Now we are ready to prove the main results of this paper. First, we solve the minimization problem for eigenvalues when potentials .
Proof of Theorem 1.1 (i) If , then . From the monotonicity property (2.1) of eigenvalues, we have that the minimizer and then by computing directly.
When . By Lemma 2.2, Lemma 2.3 and Lemma 2.4, there exists a minimizer such that and for a.e. t. Let be a positive eigenfunction corresponding to the first eigenvalue . We have from the proof of Lemma 2.4 that is also an eigenfunction corresponding to the first eigenvalue , which implies that because .
Then, we have that for each ,
Let and . Since is symmetric about and increasing in , we have that
We claim that
Notice that . Comparing (3.1) and (3.2), we know that the equality holds in (3.2) and is an eigenfunction corresponding to . Since (3.2) is an equality, we have from the proof that and for a.e. t.
Let . Now, (1.1) becomes
We can find that the solution of (3.3) is given by
where . Since and , we get from (3.4) that is the unique solution of , where is as in (1.8). □
Finally, we use the limiting approach to obtain a complete solution for the minimization problem of eigenvalues when the norm of integrable potentials is given.
Proof of Theorem 1.2 By the definitions of in (1.8) and Z in (1.11), we have that
which implies that
Since , we have that and then
On the other hand, since is continuous in , is continuous in . Notice that , where the closure is taken in the space . We have that for arbitrary and , there exist and such that
Because is arbitrary, it holds that
Therefore, we have that
since is arbitrary.
By (3.5) and (3.6), the proof is complete. □
Remark 3.1 Fix . Assume that is as in (1.7) when and as in (1.9) when . Then . Moreover, we have that
as . In fact, for any , it holds that
Notice that is not in the space. So, we get the so-called measure differential equations. For some basic theory for eigenvalues of the second-order linear measure differential equations, see .