Steklov-type 1D inequalities (a survey)
aa r X i v : . [ m a t h . C A ] J a n Steklov-type 1D inequalities (a survey)
Alexander I. Nazarov and Alexandra P. Shcheglova
Abstract.
We give a survey of classical and recent results on sharp con-stants and symmetry/asymmetry of extremal functions in 1-dimensionalfunctional inequalities.
Mathematics Subject Classification (2010).
Keywords.
One-dimensional functional inequalities, symmetry, sym-metry breaking.
This survey is an extended version of the talk given at the InternationalConference “Qualitative Theory of Differential Equations” in December 2020.It has partial intersection with the survey [27] where an extensive bibliogra-phy on multidimensional functional inequalities is given.Sharp constants in one-dimensional functional inequalities are impor-tant in various fields of mathematics such as the theory of functions (see [30],[49]), mathematical physics (see, e.g., [12], [10]), mathematical statistics (see[39, § ℓ Z u ( x ) dx ≤ (cid:16) ℓπ (cid:17) ℓ Z [ u ′ ( x )] dx, ℓ Z u ( x ) dx = 0; ℓ Z u ( x ) dx ≤ (cid:16) ℓπ (cid:17) ℓ Z [ u ′ ( x )] dx, u (0) = u ( ℓ ) = 0; ℓ Z u ( x ) dx ≤ (cid:16) ℓ π (cid:17) ℓ Z [ u ′ ( x )] dx, u (0) = u ( ℓ ) , ℓ Z u ( x ) dx = 0 . (1) The paper was supported by RFBR grant 20-01-00630.
A.I. Nazarov and A.P. ShcheglovaThese constants were found by V.A. Steklov [46] and [47], respectively (seealso [48]), and E. Almansi [2] (see also [26]). See [32, Ch. II], [27] for acomprehensive history of inequalities (1).Notice that the sharp constants in the first and the second inequality in(1) are attained by functions cos( πℓ x ) and sin( πℓ x ), respectively (up to a mul-tiplicative constant). So, the extremal functions are symmetric (respectively,odd and even) about the middle of the interval, see Fig. 1. The extremal func-tions in the third inequality are given by any linear combination of cos( π ℓ x )and sin( π ℓ x ). ℓ ℓ Figure 1.
The graphs of extremal functions for the first(thin line) and the second (bold line) inequalities in (1).We begin with a simplest extension of the second inequality in (1) k u k L q (0 ,ℓ ) ≤ λ ℓ q + p ′ k u ′ k L p (0 ,ℓ ) , u (0) = u ( ℓ ) = 0 . (2)(here and below 1 ≤ p, q ≤ ∞ , and p ′ stands for the H¨older conjugate expo-nent to p ). By dilation, it is easy to see that λ depends only on p and q . So,it is sufficient to consider ℓ = 1.Inequality (2) is equivalent to several ones. We list three examples: k u k L q (0 , ≤ λ k u ′ k L p (0 , , u (0) = 0; k u k L q (0 , ≤ λ k u ′ k L p (0 , , u (0) + u (1) = 0; k u k L q (0 , ≤ λ k u ′ k L p (0 , , u (0) = u (1) , min u + max u = 0 . (3)The following statement holds: Theorem 1.
The sharp constant in (2) is given by λ ( p, q ) = F (cid:0) p ′ + q (cid:1) F (cid:0) p ′ (cid:1) F (cid:0) q (cid:1) , (4) where F ( s ) = Γ( s +1) s s . The corresponding extremal function U can be expressedin quadratures, does not change sign and is even with respect to x = , seeFig. 2. In literature, inequalities (1) are often called the Poincar´e inequalities or the Wirtingerinequalities. teklov-type 1D inequalities 31 Figure 2.
The graphs of extremal functions in (2) for p = 2: q = 1 (thin line) and q = ∞ (bold line). Remark . For p >
1, the natural space for u in (2) is the Sobolev space ◦ W p (0 , p = 1 the sharp constant in (2) is not achieved in the Sobolevspace. However, in this case one can consider u ∈ BV (0 ,
1) and understandthe right-hand side of (2) in the sense of measures. In this case the statementof Theorem 1 is true (except for the case p = 1, q = ∞ , where both symmetricand asymmetric extremals exist).The history of Theorem 1 is given in the Table 1. Year Authors Parameters1901 Steklov [47] p = q = 21934 Hardy et al [18, Sec. 7.6] p = q = 2 k , k ∈ N p = q ∀ p, q Table 1.
