A comparison theorem for super- and subsolutions of ∇ 2 u+f(u)=0 and its application to water waves with vorticity
aa r X i v : . [ m a t h . A P ] O c t A comparison theorem for super- andsubsolutions of ∇ u + f ( u ) = Vladimir Kozlov a , Nikolay Kuznetsov b a Department of Mathematics, Link¨oping University, S–581 83 Link¨oping, Sweden b Laboratory for Mathematical Modelling of Wave Phenomena,Institute for Problems in Mechanical Engineering, Russian Academy of Sciences,V.O., Bol’shoy pr. 61, St. Petersburg 199178, RF
E-mail: [email protected] ; [email protected]
Abstract
A comparison theorem is proved for a pair of solutions that satisfy in a weak senseopposite differential inequalities with nonlinearity of the form f ( u ) with f belongingto the class L ploc . The solutions are assumed to have non-vanishing gradients in thedomain, where the inequalities are considered. The comparison theorem is appliedto the problem describing steady, periodic water waves with vorticity in the case ofarbitrary free-surface profiles including overhanging ones. Bounds for these profiles aswell as streamfunctions and admissible values of the total head are obtained. Keywords:
Comparison theorem, nonlinear differential inequality, partial hodographtransform in n dimensions, periodic steady water waves with vorticity, streamfunction In their remarkable article [7], Gidas, Ni and Nirenberg investigated various properties thatsolutions (in particular, positive solutions) of several nonlinear equations have in different(bounded as well as unbounded) domains in IR n , n ≥
1. For this purpose several formsof the maximum principle were employed along with some other methods. The authorsemphasised that their techniques could be applicable in physical situations other than thoseconsidered in the paper. During the decades past since the publication of [7], this predictionproved correct. The most spectacular results obtained in the paper deal with the equation ∇ u + f ( u ) = 0 , ∇ u = ( u x , . . . , u x n ) and u x i = ∂ i u = ∂u/∂x i , (1)in which f is a C -function.The two-dimensional version of (1) describes, in particular, periodic steady water waveswith vorticity in which case f is the given vorticity distribution. If the depth of water isfinite, the domain is a quadrangle bounded by three straight segments – two of them that1re opposite to each other are equal in view of periodicity – and a curve that is oppositethe third segment and corresponds to the smallest period of wave propagating on the freesurface; of course, one can consider a strip with periodic upper boundary and horizontalbottom as the water domain. (The relevant free-boundary problem is derived from Euler’sequations, for example, in [3].)Instead of equation (1), the present paper deals with the inequality ∇ u + f ( u ) ≤ ⊂ IR n , n ≥ , (2)and its opposite which are understood in a weak sense. In the case of (2), this means thatthe integral inequality Z X ∇ u · ∇ v d x ≥ Z X f ( u ) v d x (3)is valid for every non-negative v ∈ C ( X ), where X is any subdomain of Ω. Moreover, nosmoothness and even continuity is required from f , and the aim is to prove the followingcomparison theorem for a pair of functions that satisfy the inequalities. Theorem 1
Let f ∈ L ploc (IR) with p > n , and let u , u ∈ C (Ω) have non-vanishinggradients in Ω and satisfy in the weak sense the inequalities ∇ u + f ( u ) ≥ and ∇ u + f ( u ) ≤ in Ω , (4) respectively. If u ≤ u in Ω and these functions are equal at some point x ∈ Ω , then u and u coincide throughout Ω . Remark 1 If f ∈ L p (Ω) and u ∈ C (Ω) is such that ∇ u = 0 throughout Ω, then f ( u ( x ))is a measurable function in Ω; moreover, this superposition belongs to L ploc (Ω).It should be mentioned that Theorem 1 is not true without the assumption that thegradients of u and u are non-vanishing. Indeed, even for H¨older continuous f (the weakercondition f ∈ L p (Ω), p > n , is imposed in Theorem 1), this follows from the example onp. 220 in [7].Let Ω = IR n and u be equal to zero identically. If p >
