On Hilbert and Riemann problems. An alternative approach
aa r X i v : . [ m a t h . C V ] O c t On Hilbert and Riemann problems.An alternative approach.
Vladimir Ryazanov
Abstract -
Recall that the Hilbert (Riemann-Hilbert) boundary valueproblem was recently solved in [11] for arbitrary measurable coefficients andfor arbitrary measurable boundary data in terms of nontangential limits andprincipal asymptotic values. Here it is developed a new approach makingpossible to obtain new results on tangential limits. It is shown that thespaces of the found solutions have the infinite dimension for prescribed col-lections of Jordan arcs terminating in almost every boundary point. Similarresults are proved for the Riemann problem.
Key words and phrases :
Hilbert and Riemann problems, analyticfunctions, limits along Jordan arcs, tangential limits, nonlinear problems.
Mathematics Subject Classification (2010) : primary 31A05, 31A20,31A25, 31B25, 35Q15; secondary 30E25, 31C05, 34M50, 35F45. The Hilbert (Riemann-Hilbert) boundary value problem, the Riemann andPoincare boundary value problems are basic in the theory of analytic func-tions and they are closely interconnected, see for the history e.g. the mono-graphs [3], [9] and [16], and also the last works [11]-[14].Recall that the classical setting of the
Riemann problem in a smoothJordan domain D of the complex plane C was on finding analytic functions f + : D → C and f − : C \ D → C that admit continuous extensions to ∂D and satisfy the boundary condition f + ( ζ ) = A ( ζ ) · f − ( ζ ) + B ( ζ ) ∀ ζ ∈ ∂D (1)with prescribed H¨older continuous functions A : ∂D → C and B : ∂D → C .Recall also that the Riemann problem with shift in D was on findingsuch functions f + : D → C and f − : C \ D → C satisfying the condition f + ( α ( ζ )) = A ( ζ ) · f − ( ζ ) + B ( ζ ) ∀ ζ ∈ ∂D (2)1where α : ∂D → ∂D was a one-to-one sense preserving correspondencehaving the non-vanishing H¨older continuous derivative with respect to thenatural parameter on ∂D . The function α is called a shift function . Thespecial case A ≡ jump problem .The classical setting of the Hilbert (Riemann-Hilbert) boundaryvalue problem was on finding analytic functions f in a domain D ⊂ C bounded by a rectifiable Jordan curve with the boundary conditionlim z → ζ Re { λ ( ζ ) · f ( z ) } = ϕ ( ζ ) ∀ ζ ∈ ∂D (3)with functions λ and ϕ that are continuously differentiable with respect tothe natural parameter s on ∂D and, moreover, | λ | 6 = 0 everywhere on ∂D .Hence without loss of generality one can assume that | λ | ≡ ∂D .It is clear that if we start to consider the Hilbert and Riemann problemswith measurable boundary data, the requests on the existence of the limits atall points ζ ∈ ∂D and along all paths terminating in ζ lose any sense (as wellas the conception of the index). Thus, the notion of solutions of the Hilbertand Riemann problems should be widened in this case. The nontangentiallimits were a suitable tool from the function theory of one complex variable,see e.g. [11]–[14]. Here it is proposed an alternative approach admittingtangential limits.Given a Jordan curve C in C , we say that a family of Jordan arcs { J ζ } ζ ∈ C is of class BS (of Bagemihl–Seidel class) , cf. [1], 740–741, if all J ζ liein a ring R generated by C and a Jordan curve C ∗ in C , C ∗ ∩ C = Ø, J ζ is joining C ∗ and ζ ∈ C , every z ∈ R belongs to a single arc J ζ , and for asequence of mutually disjoint Jordan curves C n in R such that C n → C as n → ∞ , J ζ ∩ C n consists of a single point for each ζ ∈ C and n = 1 , , . . . .In particular, a family of Jordan arcs { J ζ } ζ ∈ C is of class BS if J ζ isgenerated by an isotopy of C . For instance, every curvilinear ring R one ofwhose boundary component is C can be mapped with a conformal mapping g onto a circular ring R and the inverse mapping g − : R → R maps radiallines in R onto suitable Jordan arcs J ζ and centered circles in R onto Jordancurves giving the corresponding isotopy of C to other boundary componentof R .Now, if Ω ⊂ C is an open set bounded by a finite collection of mutuallydisjoint Jordan curves, then we say that a family of Jordan arcs { J ζ } ζ ∈ ∂ Ω is of class BS if its restriction to each component of ∂ Ω is so.