The history of inequality (2)Now we consider an extension of the first inequality in (1): k u k L q (0 ,ℓ ) ≤ λ ℓ q + p ′ k u ′ k L p (0 ,ℓ ) , ℓ Z | u ( x ) | r − u ( x ) dx = 0 (5)(here 1 ≤ p, q, r ≤ ∞ ; for r = ∞ the last relation is understood in the limitsense). As in (2), λ = λ ( p, q, r ) does not depend on ℓ , and we put ℓ = 1.Under additional restriction that u is 1-periodic, the inequality (5) holdswith the sharp constant λ . In the case r = 2, (5) is equivalent to severalother inequalities. We again list three examples: k u ′ k L q (0 , ≤ λ k u ′′ k L p (0 , , u (0) = u (1) = u ′ (0) = u ′ (1) = 0; k u − u k L q (0 , ≤ λ k u ′ k L p (0 , , u (0) = u (1) = 0; k u ( k ) k L q (0 , ≤ λ k u ( k +1) k L p (0 , , u is 1-periodic , k ∈ N (6) In fact, Hardy–Littlewood–P´olya and Levin dealt with the first inequality in (3) whereasSchmidt considered the third inequality in (3).
A.I. Nazarov and A.P. Shcheglova( u stands for the mean value of u ).Notice that if the extremal function in (5) is odd w.r.t. x = then theintegral restriction is fulfilled for any r . Therefore, in this case λ does notdepend on r . However, general picture is more complicated. Theorem 3. If q ≤ (2 r − p then the following equality holds: λ ( p, q, r ) = λ ( p, q ) , see (4). The corresponding extremal function V is given by V ( x ) = ( U (cid:0) x + (cid:1) if x ≤ , − U (cid:0) x − (cid:1) if x ≥ , where U is introduced in Theorem 1. In particular, V is odd w.r.t. x = .In contrast, if q > (2 r − p then λ ( p, q, r ) > λ ( p, q ) , and the extremalfunction V has no symmetry, see Fig. 3. Figure 3.
The graphs of extremal functions in (5) for p = 2, r = 2: q = 1 (thin line) and q = ∞ (bold line). Remark . Similarly to Theorem 1, for p = 1 the statement of Theorem 3 istrue if one understands the right-hand side of (5) in the sense of measures.If r = 1 then the statement is true for q ≤ p whereas for q > p the sharpconstant in (5) is not achieved. For r = ∞ the last relation in (5) is understood in the limit sense asmin u + max u = 0. The statement of Theorem 3 is true.The history of Theorem 3 is given in the Table 2. Up to our knowledge, a unique explicit expression for λ in the asym-metry region of parameters is as follows. However, for q > p any normalized minimizing sequence converges to a non-symmetricfunction. See [16], §
2; cf. also [6]. In fact, Bohr, Dacorogna–Gangbo–Subia, Belloni–Kawohl and Croce–Dacorogna dealtwith (5) for periodic functions whereas Egorov and Buslaev–Kondratiev–Nazarov consid-ered the first inequality in (6). teklov-type 1D inequalities 5Year Authors Symmetry Asymmetry1896 Steklov [46] r = 2, q = p = 21935 Bohr [4] r = 2, q = p = ∞ r = q r = 2, q ≤ p r = 2, q >> r = 2, q > p − r = 2, q ≤ p + ε r = 2, q > p r = 2, q ≤ p + 12002 Nazarov [35] r = 2, q ≤ p q ≤ rp + ε q > r p − ( r − q ≤ rp + r − q > (2 r − p q ≤ (2 r − p q ≤ (2 r − p Table 2.