2, then u equal to (1 − | x | ) p when | x | ≤ | x | > C (IR n ). It is straightforward to check that(1) holds for u with f ( u ) = − p ( p − u − /p + 2 p ( n + 2 p − u − /p , which is H¨older continuous with the exponent 1 − /p and such that f (0) = 0. Thus, allassumptions of Theorem 1 are fulfilled for u and u except for the condition concerningtheir gradients; for both functions they vanish when | x | ≥
1. Therefore, the conclusion ofTheorem 1 is not true – these functions do not coincide.The proof of Theorem 1 is given in §
2; it is based on the so-called partial hodographtransform in n dimensions which allows us to use the weak Harnack type inequality provedin [19]. In §
3, we apply Theorem 1 to obtain bounds for solutions of the free-boundaryproblem mentioned above; it describes steady, periodic water waves with vorticity.2
Proof of Theorem 1
First, the local version of the n -dimensional partial hodograph transform is introduced. Itis used in the proof of an auxiliary lemma required for proving Theorem 1. Then a versionof Hopf’s lemma is discussed; the latter is applied in considerations of § Being defined locally, it generalises the transform introduced by Dubreil-Jacotin [6] in herstudies of water waves with vorticity in the two-dimensional case. Moreover, the transformproposed here extends that considered in [6] to the case of n > ⊂ IR n be a domain. If u is a function whose gradient does not vanish throughoutthis domain, then at any point x ∈ Ω the coordinate system can be chosen so that u x n ( x ) >
0. This allows us to introduce the following transform in a neighbourhood of x . We put q = ( q , . . . , q n − ) with q k = x k , k = 1 , . . . , n − , and p = u ( x ) , and take these as new independent variables. Furthermore, instead of u ( x ) satisfying, sayinequality (2), we consider h ( q, p ) = x n as the unknown. Then we have ∂h∂x k = h q k + h p ∂u∂x k = 0 for k = 1 , . . . , n − ∂h∂x n = h p u x n = 1 , and so h q k = − u x k u x n for k = 1 , . . . , n − h p = 1 u x n > . In view of the equalities u x k = − h q k h p , k = 1 , . . . , n − , and u x n = 1 h p , the weak formulation (3) of inequality (2) for u takes the following form in terms of h : Z Q (cid:20) −∇ q h · ∇ q w + 1 + |∇ q h | h p w p (cid:21) d q d p ≤ Z Q f ( p ) wh p d q d p . (5)Here Q – a neighbourhood of the point ( q , p ) – is the image of X which is the neighbourhoodof x that corresponds to ( q , p ) and w ( q, p ) stands instead of v ( x ( q, p )). It is also takeninto account that d x = h p d q d p and ∇ q h = ( h q , . . . , h q n − ) . Like (3), the last inequality must hold for every non-negative w ∈ C ( Q ).In the case of smooth h , a consequence of (5) is the differential inequality( Lh )( q, p ) ≥ f ( p ) h p ( q, p ) , where Lh = ∂ q k h q k − ∂ p |∇ q h | h p .
3t should be noted that the right-hand side of this inequality depending on f is linear in h ,whereas nonlinearity is present in the operator L on the left-hand side. It will be clear fromwhat follows that this is the advantage resulting from the introduced form of the hodographtransform. Moreover, this differs from what we have in (2), where the differential operatoris linear and nonlinearity is involved through the superposition operator f ( u ).Furthermore, let h , h ∈ C ( Q ), then we have Lh − Lh = ∂ q k ( h − h ) q k − ∂ p Z ∂ t |∇ q h ( t ) | h ( t ) p d t, where h ( t ) = th + (1 − t ) h . This difference can be written as an operator in divergent formwith continuous coefficients. Indeed, let W = h − h , then Lh − Lh = L W , where L W = ∂ q k W q k − ∂ p " W q k Z h ( t ) q k h ( t ) p d t + ∂ p " W p Z |∇ q h ( t ) | h ( t )2 p d t . Moreover, the inequalities n − X k =1 "Z h ( t ) q k h ( t ) p d t ≤ n − X k =1 Z h ( t )2 q k h ( t )2 p d t < Z |∇ q h ( t ) | h ( t )2 p d t show that L is an elliptic operator. This immediate corollary of Theorem 5.1 in [19] is given for the reader’s convenience. Itconcerns an inequality for the linear elliptic operator P ( ∂ ) = ∂ j ( a ij ∂ i ) + b j ∂ j + c . As usualthe ellipticity means that there exists λ > λ − | ξ | < a ij ξ i ξ j < λ | ξ | for all ξ = ( ξ , . . . , ξ n ) ∈ IR n \ { } . Here and below, the Einstein summation notation is used.