Theorem 1.
Let D be a bounded domain in C whose boundary consistsof a finite number of mutually disjoint rectifiable Jordan curves, and let λ : ∂D → C , | λ ( ζ ) | ≡ , ϕ : ∂D → R and ψ : ∂D → R be measurablefunctions with respect to the natural parameter. Suppose that { γ ζ } ζ ∈ ∂D is afamily of Jordan arcs of class BS in D .Then there exist single-valued analytic functions f : D → C such that lim z → ζ Re { λ ( ζ ) · f ( z ) } = ϕ ( ζ ) , (4)lim z → ζ Im { λ ( ζ ) · f ( z ) } = ψ ( ζ ) (5) along γ ζ for a.e. ζ ∈ ∂D with respect to the natural parameter. Remak 1.
Thus, the space of all solutions f of the Hilbert problem (4)in the given sense has the infinite dimension for any prescribed ϕ , λ and { γ ζ } ζ ∈ D because the space of all measurable functions ψ : ∂D → R has theinfinite dimension. Proof.
Indeed, set Ψ( ζ ) = ϕ ( ζ ) + iψ ( ζ ) and Φ( ζ ) = λ ( ζ ) · Ψ( ζ ) for all ζ ∈ ∂D . Then by Theorem 2 in [1] there is a single-valued analytic function f such that lim z → ζ f ( z ) = Φ( ζ ) (6)along γ ζ for a.e. ζ ∈ ∂D with respect to the natural parameter. Then alsolim z → ζ λ ( ζ ) · f ( z ) = Ψ( ζ ) (7)along γ ζ for a.e. ζ ∈ ∂D with respect to the natural parameter. (cid:3) Similar result can be formulated for arbitrary Jordan domains in termsof the harmonic measure.
Theorem 2.
Let D be a bounded domain in C whose boundary consistsof a finite number of mutually disjoint Jordan curves, and let λ : ∂D → C , | λ ( ζ ) | ≡ , ϕ : ∂D → R and ψ : ∂D → R be measurable functions withrespect to the harmonic measure. Suppose that { γ ζ } ζ ∈ ∂D is a family ofJordan arcs of class BS in D .Then there exist single-valued analytic functions f : D → C such that lim z → ζ Re { λ ( ζ ) · f ( z ) } = ϕ ( ζ ) , (8)lim z → ζ Im { λ ( ζ ) · f ( z ) } = ψ ( ζ ) (9) along γ ζ for a.e. ζ ∈ ∂D with respect to the harmonic measure. Remak 2.
Again, the space of all solutions f of the Riemann-Hilbertproblem (8) in the given sense has the infinite dimension for any prescribed ϕ , λ and { γ ζ } ζ ∈ D because the space of all functions ψ : ∂D → R that aremeasurable with respect to the harmonic measure has the infinite dimension. Proof.
Theorem 2 is reduced to Theorem 1 in the following way.First, there is a conformal mapping ω of D onto a circular domain D ∗ whose boundary consists of a finite number of circles and points, see e.g.Theorem V.6.2 in [4]. Note that D ∗ is not degenerate because isolatedsingularities of conformal mappings are removable that is due to the well-known Weierstrass theorem, see e.g. Theorem 1.2 in [2]. Applying in thecase of need the inversion with respect to a boundary circle of D ∗ , we mayassume that D ∗ is bounded.Remark that ω is extended to a homeomorphism ω ∗ of D onto D ∗ , seee.g. point (i) of Lemma 3.1 in [14]. Set Λ = λ ◦ Ω, Φ = ϕ ◦ Ω and Ψ = ψ ◦ Ωwhere Ω : ∂ D ∗ → ∂D is the restriction of Ω ∗ := ω − ∗ to ∂ D ∗ . Let us showthat these functions are measurable with respect to the natural parameteron ∂ D ∗ .For this goal, note first of all that the sets of the harmonic measure zeroare invariant under conformal mappings between multiply connected Jordandomains because a composition of a harmonic function with a conformalmapping is again a harmonic function. Moreover, a set E ⊂ ∂ D ∗ has theharmonic measure zero if and only if it has the length zero, say in view of theintegral representation of the harmonic measure through the Green functionof the domain D ∗ , see e.g. Section II.4 in [10].Hence Ω and Ω − transform measurable sets into measurable sets be-cause every measurable set is the union of a sigma-compact set and a set ofmeasure zero, see e.g. Theorem III(6.6) in [15], and continuous mappingstransform compact sets into compact sets. Thus, the functions λ , ϕ and ψ are measurable with respect to the harmonic measure on ∂D if and onlyif the functions Λ, Φ and Ψ are measurable with respect to the naturalparameter on ∂ D ∗ .Then by Theorem 1 there exist single-valued analytic functions F : D → C such that lim w → ξ Re { Λ( ξ ) · F ( w ) } = Φ( ξ ) , (10)lim w → ξ Im { Λ( ξ ) · F ( w ) } = Ψ( ξ ) (11)along Γ ξ = ω ( γ Ω( ξ ) ) for a.e. ξ ∈ ∂ D ∗ with respect to the natural parameter.Thus, by the construction the functions f = F ◦ ω are the desired analyticfunctions f : D → C satisfying the boundary conditions (8) and (9) along γ ζ for a.e. ζ ∈ ∂D with respect to the harmonic measure. (cid:3) Remark 3.
