p ( U ∩ B ) . Analogous result holds with W s,p in place of X s,p .Proof. Obviously, it suffices to estimate | e u | X s,p ( U ) . Since e u = 0 on B cδ we have Z U Z U (cid:12)(cid:12) ( e u ( x ) − e u ( y )) ( x − y ) | x − y | (cid:12)(cid:12) p | x − y | d + sp dx dy ≤ | u | p X s,p ( U ∩ B ) + 2 Z U ∩ B δ | u ( x ) | p Z U \ B | x − y | d + sp dy dx ≤ | u | p X s,p ( U ∩ B ) + c ( δ ) k u k pL p ( U ∩ B ) . k u k p X s,p ( U ∩ B ) . The proof for W s,p is identical. (cid:3) Proof of the Korn’s inequality
We will show that the Korn’s inequality holds for the epigraphs with sufficiently small Lipschitz constantand then use this fact to establish Theorem 1.1.
Theorem 3.1.
Assume that sp = 1 and that D is the epigraph of a Lipschitz function f with sufficiently smallLipschitz constant L . Then there exists C ≥ such that for every u ∈ X s,p ( D ) we have C k u k X s,p ( D ) ≥ k u k W s,p ( D ) . Consequently, X s,p ( D ) = W s,p ( D ) .Proof. Following the approach of Nitsche [4, Remark 3] we will show that there exist c = c ( L ) and c = c ( L ),such that(3.1) | u | pW s,p ( D ) ≤ c | u | p X s,p ( D ) + c | u | pW s,p ( D ) . RACTIONAL KORN’S INEQUALITY 5
We will propose an explicit form of c and c so that it will be obvious that for sufficiently small L we have c < c | u | pW s,p ( D ) .Let u ∈ ( C c ( D )) d ⊆ W s,p ( D ). If we substitute ( w ′ , w d ) = ( x ′ , x d − f ( x ′ )) and ( z ′ , z d ) = ( y ′ , y d − f ( y ′ )), thenby Lemmas 2.2 and 2.3 (see the latter for the definition of C ( L )) we get | u | pW s,p ( D ) ≤ C ( L ) d Z R d + Z R d + | u ( w ′ , w d + f ( w ′ )) − u ( z ′ , z d + f ( z ′ )) | p | ( w ′ , w d + f ( w ′ )) − ( z ′ , z d + f ( z ′ )) | d + sp dz dw ≤ C ( L ) d + sp Z R d + Z R d + | u ( w ′ , w d + f ( w ′ )) − u ( z ′ , z d + f ( z ′ )) | p | w − z | d + sp dz dw. (3.2)Let v ( w ′ , w d ) = u ( w ′ , w d + f ( w ′ )). Note that the above inequalities are in fact comparisons, in particular thedouble integral in (3.2) is finite, which means that v ∈ W s,p ( R d + ). Now, if we let C K be the constant in theKorn’s inequality of Mengesha and Scott [6, Theorem 1.1], and u d — the d -th coordinate of u , then we canestimate the double integral in (3.2) as follows: Z R d + Z R d + | v ( w ) − v ( z ) | p | w − z | d + sp dz dw ≤ C K Z R d + Z R d + (cid:12)(cid:12) ( v ( w ) − v ( z )) ( w − z ) | w − z | (cid:12)(cid:12) p | w − z | d + sp dz dw = C K Z R d + Z R d + (cid:12)(cid:12) ( u ( w ′ , w d + f ( w ′ )) − u ( z ′ , z d + f ( z ′ ))) ( w − z ) | w − z | (cid:12)(cid:12) p | w − z | d + sp dz dw ≤ p − C K Z R d + Z R d + (cid:12)(cid:12) ( u ( w ′ , w d + f ( w ′ )) − u ( z ′ , z d + f ( z ′ )))(( w ′ , w d + f ( w ′ )) − ( z ′ , z d + f ( z ′ ))) (cid:12)(cid:12) p | w − z | d + sp + p dz dw (3.3) + 2 p − C K Z R d + Z R d + (cid:12)(cid:12) ( u d ( w ′ , w d + f ( w ′ )) − u d ( z ′ , z d + f ( z ′ )))( f ( z ′ ) − f ( w ′ )) (cid:12)(cid:12) p | w − z | d + sp + p dz dw. (3.4)By going back to the old variables and by using Lemma 2.3 once more, we get that (3.3) is estimated fromabove by 2 p − C K C ( L ) d + sp + p Z D Z D (cid:12)(cid:12) ( u ( x ) − u ( y ))( x − y ) (cid:12)(cid:12) p | x − y | d + sp + p dx dy = 2 p − C K C ( L ) d + sp + p | u | p X s,p ( D ) . In (3.4) we also substitute the old variables so that it is estimated from above by2 p − C K C ( L ) d + sp + p Z D Z D (cid:12)(cid:12) ( u d ( x ) − u d ( y ))( f ( y ′ ) − f ( x ′ )) (cid:12)(cid:12) p | x − y | d + sp + p dx dy ≤ p − C K C ( L ) d + sp + p L p Z D Z D (cid:12)(cid:12) u d ( x ) − u d ( y ) (cid:12)(cid:12) p | x − y | d + sp dx dy ≤ p − C K C ( L ) d + sp + p L p | u | pW s,p ( D ) . Overall, we get (3.1): | u | pW s,p ( D ) ≤ p − C K C ( L ) d +2 sp + p | u | p X s,p ( D ) + 2 p − C K C ( L ) d +2 sp + p L p | u | pW s,p ( D ) . ARTUR RUTKOWSKI
Since C ( L ) ≈ L , the second constant can be made arbitrarily small, which yields the Korn’s inequalityfor u ∈ ( C c ( D )) d .Now let u ∈ W s,p ( D ). There exists a sequence u n ∈ ( C c ( D )) d such that k u n − u k W s,p ( D ) → n → ∞ .From this we infer that k u n − u k X s,p ( D ) → u n we get that | u | W s,p ( D ) ≤ | u n − u | W s,p ( D ) + | u n | W s,p ( D ) ≤ | u n − u | W s,p ( D ) + C | u n | X s,p ( D ) . By letting n → ∞ , we obtain the Korn’s inequality for W s,p ( D ). Since the identity mapping is a bi-Lipschitzhomeomorphism from W s,p ( D ) to X s,p ( D ) and both are Banach spaces, the former is a closed subspace of thelatter. By density, W s,p ( D ) = X s,p ( D ). (cid:3) Proof of Theorem 1.1.