Lemma 1
Let u ∈ C (Ω) be a non-negative function satisfying the inequality P ( ∂ ) u ≤ in Ω (6) in the weak sense; here all coefficients a ij are measurable in Ω , b j ∈ L q (Ω) and c ∈ L q/ (Ω) for some q > n . If there exists x ∈ Ω such that u ( x ) = 0 , then u vanishes identically in Ω .Proof. Let ρ > K ρ ( x ) ⊂ Ω; by K σ ( x ) the open cube centred at x isdenoted which has edges equal to σ and sides parallel to the coordinate axes. The lemma’sassumptions about u and P yield the inequality ρ − n/γ k u k L γ ( K ρ ( x )) ≤ C min x ∈ K ρ ( x ) u ( x )4 x n p u ( x ) u ( x ) h ( q, p ) h ( q, p )Figure 1: A sketch of the partial hodograph transform with fixed x = q , . . . , x n − = q n − .It demonstrates the relationship between inequalities for two pairs of corresponding func-tions. If u ( x ) < u ( x ) near ( x , . . . , x n − , x n ), then h ( q, p ) > h ( q, p ) in a neighbourhoodof ( q, p ).for u satisfying (6); here C is a positive constant and γ is an arbitrary number from theinterval (1 , n/ ( n − x ∈ K ρ ( x ) u ( x ) = u ( x ) = 0 , there exists a neighbourhood of x , where u vanishes identically. It is clear that the maxi-mal such neighbourhood is Ω because otherwise the same argument can be applied to anyboundary point of the maximal neighbourhood which is an interior point of Ω, thus leadingto a contradiction. Remark 2
For a non-positive u satisfying the inequality P ( ∂ ) u ≥ the assertion of Lemma1 remains valid. Let us apply the partial hodograph transform to u and u in a neighbourhood of x . Thenthere exists a neighbourhood Q of ( q , p ) – the image of x – in which the inequalities( − j (cid:2) ( Lh j )( q, p ) − f ( p ) ∂ p h j ( q, p ) (cid:3) ≥ , j = 1 , , (7)follow from (2); here h and h are the functions corresponding to u and u , respectively.It is clear that h ( q , p ) = h ( q , p ), and Q can be taken so that h ≥ h in it because u ≤ u in Ω (see Figure 1).From (7) one obtains that W = h − h (it is non-negative in Q ) satisfies the inequality( L W )( q, p ) − f ( p ) W ( q, p ) ≤ q, p ) ∈ Q. W vanishes identically in Q , that is, h = h throughout Q , andso u = u in some neighbourhood of x . Thus the set, say E , where u coincides with u , isnon-empty. It is clear that E is closed in Ω, and so if E = Ω, then there exists x ∗ ∈ ∂E ∩ Ω.Applying the same considerations, we see that x ∗ has a neighbourhood belonging to E whichis a contradiction. In § Lemma 2
Let x be a point on a part of ∂ Ω belonging to the class C ,α , α ∈ (0 , , and let u ∈ C ( ¯ X ) satisfy (6) in the weak sense in X which is the intersection of a neighbourhoodof x with Ω , whereas P is such that all a ij ∈ C ,α ( X ) , b j ∈ L q ( X ) and c ∈ L q/ ( X ) forsome q > n .If u is non-negative in X and u ( x ) = 0 , then either u vanishes identically in X or ∂ n u ( x ) < , where ∂ n denotes the normal derivative on ∂ Ω directed to the exterior of Ω .Proof. The proof is essentially the same as that of Theorem 1.1 in [18], where the assumptionsimposed on b j and c (boundedness and non-negativity of the last coefficient) are superfluous.Therefore, we restrict ourselves to some necessary remarks and begin with a couple of minornotes. First, it is sufficient to prove the assertion in the case when ∂ Ω is flat near x , towhich the general case reduces by a change of variables. Second, in view of Lemma 1, onehas just to show that ∂ n u ( x ) < u > X .The only amendment that needs more details concerns the proof of Lemma 3.4 in [18],in which ∇ w should be estimated as follows (we keep the notation used in [18]): |∇ w ( x ) | ≤ C k∇ v k L ∞ Z Ω ρ | b ( y ) | + c ( y ) | x − y || x − y | n − d y ≤ C k∇ v k L ∞ (cid:0) ρ α k b k L q + ρ α k c k L q/ (cid:1) , where α = ( q − n ) / ( q − α = 2( q − n ) / ( q − Remark 3
For a non-positive u satisfying the inequality P ( ∂ ) u ≥ ∂ n u ( x ) < ∂ n u ( x ) > . In this section, we consider the two-dimensional nonlinear problem of steady, periodic wavesin an open channel occupied by an inviscid, incompressible, heavy fluid, say water. Thewater motion is assumed to be rotational which, according to observations, is the type of6otion commonly occurring in nature. A brief characterization of results obtained earlierfor this and other related problems is given in our paper [10]. Further details can be found inthe survey article [17] by Strauss; see also the recent papers [5], [12] and [13]. Here, our aimis to apply Theorem 1 in order to generalise conditions guaranteeing the validity of boundsfor solutions to the problem that were obtained in [11].