The history of inequality (5)
Proposition 5 ( [5] , §
4; see also [45] ). λ ( p, ∞ ,
2) = ( p ′ + 1) − p ′ . Further, we consider a higher-order extension of the second inequalityin (1): k u ( k ) k L q (0 ,ℓ ) ≤ λ ℓ n − k + q − p k u ( n ) k L p (0 ,ℓ ) , u ∈ ◦ W np (0 , ℓ ) (7)(here n, k ∈ Z + , n > k ). Remark . As earlier, λ = λ ( n, k, p, q ) does not depend on ℓ , and we canput ℓ = 1. For p = 1 the right-hand side of (7) is understood in the senseof measures. Evidently, λ (1 , , p, q ) = λ ( p, q ). Moreover, by (6) we have λ (2 , , p, q ) = λ ( p, q, Conjecture : If k is even then the extremal in the problem (7) is afunction even w.r.t. x = for all admissible n, p, q (except for the case p = 1, q = ∞ , n = k + 1). If k is odd then for all admissible n and p there exists b q ( n, k, p ) > p such that the extremal is even w.r.t. x = for q ≤ b q and isnon-symmetric for q > b q .The known values of λ , besides n = 1, k = 0, and n = 2, k = 1, concernthe cases p = q = 2 and q = ∞ .The following statement was proved in [22] for n = k + 1 and in [41] in general case. A technical gap was fixed in [25]. In [16], a computer-assisted proof was given while Ghisi–Gobbino–Rovellini succeeded inpure analytical proof. Stechkin considered (5) and some higher-order inequalities for periodic functions. The case n = 2 was considered in [20]. The announcement without proof was given in[21]. See also [37]. See also [7] and [23] (without proof), and [44] for n = k + 2. A.I. Nazarov and A.P. Shcheglova
Theorem 7. ω = λ − n − k ( n, k, , is the least positive root of the function Φ ( n,k ) ( ω ) = det (cid:2) D ( n,k ) ( ω ) (cid:3) , where D ( n,k ) ( ω ) is the ( n − k ) × ( n − k ) -matrix with entries D ( n,k ) jm ( ω ) = ( ωe imπn − k ) k +2 j − J k +2 j − ( ωe imπn − k ) , j, m = 0 , . . . , n − k − (here J ν is the Bessel function of the first kind). The corresponding extremalfunction is even w.r.t. x = , see Fig. 4. Figure 4.
The graphs of extremal functions in (7) for p = 2, q = 2: n = 4, k = 2 (thin line) and n = 3, k = 1 (bold line).For q = ∞ and general p , the answers are known only for n = 2, k = 0[40, 51] and n = 3, k = 0 [51]. Theorem 8.
The following equalities hold: λ (2 , , p, ∞ ) = 18 ( p ′ + 1) − p ′ .λ (3 , , p, ∞ ) = 116 · min α ∈ (0 , (cid:16) Z x p ′ | x − α | p ′ dx (cid:17) p ′ . The corresponding extremal function is even w.r.t. x = . To deal with the case p = 2, q = ∞ , it is convenient to introduce thefunction (see [24]) A n,k ( x ) = max {| u ( k ) ( x ) | : u ∈ ◦ W n (0 , , k u ( n ) k L (0 , ≤ } , x ∈ (0 , . It is easy to see that λ ( n, k, , ∞ ) = max x ∈ (0 , A n,k ( x ). Moreover, the extremalfunction in (7) is even w.r.t. x = if and only if max x ∈ (0 , A n,k ( x ) = A n,k ( ). Remark . It is shown in [24] that A n,k ( x ) is a degree 2 n − t = x − x ∈ (0 , ).teklov-type 1D inequalities 7The following statement holds: Theorem 10.
Let k be even. Then λ ( n, k, , ∞ ) = A n,k ( ) = ( k − n − k − ( n − k − √ n − k − . The corresponding extremal function is even w.r.t. x = .In contrast, if k is odd then λ ( n, k, , ∞ ) > A n,k ( ) , and the extremalfunction has no symmetry, see Fig. 5. Figure 5.
The graphs of extremal functions in (7) for p = 2, q = ∞ : n = 5, k = 2 (left) and n = 5, k = 3 (right).The history of Theorems 7–10 is given in Table 3.Year Authors n k p Symm. Asymm.1940 Schmidt [42] 1 0 ∀ ∀ q ∀ q > p ∀ q ≤ p n = k + 1 2 q = 22017 Yu. Petrova [41] ∀ ∀ q = 22008 Oshime [40] ∀ q = ∞ ∀ q = ∞ ∀ , q = ∞ ∀ q = ∞ ∀ , q = ∞ ∀ odd 2 q = ∞ ∀ even 2 q = ∞ Table 3.
The history of inequality (7) See also [37]. See also [44] for the case n = k + 2, and [31]. See also [51]. See also [52] for the case k = 0. A.I. Nazarov and A.P. ShcheglovaIf k is odd then explicit expressions for λ ( n, k, , ∞ ) are known onlyfor k = 1 [24], k = 3 , Theorem 11.
