Asymptotic for a Riemann-Hilbert Problem Solution 1

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1 Int Journal of Math Analysis, Vol 7, 203, no 34, HIKARI Ltd, wwwm-hikaricom htt://dxdoiorg/02988/ijma Asymtotic for a Riemann-Hilbert Problem Solution M P Árciga-Alejandre, J Sánchez-Ortiz Universidad Autónoma de Guerrero Unidad Académica de Matemáticas Av Lázaro Cárdenas S/N, Ciudad Universitaria Chilancingo de los Bravo, Guerrero 39087, Mexico M A Taneco-Hernández Universidad Autónoma de Guerrero Unidad Académica de Matemáticas Av Lázaro Cárdenas S/N, Ciudad Universitaria Chilancingo de los Bravo, Guerrero 39087, Mexico and Deartamento de Ciencias Básicas, UAM-A, Av San Pablo 80 Col Reinosa Tamaulias, México, D F, Mexico Coyright c 203 M P Árciga-Alejandre et al This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited Abstract We solve an homogeneous Riemann-Hilbert roblem, such that the density does not satisfy the zero index condition Moreover, we find an asymtotic reresentation for the solution to this roblem As an alication, the roblem mentioned above aears in the construction of a Green oerator for a linear artial differential equation with Riesz fractional derivative Also, in order to estimate such oerator the asymtotic reresentation is used Mathematics Subject Classification: 30E5, 30E20, 30E25 Keywords: Riemann-Hilbert roblem, comlex analysis Research suorted in art by PROMEP, Mexico

2 668 M P Árciga-Alejandre et al Introduction As a motivation for the main result of this aer we give an alication in the theory of artial differential equations In the study of initial boundary-value roblems for linear evolution equations with Riesz fractional derivatives, u t + x α u =0, ux, 0) = u 0 x), x > 0; u0,t)=ht), t > 0; t > 0, x>0; ) where x α is a Riesz fractional derivative defined by x α u := R α+[α] [α]+ x u, [α] denotes the integer art of the number 0 <α<2, α, and R α is the modified Riesz otential see [3]), R α u = 2Γα) sin π 2 α) 0 sgnx y) uy)dy x y α In articular, in the construction of the Green function for the linear roblem ), an homogeneous Riemann-Hilbert roblem has to be solved see []) Also, in order to rove the existence and uniqueness of solutions, it is imortant to estimate such Green function, this can be done using the asymtotic reresentation for the solution to the Riemann-Hilbert roblem In this aer we solve an generalized homogeneous Riemann-Hilbert roblem and we construct their asymtotic reresentation 2 Preliminaries First, we enunciate some results from the theory of functions Definition 2 A function f of a comlex variable z is said to satisfy a Hölder condition or to be Hölder continuous on a set D if there exists A>0and 0 <α< such that fz ) fz 2 ) A z z 2 α for all z,z 2 D A = A α f) =Af; D, α) is called the Hölder constant, α the Hölder exonent The set of Hölder continuous functions on D is denoted by C α D) Definition 22 Let L be simle smooth curve, ϕ CL) and ϕζ) 0on L Then the index β of ϕ with resect to L is the mean variation or arg ϕζ) while ζ varies on L in the ositive direction assing any oint once, β := Ind ϕ = 2π L d arg ϕζ) = L d ln ϕζ) 2)

3 Asymtotic for a Riemann-Hilbert roblem solution 669 Suose that we are given a simle smooth contour L dividing the lane of the comlex variable into the left-hand semi-lane D + and the right-hand semi-lane D, deending on the curve orientation Theorem 23 see [2]) Let F a function given on the contour L, satisfying the Hölder condition and having zero index, then F is reresentable as a ratio of functions Φ + ) and Φ ) constituting the boundary values of functions analytic in the domains D + and D, resectively; and having in the domains no zeros These functions are determined to within an arbitrary constant factor A and are given by formulae Φ ± ) =Ae Γ± ), where Γ ± ) = lim z Γz), Γz) = Rz>0 L ln F q) dq 22) q z 3 Problem formulation and main results We consider the contour L := iir and a not vanishing comlex-valued function G) defined on L which satisfy the Hölder condition on L, G) =+O ɛ ), when ±i, ɛ>0, and with Ind G = β It is required to find two functions Y + z) analytic on the left-hand comlex lane D +, and Y z) analytic on the right-half comlex lane D, which satisfy on L the relation Y + ) =G)Y ) 3) This roblem is know as homogeneous Riemann-Hilbert roblem Definition 3 We define ln ± z) =ln z +i arg ± z), where 3 2 π<arg z) π 2 and 3 2 π arg+ z) < π 2 We note that ln z) and ln + z) are the analytic functions on the right and left comlex semi-lanes, resectively The main result of this aer is the following Theorem 32 Main result) Let G) C α iir), G) =+O ɛ ), when ±i, ɛ>0, and Ind G = β Then, the Riemann-Hilbert roblem 3) has the solution Y ± ) =Ae Γ± ),

