课程名称︰应用分析一
课程性质︰数学系选修、数学研究所选修、应用数学科学研究所必修
课程教师︰薛克民
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰01/15, 2015
考试时限（分钟）：15:30~17:20
试题 :
Instructions:
> Total points 70
> Open books, notes, and laptops
> Answer the questions thoroughly and justify all your answers. Show
all your work to maximize partial credit.
1 (20)
Find the curve joining the points (-1, -1) and (1, 1) which minimize the
functional
1
J(y) = ∫ [x(y')]^2 + 12 y^2 dx .
-1
2 Consider the Sturm-Liouville problem of the form
A(u) = - (ε(x) * u')' + αu = f(x), 0 < x < 1,
u(0) = 0, u'(1) = 0 (1)
where ε(x) > 0 on x ∈[0, 1] and α∈R.
(a) (30) Find the Green function and hence the solution of (1).
(b) (10) In the case of the Sturm-Liouville eigenvalue problem of the form
A(u) = - (ε(x) * u')' + αu = λu, 0 < x < 1,
u(0) = 0, u'(1) = 0 (2)
formlate a variational problem leading to (2).
(c) (10) In view of the variational form derived in (b), find an expression
for the first eigenvalue of (2).