Let an open channel of uniform rectangular cross-section be bounded below by a horizontalrigid bottom and let water occupying the channel be bounded above by a free surface nottouching the bottom. The surface tension is neglected and the pressure is constant on thefree surface. Since the water motion is supposed to be two-dimensional and rotational andin view of the water incompressibility, we seek the velocity field in the form ( ψ y , − ψ x ), wherethe unknown function ψ ( x, y ) is referred to as the streamfunction (see, for example, [14] fordetails of this model). It is also supposed that the vorticity distribution ω (it is a functionof ψ as is explained in [14], §
1) is a prescribed function belonging to L ploc (IR) with p > ω was assumed to bea locally Lipschitz function.Non-dimensional variables are chosen so that the constant volume rate of flow per unitspan and the constant acceleration due to gravity are scaled to unity in our equations. Inappropriate Cartesian coordinates ( x, y ), the bottom coincides with the x -axis and gravityacts in the negative y -direction. The frame of reference is taken so that the velocity fieldis time-independent as well as the unknown free-surface profile. The latter is assumed tobe a simple C -curve, say Γ, which is Λ-periodic along the x -axis for some Λ >
0, but notnecessarily representable as the graph of an x -dependent function. (It was found numericallyby Vanden-Broeck [20] that there are such overhanging profiles bounding rotational flowswith periodic waves, whereas Constantin, Strauss and Varvaruca [5] recently investigatedthem rigorously; the relevant figures are presented in these papers.) Thus, the longitudinalsection of the water domain is the strip D Γ that lies between the x -axis and Γ, and ψ ( x, y )is assumed to be a Λ-periodic function of x in D Γ .Since the surface tension is neglected, the pair ( ψ, Γ) must be found from the followingfree-boundary problem: ψ xx + ψ yy + ω ( ψ ) = 0 , ( x, y ) ∈ D Γ ; (8) ψ ( x,
0) = 0 , x ∈ IR; (9) ψ ( x, y ) = 1 , ( x, y ) ∈ Γ; (10) |∇ ψ ( x, y ) | + 2 y = 3 r, ( x, y ) ∈ Γ . (11)Here r is a constant considered as the given problem’s parameter. Notice that the boundarycondition (10) allows us to write relation (11) (Bernoulli’s equation) as follows:[ ∂ n ψ ( x, y )] + 2 y = 3 r, ( x, y ) ∈ Γ . (12)Here and below ∂ n denotes the normal derivative on Γ; the normal n = ( n x , n y ) has unitlength and points out of D Γ . 7n this section, we keep the notation adopted in our previous papers, and so ψ and ω stand in (8) instead of u and f , respectively, used in (1) and (2). To give the precisedefinition how a solution of problem (8)–(11) is understood we need the following set Γ ψ = { ( x, y ) ∈ D Γ : ∇ ψ ( x, y ) = 0 } . Definition 1
The pair ( ψ, Γ) is called a solution of problem (8)–(11) with the vorticitydistribution ω ∈ L ploc (IR), p >
