The Willmore boundary problem

We consider the Euler–Lagrange equation of the Willmore functional coupled with Dirichlet and Neumann boundary conditions on a given curve. We prove existence of a branched solution, and for Willmore energy < 8π, we prove the existence of a smooth, embedded solution.     

Uniqueness for an overdetermined boundary value problem for the p-Laplacian

For set , and let be a measure with compact support. Suppose, for , there are functions and (bounded) domains , both containing the support of with the property that in (weakly) and in the complement of . If in addition is convex, then and .

     

Uniqueness theorems for an impulsive Sturm-Liouville boundary value problem

In this study, an impulsive boundary value problem, generated by Sturm-Liouville differential equation with the eigenvalue parameter contained in one boundary condition is considered. It is shown that the coefficients of the problem are uniquely determined either by the Weyl function or by two given spectra.     

Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences

This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.     

Uniqueness for weak solutions of a one-dimensional boundary value problem for the Vlasov-Poisson system

The Vlasov-Poisson system models a collisionless plasma. When a boundary condition is included in the problem it is known that singularities can occur but that weak solutions exist globally in time. This article shows that the weak solution is unique for a problem in one dimension with specular reflection at the boundary.     

The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow

We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

Uniqueness and existence for bounded boundary value problems

The existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established. We assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems.     

Uniqueness of the Dirichlet problem for a partial differential equation of the fourth-order

We give some uniqueness theorems for regular solutions to the Dirichlet problem associated with a partial differential equation of the fourth-order in an unbounded domain. Moreover, by means of a counterexample, we show that one of these result is sharp.