Many investigations were devoted to the nonlinear Hilbert(Riemann-Hilbert) boundary value problems with conditions of the typeΦ( ζ, f ( ζ ) ) = 0 ∀ ζ ∈ ∂D , (12)see e.g. [6], [7] and [17]. It is natural also to weaken such conditions toΦ( ζ, f ( ζ ) ) = 0 for a.e. ζ ∈ ∂D . (13)It is easy to see that the proposed approach makes possible also to reducesuch problems to the algebraic and measurable solvability of the relationΦ( ζ, v ) = 0 (14)with respect to a complex-valued function v ( ζ ), cf. e.g. [5].Through suitable modifications of Φ under the corresponding mappingsof Jordan boundary curves onto the unit circle S = { ζ ∈ C : | ζ | = 1 } , wemay assume that ζ belongs to S . Theorem 3.
Let D be a domain in C whose boundary consists of a finitenumber of mutually disjoint rectifiable Jordan curves, A : ∂D → C and B : ∂D → C be measurable functions with respect to the natural parameter.Suppose that { γ + ζ } ζ ∈ ∂D and { γ − ζ } ζ ∈ ∂D are families of Jordan arcs of classBS in D and C \ D , correspondingly.Then there exist single-valued analytic functions f + : D → C and f − : C \ D → C that satisfy (1) for a.e. ζ ∈ ∂D with respect to the naturalparameter where f + ( ζ ) and f − ( ζ ) are limits of f + ( z ) and f − ( z ) az z → ζ along γ + ζ and γ − ζ , correspondingly.Furthermore, the space of all such couples ( f + , f − ) has the infinite di-mension for every couple ( A, B ) and any collections γ + ζ and γ − ζ , ζ ∈ ∂D . Theorem 3 is a special case of the following lemma on the generalizedRiemann problem with shifts that can be useful for other goals, too.
Lemma 1.
Under the hypotheses of Theorem 3, let in addition α : ∂D → ∂D be a homeomorphism keeping components of ∂D such that α and α − have the ( N ) − property of Lusin with respect to the natural parameter.Then there exist single-valued analytic functions f + : D → C and f − : C \ D → C that satisfy (2) for a.e. ζ ∈ ∂D with respect to the naturalparameter where f + ( ζ ) and f − ( ζ ) are limits of f + ( z ) and f − ( z ) az z → ζ along γ + ζ and γ − ζ , correspondingly.Furthermore, the space of all such couples ( f + , f − ) has the infinite di-mension for every couple ( A, B ) and any collections γ + ζ and γ − ζ , ζ ∈ ∂D . Proof.