2, provided the following conditions are fulfilled for someΛ > • Γ is a simple, Λ-periodic along the x -axis C -curve; • ψ ( x, y ) is a Λ-periodic function of x belonging to C ( D Γ ); • the boundary conditions (9)–(11) are fulfilled pointwise; • the two-dimensional measure of Γ ψ is equal to zero and D Γ \ Γ ψ is a domain; • for all v ∈ C ∞ ( D Γ \ Γ ψ ) the following identity holds: Z D Γ ∇ ψ · ∇ v d x d y = Z D Γ ω ( ψ ) v d x d y. The last condition means that ψ is a weak solution of (8) in D Γ \ Γ ψ . The results presented in this section were obtained in [9] under the assumption that ω is aLipschitz function. Since they are essential for our considerations, what follows is a digestof these results valid under the assumption that ω ∈ L loc (IR).First, let s >
0, then by U ( y ; s ) we denote a strictly monotonic solution of the followingCauchy problem: U ′′ + ω ( U ) = 0 , y ∈ IR; U (0; s ) = 0 , U ′ (0; s ) = s ;here and below ′ stands for d / d y . It is straightforward to obtain the implicit formula y = Z U d τ p s − τ ) , Ω( τ ) = Z τ ω ( t ) d t, (13)that defines U on the maximal interval of monotonicity ( y − ( s ) , y + ( s )), where y ± ( s ) = Z τ ± ( s )0 d τ p s − τ ) , and the definition of τ ± ( s ) is as follows. By τ + ( s ) and τ − ( s ) we denote the least positiveand the largest negative root, respectively, of the equation 2 Ω( τ ) = s . If this equation hasno positive (negative) root, we put τ + ( s ) = + ∞ ( τ − ( s ) = −∞ respectively) . Second, we consider the problem u ′′ + ω ( u ) = 0 on (0 , h ) , u (0) = 0 , u ( h ) = 1 , (14)8n the class of monotonic functions. It is clear that formula (13) gives a solution of problem(14) on the interval (0 , h ( s )), where h ( s ) = Z d τ p s − τ ) and s > s = r τ ∈ [0 , Ω( τ ) ≥ . (15)Moreover, all monotonic solutions of problem (14) have the form (13) on the interval (0 , h ).This remains valid for s = s with h = h = Z d τ p s − τ ) < ∞ , that is , h = lim s → s h ( s ) . It is clear that h ( s ) decreases strictly monotonically from h and asymptotes zero as s → ∞ .Furthermore, the pair ( u, Γ) with u ( y ) = U ( y ; s ) and Γ = { ( x, y ) : x ∈ IR , y = h ( s ) } is a solution of problem (8)–(11) provided s is found from the equation R ( s ) = r , where R ( s ) = [ s − h ( s )] / . (16)The latter function has only one minimum, say r c >
0, attained at some s c > s . Hence if r ∈ ( r c , r ), where r = lim s → s +0 R ( s ) = 13 (cid:2) s − h (cid:3) , then equation (16) has two solutions s + and s − such that s < s + < s c < s − . By substitut-ing s + and s − into (13) and (15), one obtains the so-called stream solutions ( u + , H + ) and( u − , H − ), respectively. Indeed, these solutions satisfy Bernoulli’s equation[ u ′± ( H ± )] + 2 H ± = 3 r along with relations (14). It should be mentioned that s − and the corresponding H − existfor all values of r greater than r c , whereas s + and H + exist only when r is less than or equalto r ; in the last case s + = s . ( ψ, Γ) To express bounds for non-stream solutions of problem (8)–(11) we use solutions of problem(14) and the values r c , H − and H + ; the last two serve as bounds forˆΓ = max ( x,y ) ∈ Γ y and ˇΓ = min ( x,y ) ∈ Γ y. Now we formulate results generalising Theorems 1.1 and 1.2 in [11] for periodic solutions.
