Some New Results On Elliptic And Parabolic Equations

In this thesis,we mainly discuss the singular limit problems of two kinds of partial differential equations,i.e.the convergence of solutions to parabolic Allen-Cahn equation and elliptic Sinh-Poisson equation when the parameter tends to zero.The convergence of the solutions to the Allen-Cahn equation is established from the point of view of geometric measure theory.And the existence of the concentrating solutions of the Sinh-Poisson equation is studied by using the Lyapunov-Schmit finite dimensional reduction method.The present thesis consists of three chapters:In chapter one,we summarize the background of the related problems and state the main results of the present thesis.In chapter two,we study the following parabolic Allen-Cahn equation with Dirichlet boundary conditions where Ω(?)Rn(n≥2)is a bounded,strictly convex domain with smooth boundary,ε is a small positive parameter,the initial value vo satisfies some suitable conditions and F is a bi-stable potential.We define the energy measures μtε of the solutions to equation(0.0.3)as dμtε=(ε/2|▽vε|2+F(vε)/ε)dx,and define the discrepancy measures ξtε as dξtε=(ε/2|▽vε|2-F(vε)/ε)dx.When considering vε satisfies the Dirichlet or dynamic boundary conditions,Y.Giga,F.Onoue and K.Takasao(arXiv:1810.09107)prove that the varifold Vt associated to limit energy measures μt is a Brakke’s mean curvature flow under the assumptions that a.limit discrepancy measures |ξ| vanish up to the boundary of the domain;b.the Dirichlet boundary energy is bounded locally with respect to time t(i.e.supε∈(0,1)∫t1 t2 ∫(?)Ωε/2|(?)vε/(?)v|2 dHn-1dt<C(t1,t2),where v is the outward unit normal vector field of(?)Ω).The main purpose of this chapter is to prove the above two hypotheses rigorously and then obtain the same conclusions by putting forward some necessary requirements for the initial value v0ε when vε satisfies the Dirichlet boundary conditions.More precisely,if the initial value v0ε satisfies some suitable conditions,we can obtain the asymptotic behavior of vε near the boundary by constructing a sub-solution of the equation(0.0.3),and then obtain the boundary gradient estimate of the solution by using the method of barrier functions.By using the boundary gradient estimate and the monotone formula involving the discrepancy function ξtε,we prove the two hypotheses mentioned above.Finally,as the main conclusion of this chapter,we give a varifold characterization of the limit energy measure μt.In chapter three,we consider the following Sinh-Poisson equation with Henon term where Ω(?)R2 is a bounded domain with smooth boundary,ε is a small positive constant,the points q1,…,qn ∈Ω,α1,……αn ∈(0,∞)\N and v is the outward unit normal vector field on(?)Ω.We prove the existence of mixed interior-boundary concentrating solutions for equation(0.0.4)by using the Lyapunov-Schmit finite dimensional reduction method,and the locations of concentrating points can be characterized by the critical point problem of a functional.More precisely,for nonnegative integers n,k,l with n≥1 and k+l=m≥1,there exists a solution μεto equation(0.0.4)with n internal singular points q1,…,qn,k interior bubbling pointsξ1,…,ξk and l boundary bubbling points ξk+1,…,ξk+l,such that the term ε2|x-q1|2α1…|x-qn|2αn(eue-e-uε)converges to some Dirac deltas with weight 4π,8π and 87r(1+αi)in the sense of measures in Ω.Moreover,the locations of k+l concentrating points can be characterized by solving the critical points of the following functional where dj,j=1,…,m,are some constants with values 4π or 8π,k(ξj)=|ξj-q1|2α1...|ξj-qn|2αn,Green’s function G(x,y)satisfiesΔxG-G=-δy,x∈Ω,(?)G/(?)v=0,x∈(?)Ω,(0.0.5)H(x,y)denotes the regular part of G(x,y).