J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 95
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The transformation S thus extended will be referred to by thesymbol S. S is easily seen to be a linear transformation and the.1inequality (12.2.25) holds forholds for all the functions g nOnce\1\with the same value ofKsinceitindependent of n.the definition of the transformation S is extended so thatitit,applies to functions in B 2 it becomes possible to widen the class offunctions within which a (generalized ) solution of our linear problem,can be sought, and at the same time to reformulate the boundaryproblem in terms of the the following functional equation:(12.2.26)inwhicho>(0>)Sg(<p) refers to the=Sg(<p)+ ia^r***,above extension of Sg((p), withg(q>)C1? 2 ,evaluated on the boundary0. By virtue of (12.2.20) and the\pdefinition of Sg(y>) for g(q>) in J5 2 one might expect that <*>(<p) wouldyield correct boundary values for the solution a>(q> 9 y).proceedto show that this is indeed the case, i.e.
that any continuous function,WegCBwhich is given by (12.2.26) for2 furnishes the boundaryvalues of a function c*)((p,y)) defined and continuous for ipwhich isahasrealinthelowerawithcontinuoushalf plane,part Q(<p,y)analyticnormal derivative Qv in the closed half plane, and such that the boundary<*>($)^condition (12.2.14) with F((p) == g(<p) is satisfied.LEVI-CIVITA'STHEORY535Theregularity properties of the function co((p 9 y) on the boundarycome about because of certain smoothing properties of theytransformation S, which we proceed to discuss. Consider first thewhich g(<p) is given, as insum. The function a(<p) + $(<?) = Sg:special case inZa+ ?pp =^_L. .-*= ^7o?8(12.2.23),--b * sin yfo.~rv^v7by afiniteFourier6 * coshas in this case the following property:-&-g+ A5.Thevalidity of this formula for *S follows from the fact that Sg(<p)^(9?, y;) as given by (12.2.24) is an analytic function in theclosed half plane y0, and that Sg satisfies the nonhomogeneousa (9 V)?>+^=(12.2.14) with Fg, when 6(<p, \p) is identifiedThis implies, because of (12.2.26) and the triangle in-boundary conditionwitha(<p, \p).equality, the inequality||,11^AMIgH, #! =constant,as one easily sees.
If g is any function in J3 2 it now can be proved thatisin the manner described abovedefined for functions g in2,BSgsuch that its imaginary part /9 has a continuous derivative with respectto (p. This is done by approximating g uniformly by a sequence g n offinite Fourier sums in B^ The corresponding derivatives f$ ntp form aCauchy sequence because of the above inequality and hence wouldg+Aoc alsoconverge to a continuous function. The relationis again seen to bewould hold in the limit for the derivative /^; thuscontinuous.
It follows, therefore, that a continuous function co((p) =^^0(<p)+&(v) gi yen by (12.2.26) has the property that r((p) has a conti= g(q>)+AO(q>)- We observenuous derivative, and in addition^(9?)next that 6+ir furnishes the boundary values of an analytic functionand continuous in the closed halfdefined foroo < \p <00(9?, y>)plane: this follows again by approximating g(<p)J?!, as in (12.2.23), defining the correspondingand making the passage to thethat the functionsco n ((p, ip)limit to obtainby functions g n (<p)o> n (<p,iny) by(12.2.20),)=0(<p,y) ) +ir((p,ip );to a continuous function fora>(<p,\pconvergeis analytic at interior points followslimitthatthey>since it is the uniform limit of analytic functions. Sincer^((p, ip)O v (<p, y))fory)0, and since r^ is itself a harmonic function with con-^we know, and<=WATER WAVES586tinuous boundary values r^ it follows that rv (<p y>) -> r^ as y -> 0,and hence that V is also continuous for \p0, i.e.
6 v ((jp 0)V;9==9= g(<p) + A0, which we have proved above= g(g?), and this is our boundary condition.hence the conditionrvto hold becomes VA0have therefore shown that a continuous functionWea)(<f>)whichisgiven by (12.2.26), with g(^) any function in B 2 furnishes the bounwhichit in \p <dary values of an analytic function co(^, y) = 6ahascontinuousderiwhoserealis continuous for0,part 0(^, ip)y,+^=vative V in the closed lower half plane with V (^, 0)A0(<, 0)g(^)Inthesubsectionaswealsowriteit:A0or,g.immediatelyV=weshall establish the existence of a continuous solutionfollowingof (12.2.26)o}(<f>)linearway onwhen g(^)c5(<)-isnot given a= 0(g) +ir(<f>),priori,i.e.but dependswhen g(<)ina non-= F(<f>) =*sin*sin00) +(cf. (12.2.12)). Assuming this to havebeen proved, we proceed to draw at once further conclusions regardingthe properties of oj(<f>) and its continuation a)(<f> y>) as an analytic3A(tf-?- 39function in the lower half plane ip0.