Theorem 2
Let ( ψ, Γ) be a non-stream solution of problem (8) – (11) in the sense of Defi-nition 1. Then the following two assertions are true provided ψ ≤ on D Γ . . If ˇΓ < h , then ψ ( x, y ) < U ( y ; ˇ s ) in the strip IR × (0 , ˇΓ) , (17) where U is defined by formula (13) and ˇ s > s is such that h (ˇ s ) = ˇΓ . Moreover, theinequalities ( A ) r ≥ r c , ( B ) H − ≤ ˇΓ hold, and if r ≤ r , then ( C ) ˇΓ ≤ H + .2. If h = + ∞ and ˇΓ = h , then inequality (17) is nonstrict, whereas inequalities ( A ) – ( C ) are true. Theorem 3
Let ( ψ, Γ) be a non-stream solution of problem (8) – (11) in the sense of Defi-nition 1. If ˆΓ < h and ψ ≥ on D Γ , then ψ ( x, y ) > U ( y ; ˆ s ) in D Γ , (18) where U is defined by formula (13) and ˆ s > s is such that h (ˆ s ) = ˆΓ . Moreover, ˆΓ ≥ H + provided r ≤ r and ψ ≤ on D Γ . It occurs that some inequalities in Theorems 2 and 3 are strict under the assumptionthat the first derivatives of ψ are H¨older continuous near the points, where the values ˇΓ andˆΓ are attained, as the following assertions demonstrate. Proposition 1
Let ( ψ, Γ) be a non-stream solution of problem (8) – (11) in the sense ofDefinition 1 and such that ψ ≤ on D Γ . Also, let Γ be of the class C ,α , α ∈ (0 , , nearsome point ( x , ˇΓ) ∈ Γ . If ψ ∈ C ,α ( ¯ X ) , where X is the intersection of D Γ with a sufficientlysmall neighbourhood of ( x , ˇΓ) , then the inequalities are strict in ( A ) and ( B ) , and if r ≤ r ,then the inequality in ( C ) is also strict. Proposition 2
Let ( ψ, Γ) be a non-stream solution of problem (8) – (11) in the sense ofDefinition 1 and such that ψ ≥ on D Γ . Also, let Γ be of the class C ,α , α ∈ (0 , , nearsome point ( x , ˆΓ) ∈ Γ . If ψ ∈ C ,α ( ¯ X ) , where X is the intersection of D Γ with a sufficientlysmall neighbourhood of ( x , ˆΓ) , then ˆΓ > H + provided r ≤ r and ψ ≤ on D Γ . First, let ˇΓ < h , and so there exists ˇ s > s such that h (ˇ s ) = ˇΓ, whereas the function U ( y ; ˇ s )solves problem (14) on (0 , ˇΓ). Moreover, formula (13) defines U ( y ; ˇ s ) on the half-axis y ≥ ω ( t ) is extended by − t >
1. This implies that ˇ s > τ ≥ Ω( τ ), and wehave U ′ ( y ; ˇ s ) = p ˇ s − U ( y ; ˇ s )) > y ≥ . Hence U ( y ; ˇ s ) is a monotonically increasing function of y , U ( y ; ˇ s ) > y > h and U ( y ; ˇ s ) → + ∞ as y → + ∞ .Putting U ℓ ( y ) = U ( y + ℓ ; ˇ s ) for ℓ ≥
0, we see that U ℓ ( y ) > , ˇΓ] when ℓ > ˇΓ.Therefore, U ℓ − ψ > D Γ for ℓ > ˇΓ. Let us show that there is no ℓ ∈ (0 , ˇΓ) such thatmin ( x,y ) ∈ D Γ { U ℓ ( y ) − ψ ( x, y ) } = 0 . (19)10ssuming that such a value exists (in the case when there are several such values, wedenote by ℓ the largest of them), we see that (19) holds only when U ℓ ( y ) − ψ ( x , y ) = 0 for some ( x , y ) ∈ IR × (0 , ˇΓ)because U ℓ − ψ is separated from zero on D Γ \ [IR × (0 , ˇΓ)]. Moreover, (19) implies that ∇ ψ = ∇ U ℓ at ( x , y ) , and so ( x , y ) ∈ D Γ \ Γ ψ . Since D Γ \ Γ ψ is a domain, Theorem 1 is applicable in D Γ \ Γ ψ , which yields that ψ coincideswith U ℓ there. Hence these functions coincide in D Γ because Γ ψ has the zero measure.However, this contradicts to the fact that U ℓ − ψ is separated from zero on D Γ \ [IR × (0 , ˇΓ)].The obtained contradiction shows that U ( y ; ˇ s ) ≥ ψ ( x, y ) on D Γ and vanishes when y = 0.Moreover, Theorem 1 implies that U ( · ; ˇ s ) and ψ cannot be equal at an inner point of D Γ \ Γ ψ because the latter function is not a stream solution. Furthermore, ∇ U = 0 on Γ ψ , and sothe extended U is strictly greater than ψ on D Γ which completes the proof of (17).To show that (A)–(C) are valid, we consider a point, say ( x , ˇΓ), at which the curve Γis tangent to y = ˇΓ, and so U (ˇΓ; ˇ s ) − ψ ( x , ˇΓ) = 0 because both