EXERCISES 2.2:

(1) Compute the formal adjoint for each of the following:

(a) L(y) = x2 y'' + x y' + y (b) L(y) = y'' + 9 \pi2 y

(c) L(y) = (ex y'(x))' + 7 y(x) (d) L(y) = y'' + 3y' + 2y

(2) Argue that L is formally self adjoint if it has constant coefficients and derivatives of even order only.

(3) Suppose that L(y) = y'' + 3y' + 2y and y(0)= y'(0) = 0.

Find conditions on v which assure that

I(0,1, )[vL(y) - L*(v) y] = 0.

(4) Let L(u) = u'' + u. The formal adjoint of L is given by L*(v) = v''+ v. For each manifold M given below, find M* such that L* on M* is the adjoint of L on M.

(a) M = {u: u(0)=u(1)=0}, ans: M = M*

(b) M = {u: u(0)=u'(0)=0} ans: M* = {z: :z(1) = z'(1) =0}

(c) M = {u: u(0)+3u'(0)=0, u(1)-5u'(1)=0}, ans: M = M*

(d) M = {u: u(0)=u(1), u'(0)=u'(1) }. ans: M = M*

(5) Let L and M be as given below; find L* and M*.

(a) L(u)(x) = u''(x) + b(x) u'(x) + c(x) u(x),

M = {u: u(0)=u'(1), u(1) = u'(0) }.

(b) L(u)(x) = -(p(x) u'(x))' + q(x) u(x);

M = {u: u(0) = u(1), u'(0) =u'(1) }.

(c) L(u)(x) = u''(x);

M = {u: u(0) + u(1) = 0, u'(0) - u'(1) = 0 }

ans: M* = {z: z(0) = z(1), z'(0) = - z'(1)}

(6) Verify that for L, M, and u as given in the TYPICAL PROBLEM above, u is in M and L(u) = f. (Recall Exercise 3 in the section AN INTRODUCTION TO THE PROBLEMS OF GREEN'S FUNCTIONS of these notes.)

Suppose G is as in TYPICAL PROBLEM and

z(x) = I(0,1, )G*(x,t) h(t) dt.

Show z solves L* on M*.

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