We show, in fact, that thesolution a>(<f>) in B of our nonlinear functional equation will not onlyfurnish the boundary values of an analytic function a)(<f> ip ) in \p0,^<9withcoasboundary values, but that^ 0,a)has continuousfirst derivativesandis as a consequence then seenin particular, cfl(^) wouldThus,\paFourierConsiderseries.the analytic functionpossessconvergentinlowerthehalf plane ywith boundary valuesF(x) defined=withforandat oo. The bounbounded<%e fi(%)\p&(%)g(y)in the closed half planeipactually to be analytic for0.<=dary conditionVWA0= g(^)satisfiedby oursolu-also extended analytically into the lower half planemeans of the relation 3le(a)^3te &(%) in theAco)tion a)((p<y>)is< bymanner used frequentlyifr^ia>=in Chapters 3Acoand4;hence we have=oo.
Weco and co both vanish for y =xhave just seen above that co has an imaginary part r with a continuousderivative T on the boundary y = 0. The fact that r^ is continuousVthen makes it possible to show that co((p, y) is Holder continuous for= 0. This follows, in fact, from a classical theorem of Privaloff (Bull.y>Soc. Math. France, Vol. 44, 1916) which states that a function which isdefined and continuous in the unit circle, analytic in the interior ofthe circle, and has an imaginary part which is Holder continuous onsince the imaginary parts ofLEVI-CIVITA'STHEORY537the boundary of the circle, is itself Holder continuous in the closedunit circle: in other words, Holder continuity of the imaginarypartbrings with it the Holder continuity of the real part of the function.This theorem is made applicable in the present case by mapping oneof the period strips of the solution a)(%) in the #-plane conformally onthe unit circle of a -plane, say: we know, in fact, that CD has the realperiod 2n/L The part of the boundary of the strip given by y)(i.e.
a full period interval on the boundary) is mapped on the boundaryof the unit circle. (Since co ->as \p -+oo, the infinity of the=||+=mapped onthe center of the circle.) Thus coiv has an0, and itimaginary part with a continuous derivative rv on y>follows that r is certainly Holder continuous for \p0. Consequentlystripis=the real part of co(), hence co() itself, is Holder continuous in theclosed unit circle, since this property is not destroyed by the conformalmapping.
Thebyg(<p), is=0, which is givenF(%) on the boundary \pseen to be Holder continuous, simply because of thereal part ofnowis given in terms ofand r, and the Holder continuity ofg(<f>)the latter functions. A second application of Privaloff 's theorem, thistime to F(%), then leads to the Holder continuity of P(%) for \p0.way^= %*(%) thus holds for = and it shows that= 0, since both co and F have this property. Inco is continuous forx= 0. Finally, onceother words co^ and co v are both continuous forTherelationico xAcoy>\p\pco(%)is shown to have a continuous derivative with respect to % on theboundary, we could make use of a theorem of H.
Lewy [L.9] to showthat co(%)12.2d.Theisactually analytic on the boundary.solution of the nonlinear boundary value problemThe nonlinear problemto be solved here differsfrom thelinearproblems discussed above because of the fact that the function g(<p)the nonhomogeneous term in the free surface condition, is not givena priori, but rather becomes known only when the solution co(q>, \p)9itself isdetermined.Onthe other hand, we have seen that the equaboundary values <o(<p) for co(<p, y), in casetion (12.2.26) furnishes thea known function in the spaceBTosolve the nonlinear problem we now reverse this process: we regard the equation (12.2.26) asa functional equation for the determination of the function 6>(<p) wheng((jp)is2.with the function F(<p) in equation (12.2.12), i.e.when g(<p) itself depends in an explicitly given way on co(<p).
The discussion of the preceding subsection shows that we have to proveg(y>) is identifiedWATER WAVES538only that the functional equation has a solution o> (y) in the Banachspace B.The existence of the solution 6>((p) of the functional equation willbe carried out, as we have stated earlier, by an iteration processapplied to the functional equation. To this end it is convenient tointroduce a nonlinear transformation R defined for any functionin the whole space B by means of the relationagifi=+(12.2.27)= X(e-M sin a -Rg(<p)a)+ ee~^ sin a.In order that the transformation S defined in the preceding subsectionshould be applicable to Rg(<p) we require, as part of the definition ofR, that e should be so determined that Rg(<p) lies in B 2 and thus lacks"its firstFourier coefficient,such thati.e.Rg((p) sin h<pd(pJonthis leads to the following condition"*(12.2.28)= -A(~I3^sine-wasin=0;e:a) sin hpdcpa sin/.<pd<pThis implies that R is defined only if the denominator of (12.2.28)does not vanish.
Clearly, this nonlinear transformation yields alwaysrealodd functions.Considernow(12.2.29)the functional equation<b(<p)= SRco((p) + iaf**>whichSg((p) refers to the extension of Sg(y>) as defined in the preceding subsection for functions g(<p) in the space I? 2 , on the boundaryand a x is a given real constant. Because of (12.2.12), (12.2.14),yin9=and the discussion of the preceding paragraph, it follows that a solution co(%) of the nonlinear boundary value problem will be establishedonce a functionc5(<p)inB is found which satisfies the functional equa-tion (12.2.29).In carrying out the existence proof it is convenient to introduce afew new notations.
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