terms on the left-hand sideare equal to one. It was proved that U ( y ; ˇ s ) − ψ ( x, y ) ≥ × [0 , ˇΓ], which implies (cid:2) U ′ ( y ; ˇ s ) − ψ y ( x, y ) (cid:3) ( x,y )=( x , ˇΓ) ≤ . (20)Since Bernoulli’s equation at ( x , ˇΓ) has the form ψ y ( x , ˇΓ) = p r − U ′ (ˇΓ; ˇ s ) ≤ p r − ⇐⇒ ˇ s − ≤ r − . Hence R (ˇ s ) ≤ r in view of (16), and combining the latter inequality and h (ˇ s ) = ˇΓ, oneobtains that (A) and (B) are true in assertion 1. Moreover, (C) is also true provided r ≤ r .Now we turn to assertion 2 and begin with the case when ˇ s = s >
0; here the equalityis a consequence of the assumption that ˇΓ = h . Let us introduce ω ( ǫ ) ( τ ) = ω ( τ ) − ǫ , where ǫ > ( ǫ ) ( τ ) and U ( ǫ ) ( y ; s ) be defined by formulae (13) with ω changedto ω ( ǫ ) ; similarly, we define h ( ǫ ) ( s ) using (15), whereas to obtain H ( ǫ )+ and H ( ǫ ) − one has tocombine (16) and (15).Since s >
0, we have that s ( ǫ )0 = r τ ∈ [0 , Ω ( ǫ ) ( τ ) < s . Furthermore, it is straightforward to verify the inequalities h ( ǫ ) ( s ) < h ( s ) and U ( ǫ ) ( y ; s ) > U ( y ; s ) for s ≥ s . Therefore, U ( ǫ ) ( y ; s ) solves the problem on (0 , ˇΓ) analogous to (14), but with ω is changedto ω ( ǫ ) and with the value U ( ǫ ) (ˇΓ; s ) greater than one. Moreover, in view of the inequality U ( ǫ ) ′′ + ω (cid:16) U ( ǫ ) (cid:17) ≥ , ˇΓ) , U ( · ; ˇ s ) are valid for U ( ǫ ) ( · ; s ) as well, thus yielding ψ ( x, y ) < U ( ǫ ) ( y ; ˇ s ) for ( x, y ) ∈ IR × (0 , ˇΓ) , (21)which is similar to (17); here it is also taken into account that ˇ s = s . Letting ǫ → ψ ( x, y ) ≤ U ( y ; ˇ s ) for ( x, y ) ∈ IR × (0 , ˇΓ) , form which the inequalities in (A)–(C) follow in the same way as above.Now let s = 0. First we assume that Ω( τ ) < τ ∈ (0 , U ′ ( y ; s ) > y > U ′ (0; s ) = 0for the function defined by formulae (13). Then the considerations used in the case whenˇΓ < h are applicable. Otherwise, the considerations based on U ( ǫ ) ( y ; s ) yield (21), and theresults follow letting ǫ → At its initial stage the proof of this theorem is similar to that of Theorem 2. Namely, weconsider the case when ˆΓ < h first. Since there exists ˆ s > s such that h (ˆ s ) = ˆΓ, thefunction U given by formula (13) with s = ˆ s solves problem (14) on (0 , ˆΓ). Moreover, thesame formula defines this function for all y ≤ ˆΓ provided ω ( t ) is extended to t < s > τ ). Then U ′ ( y ; ˆ s ) = p ˆ s − U ( y ; ˆ s )) > y ≤ ˆΓ , and so U ( y ; ˆ s ) is a monotonically increasing function of y such that U ( y ; ˆ s ) < y < U ℓ ( y ) = U ( y − ℓ ; ˆ s ) for ℓ ≥
0, which implies that U ℓ ( y ) < , ˆΓ] provided ℓ > ˆΓ.Therefore, U ℓ − ψ < D Γ for ℓ > ˆΓ. Similarly to the proof of Theorem 2, one obtainsthat there is no ℓ ∈ (0 , ˆΓ) such thatmax ( x,y ) ∈ IR × [0 , ˆΓ] { U ℓ ( y ) − ψ ( x, y ) } = 0 . Hence U ( y ; ˆ s ) − ψ ( x, y ) is non-positive on D Γ and vanishes when y = 0. Now, applyingTheorem 1 in the same way as in the proof of Theorem 2, we arrive at inequality (18).To prove that ˆΓ ≥ H + , we argue by analogy with the proof of Theorem 2. In view ofperiodicity of Γ, there exists ( x , y ) ∈ Γ such that y = ˆΓ (it is clear that Γ is tangent to y = ˆΓ at this point). Then U (ˆΓ; ˆ s ) − ψ ( x , ˆΓ) = 0 because both terms on the left-hand sideare equal to one. Since ψ is a non-stream solution, then (cid:2) U ′ ( y ; ˆ s ) − ψ y ( x, y ) (cid:3) ( x,y )=( x , ˆΓ) ≥ . ψ at ( x , ˆΓ), we show that U ′ (ˆΓ; ˆ s ) ≥ q r − . (22)Indeed, it follows from the boundary condition ψ ( x , ˆΓ) = 1 and the assumption that ψ ≤ D Γ that ψ y ( x , ˆΓ) is non-negative, and so ψ y ( x , ˆΓ) = q r − . Combining this and the inequality preceding (22), we see that (22) is true, which impliesthe required inequality.