The following equalities hold: λ ( n, , , ∞ ) = A n, ( x , ) = 1( n − (cid:16) n − n − (cid:17) n − r n − n − , where x , = (cid:0) ± √ n − (cid:1) ; λ ( n, , , ∞ ) = A n, ( x , ) = (cid:16) ( n − n − √ ( n − n − n − n − (cid:17) n − × q n − n − − (2 n − √ n − n − n − n − q n − n − , where x , = (cid:16) ± r − ( n − n − √ n − n − n − n − (cid:17) ; λ ( n, , , ∞ ) = A n, ( x , ) = A n, (cid:0) (cid:0) ± √ − t (cid:1)(cid:1) , where t = n − n − + √ n − n − √ n − cos (cid:16) arccos (cid:16) − n − n − q n − n − (cid:17)(cid:17) . The explicit expression for λ ( n, , , ∞ ) is quite complicated, and we omitit. Figure 6 shows examples of functions A n,k ( x ) for even and odd k . x x Figure 6.
The graphs of the functions A n,k ( x ) for n = 6: k = 4 (left) and k = 3 (right).Finally, we consider an estimate related to the last inequality in (1): k u k pL q (0 ,ℓ ) ≤ µ k u k pW p (0 ,ℓ ) ≡ µ ℓ Z (cid:0) | u ′ ( x ) | p + | u ( x ) | p (cid:1) dx, u (0) = u ( ℓ ) . (8)teklov-type 1D inequalities 9By dilation, we can reduce (8) to the case ℓ = 1: k u k pL q (0 , ≤ e µ Z (cid:0) | u ′ ( x ) | p + a | u ( x ) | p (cid:1) dx, u (0) = u (1) . (9)Here a >
0, and e µ depends on p , q and a .For q ≤ p , using the H¨older inequality we conclude that the extremalfunction in (9) is constant and thus e µ ( p, q, a ) ≡ a − .For q > p , the problem of sharp constant in (9) is more delicate anddepends on p . It was shown in [34] that for p > q > p the extremalfunction in (9) is non-constant and thus e µ ( p, q, a ) > a − . In contrast, if p < local extremal in (9) for arbitrary q . However,given q > p , for sufficiently small a the extremal function is constant whereasfor sufficiently large a the extremal function is non-constant. The most interesting case is p = 2. In this case we consider a moregeneral inequality with “magnetic term” k u k L q (0 , π ) ≤ λ π Z (cid:0) | ( e iαx u ( x )) ′ | + a | u ( x ) | (cid:1) dx, u (0) = u (2 π ) (10)(here λ = λ ( α, a, q )).It is easy to see that both sides of (10) are invariant w.r.t. replacement α α + k and u ( x ) u ( x ) e − ikx , k ∈ Z . Therefore, without loss of generalitywe may assume that | α | ≤ . Then the necessary and sufficient condition ofthe validity of inequality (10) is a + α > Theorem 12.
Let α ( q + 2) + a ( q − ≤ . Then λ ( α, a, q ) = (2 π ) q − a + α . The corresponding extremal function is constant.In contrast, if α ( q +2)+ a ( q − > then λ ( α, a, q ) > (2 π ) q − / ( a + α ) ,and the extremal function is non-constant.Remark . In the borderline case α ( q + 2) + a ( q −
2) = 1 we conclude from | α | ≤ q = 1 + 2( a − α ) a + α ≥ . In particular, this means that for q ≤ This problem was considered earlier in [25], but the conclusion in this paper is notcorrect. For convenience we choose ℓ = 2 π . α = 02004 Nazarov [36] α = 02018 Nazarov, Shcheglova [38] a = 02018 Dolbeaut et al [10] ∀ α, a Table 4.
The history of inequality (10)Up to our knowledge, a unique explicit expression for λ in the non-constancy region of parameters is as follows. Theorem 14 ( [13] , §
4, and [14] ). Let a + α > . Then λ ( a, α, ∞ ) = √ a sinh(2 π √ a )cosh(2 π √ a ) − cos(2 πα ) , a > π − cos(2 πα ) , a = 0; √− a sin(2 π √− a )cos(2 π √− a ) − cos(2 πα ) , a < . Acknowledgements.
We are grateful to Prof. N.G. Kuznetsov for valuablecomments and suggestions.
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Takemura, The bestconstant of Sobolev inequality corresponding to clamped boundary value prob-lem. Bound. Value Probl. (2011), Article ID 875057, 17 pp.Alexander I. NazarovSt. Petersburg Dept of Steklov Institute,St. Petersburg State University,St. Petersburg, Russia.e-mail: [email protected]