4 670 M P Árciga-Alejandre et al where A is a constant and Γ ± ) = ln q )d ln Gq) 32) Moreover, Y ± has the asymtotic reresentation, as ±, ) Y ± ) =A β ± + O, ɛ > 0, where β ± = e β ln± ) β+ɛ ± 4 Proof of main theorem First, we note that Ind z 0) β +z 0 ) β by Theorem 23 we get where X ± ) =e Γ± 0 ), Γ ± 0 = β, z 0 IR +, then Ind G) z 0) β +z 0 ) β = 0 and φ) :=G) z 0) β + z 0 ) β = X + ) X ), 4) ) = lim Γ z 0 z), Γ 0 z) = Rz>0 Using 3) and 4) we obtain Y + ) X + ) z 0 ) β = Therefore by Liouville s Theorem see [2]) where A is a comlex constant Riemann-Hilbert roblem 3) reresentation for Y ± ) 4 Asymtotic for Y ± ) q z Y ) X ) + z 0 ) β, L Y ± ) =AX ± ) z 0 ) β, ln φq) dq 42) The revious formula is a solution to the Now, we are going to find an asymtotic First, we obtain an equivalent formula for Γ 0 Integrating by arts we get Γ 0 = lim R lnq z)lnφq) ir ir lnq z)d ln φq)

5 Asymtotic for a Riemann-Hilbert roblem solution 67 Using that ln φ±ir) 0, as R, we obtain Γ 0 z) = z)d ln φq) Then, for R =0, Γ ± 0 ) = i lnq ln q )d ln φq), 43) where ln ± z) was defined in 3) Now, since d ln φq) =d ln Gq)+ 2βz 0 q 2 z 2 0 dq, and by Cauchy s Residue Theorem, from 43) follows ln ± 2β q ) dq = iπβ +ln ± ± z q 2 z0 2 0 ) β, Γ ± 0 ) = iπβ +ln z 0 ) β Therefore, by last equation ln q )d ln Gq) Y ± ) =A ) β e Γ± ), 44) where Γ ± is defined in 32) Now, we estimate the function Y + Using the relations ln q ) =iπ +ln + )+ln + q ), q <, ln q ) =iπ +ln + )+ln + q ) θi)θiq + I), q >, where θ is the Heaviside function, we obtain Γ + ) = β iπ +ln + ) ) [ ln + q ) h, q) ] d ln Gq), where hq, ) =θi)θi + Iq)θ q ) Then, from ln z) =O z γ ), 0 <γ<, z <, and using that Gq) =+Oq ɛ ), when q ±i, ɛ>0, we get [ ln + q ) h, q)] d ln Gq) =O ) ɛ q <

6 672 M P Árciga-Alejandre et al On the other hand, since ln + q ) = iπ +ln + q) ln + )+ln + ) + θi))θiq + I), q for q >, then q > [ ln + q ) h, q) ] d ln Gq) =O ) ɛ γ Therefore, ) Γ + ) = βiπ +ln + )) + O, ɛ ɛ = ɛ γ Then, from 44) and equation above follows In the same way we obtain Y + ) =A β + + O Y ) =A β + O β+ɛ + β+ɛ We note that in the last formulas β ± = e β ln± ) Therefore, the Theorem 32 has been roved References [] M P Árciga-Alejandre, Asymtotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line, Boundary Value Problems, Volume 20, Article ID 94643, 29 ages, doi:055/20/94643 ) ) [2] F D Gakhov, Boundary value roblems, Dover, 966 [3] S G Samko, A A Kilbas, O I Marichev, Fractional Integrals and Derivatives Theory and Alications, Gordon and Breach, Yverdon, 993 MR d:2602) Received: March 9, 203