First, let D be bounded and let g − : ∂D → C be a measurablefunction. Note that the function g + := { A · g − + B } ◦ α − (15)is measurable. Indeed, E := { A · g − + B } − (Ω) is a measurable subset of ∂D for every open set Ω ⊆ C because the function A · g − + B is measurableby the hypotheses. Hence the set E is the union of a sigma-compact setand a set of measure zero, see e.g. Theorem III(6.6) in [15]. However,continuous mappings transform compact sets into compact sets and, thus, α ( E ) = α ◦ { A · g − + B } − (Ω) = ( g + ) − (Ω) is a measurable set, i.e. thefunction g + is really measurable.Then by Theorem 2 in [1] there is a single-valued analytic function f + : D → C such that lim z → ξ f + ( z ) = g + ( ξ ) (16)along γ + ξ for a.e. ξ ∈ ∂D with respect to the natural parameter. Note that g + ( α ( ζ )) is determined by the given limit for a.e. ζ ∈ ∂D because α − alsohas the ( N ) − property of Lusin.Note that C \ D consists of a finite number of (simply connected) Jordandomains D , D , . . . , D m in the extended complex plane C = C ∪ {∞} . Let ∞ ∈ D . Then again by Theorem 2 in [1] there exist single-valued analyticfunctions f − l : D l → C , l = 1 , . . . , m, such thatlim z → ζ f − l ( z ) = g − l ( ζ ) , g − l := g − | ∂D l , (17)along γ − ζ for a.e. ζ ∈ ∂D l with respect to the natural parameter.Now, let S be a circle that contains D and let j be the inversion of C with respect to S . Set D ∗ = j ( D ) , g ∗ = g ◦ j, g − := g − | ∂D , γ ∗ ξ = j (cid:16) γ − j ( ξ ) (cid:17) , ξ ∈ ∂D ∗ . Then by Theorem 2 in [1] there is a single-valued analytic function f ∗ : D ∗ → C such that lim w → ξ f ∗ ( w ) = g ∗ ( ξ ) (18)along γ ∗ ξ for a.e. ξ ∈ ∂D ∗ with respect to the natural parameter. Note that f − := g ∗ ◦ j is a single-valued analytic function in D and by constructionlim z → ζ f − ( z ) = g − ( ζ ) , g − := g − | ∂D , (19)along γ − ζ for a.e. ζ ∈ ∂D with respect to the natural parameter.Thus, the functions f − l , l = 0 , , . . . , m, form an analytic function f − : C \ D → C satisfying (2) for a.e. ζ ∈ ∂D with respect to the naturalparameter.The space of all such couples ( f + , f − ) has the infinite dimension for everycouple ( A, B ) and any collections γ + ζ and γ − ζ , ζ ∈ ∂D , in view of the aboveconstruction because of the space of all measurable functions g − : ∂D → C has the infinite dimension.The case of unbounded D is reduced to the case of bounded D throughthe complex conjugation and the inversion of C with respect to a circle S insome of the components of C \ D arguing as above. (cid:3) Remark 4.
Some investigations were devoted also to the nonlinearRiemann problems with boundary conditions of the formΦ( ζ, f + ( ζ ) , f − ( ζ ) ) = 0 ∀ ζ ∈ ∂D . (20)It is natural as above to weaken such conditions to the followingΦ( ζ, f + ( ζ ) , f − ( ζ ) ) = 0 for a.e. ζ ∈ ∂D . (21)It is easy to see that the proposed approach makes possible also to reducesuch problems to the algebraic and measurable solvability of the relationsΦ( ζ, v, w ) = 0 (22)with respect to complex-valued functions v ( ζ ) and w ( ζ ), cf. e.g. [5].Through suitable modifications of Φ under the corresponding mappingsof Jordan boundary curves onto the unit circle S = { ζ ∈ C : | ζ | = 1 } , wemay assume that ζ belongs to S . Example 1.
For instance, correspondingly to the scheme given above,special nonlinear problems of the form f + ( ζ ) = ϕ ( ζ, f − ( ζ ) ) for a.e. ζ ∈ ∂ D (23)in the unit disk D = { z ∈ C : | z | < } are always solved if the function ϕ : S × C → C satisfies the Caratheodory conditions : ϕ ( ζ, w ) is continuousin the variable w ∈ C for a.e. ζ ∈ S and it is measurable in the variable ζ ∈ S for all w ∈ C .Furthermore, the spaces of solutions of such problems always have theinfinite dimension. Indeed, the function ϕ ( ζ, ψ ( ζ )) is measurable in ζ ∈ S for every measurable function ψ : S → C if the function ϕ satisfies theCaratheodory conditions, see e.g. Section 17.1 in [8], and the space of allmeasurable functions ψ : S → C has the infinite dimension. Problems.
Finally, it is necessary to point out the open problems onsolvability of Hilbert and Riemann problems along any prescribed familiesof arcs but not only along families of the Bagemihl–Seidel class and, moregenerally, along any prescribed families of paths to a.e. boundary point.
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Vladimir Ryazanov