First, suppose that u ∈ W s,p ( D ). We may assume that D is a Lipschitz set withthe constant L satisfying the assumptions of Theorem 3.1. Let B , . . . , B n , B n +1 , R , . . . , R n , f , . . . , f n , and U , . . . , U n be as in Definition 2.1, and let U n +1 = R d and R n +1 = I . We consider a smooth partition of unity ψ , . . . , ψ n +1 subordinate to B , . . . , B n +1 , i.e., 0 ≤ ψ i ≤
1, supp( ψ i ) ⊂ B i , and P ψ i = 1 on D .We define u i = uψ i , i = 1 , . . . , n + 1. By the triangle inequality we have | u | W s,p ( D ) . n +1 X i =1 | u i | W s,p ( D ) . Furthermore, | u i | W s,p ( D ) . k u i k W s,p ( B i ∩ D ) by Lemma 2.5.Now, for i = 1 , . . . , n + 1 we extend R i ( u i )( R − i ( · )) from R i ( B i ∩ D ) to U i by 0 and we call the resultingvector fields e u i . Unlike k u k X s,p ( D ) , the norm k u k W s,p ( D ) is invariant under the rotations of u , so it is crucialthat we rotate u , . . . , u n at this point, so that they agree with the new coordinate system. Obviously, we have k u i k W s,p ( B i ∩ D ) ≤ k e u i k W s,p ( U i ) and by Lemma 2.5 the right-hand sides are finite for all i , hence e u i ∈ W s,p ( U i ).By using Theorem 3.1 for e u , . . . , e u n and the Korn’s inequality for the whole space [6, Theorem 1.1] for e u n +1 ,we obtain | u | W s,p ( D ) . n +1 X i =1 k e u i k X s,p ( U i ) . By the definition of e u i and Lemmas 2.4 and 2.5, we get that for every i = 1 , . . . , n + 1, k e u i k X s,p ( U i ) . k R i ( u i )( R − i ( · )) k X s,p ( R i ( B i ∩ D )) = k u i k X s,p ( B i ∩ D ) . k u k X s,p ( D ) . This concludes the proof of the Korn’s inequality for W s,p ( D ). The result for X s,p ( D ) is obtained as in the lastpart of the proof of Theorem 3.1. (cid:3) Application of the Hardy’s inequality
In [2, Theorem 2.3] Mengesha gives a Hardy-type inequality for the half-space R d + , p ≥ s ∈ (0 , sp = 1,and u ∈ C c ( R d + ): Z R d + | u ( x ) | p x spd dx . | u | p X s,p ( R d + ) . In this section we give a simple framework which allows to obtain the Korn’s inequality for open sets D directlyfrom the Korn’s inequality for the whole space and the Hardy’s inequality for D . RACTIONAL KORN’S INEQUALITY 7
Proposition 4.1.
Let p > , s ∈ (0 , . Assume that the open set D ⊂ R d admits the following Hardy’sinequality for u ∈ ( C c ( D )) d : Z D | u ( x ) | p d ( x, D c ) sp dx . | u | p X s,p ( D ) . Then the Korn’s inequality holds for D , that is, there exists C ≥ such that for u ∈ ( C c ( D )) d , C | u | X s,p ( D ) ≥ | u | W s,p ( D ) . Proof.
Let u ∈ ( C c ( D )) d and let e u be the vector field u extended to the whole of R d by 0. First, by the Korn’sinequality for the whole space [6, Theorem 1.1] we obtain | u | W s,p ( D ) ≤ | e u | W s,p ( R d ) . | e u | X s,p ( R d ) . We estimate the right-hand side as follows: | e u | p X s,p ( R d ) ≤ | u | p X s,p ( D ) + 2 Z D | u ( x ) | p Z D c dy | x − y | d + sp dx. By using the polar coordinates we see that for every x ∈ D , Z D c | x − y | − d − sp dy ≤ Z B (0 ,d ( x,D c )) c | y | − d − sp dy . d ( x, D c ) − sp . Therefore, by the Hardy’s inequality we get Z D | u ( x ) | p Z D c dy | x − y | d + sp dx . Z D | u ( x ) | p d ( x, D c ) sp dx . | u | p X s,p ( D ) , which ends the proof. (cid:3) Remark 4.2.
Thanks to the above result, we can significantly simplify the proof of the Korn’s inequality forthe half-space by Mengesha by omitting the discussion of the extension operator [2, Section 4.1]. If we had atour disposal the Hardy’s inequality for the sets discussed in Theorem 1.1, we would obtain a slightly strongerinequality: | u | W s,p ( D ) . | u | X s,p ( D ) , that is, the estimate without the L p norm of u on the right-hand side. References [1] P. Haj lasz. Change of variables formula under minimal assumptions.
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Faculty of Pure and Applied Mathematics, Wroc law University of Science and Technology, Wyb. Wyspia´nskiego27, 50-370 Wroc law, Poland
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