Since the proof of Proposition 2 is similar to that of Proposition 1, we restrict ourselves toproving the latter assertion only.To prove Proposition 1, we notice that there exists ˇ s > s such that h (ˇ s ) = ˇΓ and U ′ ( y ; ˇ s ) > y ≥
0; here U ( · ; ˇ s ) is defined by formula (13) provided ω ( t ) is extended to t > ψ y ( x , ˇΓ) > x , ˇΓ) so thatin both cases the image of this point is ( q ,
1) on the ( q, p )-plane. In the first case, some X ⊂ D Γ is mapped to a neighbourhood Q ⊂ { ( q, p ) : q ∈ IR , p < } , and h ( q, p ) in Q corresponds to ψ ( x, y ) defined in X . In the second case, some X ⊂ IR × (0 ,
1) is mapped to another neighbourhood Q on the same plane as Q , whereas h ( q, p )in Q corresponds to U ( y ; ˇ s ). Since inequality (17) holds for ψ and U ( · ; ˇ s ), we have that h − h > Q ∩ Q which is the intersection of { ( q, p ) : q ∈ IR , p < } with aneighbourhood of ( q , h − h vanishes at ( q ,
1) because ψ ( x , ˇΓ) = U (ˇΓ; ˇ s ) = 1.Then it follows from Lemma 2 that[ ∂ p ( h − h )] ( q,p )=( q , < , which implies that inequality (20) is strict. Using this fact in the considerations that follow(20), we obtain that the inequalities in (A)–(C) are strict. In his renown book [2], the first edition of which was published in 1918, Carath´eodory hadproved a quite general theorem for the first order ordinary differential equation. It concernsthe existence of a solution which satisfies the equation on an interval up to a set of Lebesgue-measure zero. The proof is based on assumptions whose general form is now referred to asthe Carath´eodory condition (see [1], § f is supposed to be continuous in almost all papers, dealing with equation (1),inequality (2) and their generalisations (see, for example, the notes [8] and [16] by Kellerand Osserman, respectively, dating back to 1957, and numerous papers citing these notes).It is also worth mentioning in this connection, that non-uniqueness takes place for the firstorder ordinary differential equation when the smoothness of a nonlinear term is less thanLipschitz with respect to the unknown function.Furthermore, considering equation (1) in [7] (see Theorem 1 on p. 209), the authorsrequire even more, namely, that f is of class C . This substantially simplifies treatment ofthe equation comparing with Theorem 1 in the present paper, where f ∈ L ploc (IR) for p > n .On the other hand, the assumption imposed on solutions in our theorem, namely, that theirgradients are non-vanishing, is essential. This condition allows us to avoid non-uniquenesseven without the Carath´eodory condition.Turning to the problem of periodic water waves with vorticity, the papers [4] and [15]should be mentioned. Discontinuous vorticity distributions from L ∞ are considered in thefirst of them, whereas the distribution is merely L p -integrable with an arbitrary p ∈ (1 , ∞ )in the second one. However, only unidirectional flows (they have no stagnation points withinthe fluid) are studied in both papers, and in this case the global partial hodograph transformcan be applied to simplify the problem, thus reducing the effect of non-smooth vorticity.It is worth mentioning that the assumptions on a solution of the water wave problemhere are weaker not only than those imposed in our recent paper [11], but also than those in[12]. Indeed, the most restrictive condition in [12] is that the horizontal component of thevelocity field is bounded from below by a positive constant. Acknowledgements.
V. K. was supported by the Swedish Research Council (VR). N. K.acknowledges the support from the G. S. Magnuson’s Foundation of the Royal SwedishAcademy of Sciences and Link¨oping University.
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