The Riemann Boundary Problem Of The Modified Helmholtz Equation In R~2

In this thesis:we investigate the Riemann boundary value problems for the first order modified Helmholtz equation. Let G be a bounded simply connected domain with a single smooth Jordanian curve boundaryΓ.f(x,y)=((?)) is a real function in G.The first order modified Helmholtz equation D f=g yields where g(x,y)=((?)) is a known real vector function defined in G,g(x,y)∈Lp(G),p>2.Eq.(1)can be rewritten as Add the first equation in Eq.(2)with the second one multiplied by i,we can write Eq. (2)as a first order elliptic complex equation (?)/((?)z)w(z)=-k/2w(z)+c(z), (3) where (?)/((?)z)=1/2((?)/((?)x)+i(?)/((?)y)),w(z)=f1(x,y)-if2(x,y),and c(z)=1/2(x,y)+ig2(x,y)). In this way,we convert the first order modified Helmholtz equation to a first order elliptic complex equation.It is easy to verify that the solution of the modified Helmholtz equation can be expressed as w(z)=Φ(z)+ψ(z),ψ(z)=Tw,w=uz∈Lp(D),Φ(z) is an analytic function in D.Using W'(z)=w(z)-ψ(z),Eq.(3)can be simplified as W2'=-k/2W'(z). (4) Therefore,one only needs to consider the Riemann boundary value problem R for Eq.(4).Problem R is denoted as R' when g(t)=0. To solve problem R for the modified Helmholtz equation, we first obtain a special solution under the inhomogeneous boundary condition through the generalized Cauchy type integral. Then, by transferring problem R to the jump problem, we work out the general solution under the homogeneous boundary condition. That is, we obtain the solvability and expression of the solution for the complex equation under both homogeneous and inhomogeneous boundary conditions.