Why is Duke Orsino's love not reciprocated by Olivia ? How does he take her rejection?

Valentine tells Orsino in Act 1, Scene 1 that Olivia is uninterested in Orsino's romantic feelings for her, and that she doesn’t even bother to read the letters he sends expressing his devotion to her.
The reason she is uninterested is because she is grieving the recent loss of her beloved brother. His death has so devastated her that she has pledged to wear a veil for seven years in mourning, and that she will eschew romance and marriage until this interval has passed.
Of course, this reasoning is not enough to satisfy Orsino, who is determined to win Olivia’s affections. He enlists the help of Cesario/Viola, sending the young servant on an errand to once again profess Orsino’s love for Olivia. This backfires, of course, since Olivia instead falls in love with Cesario.
When Cesario returns the message that Olivia still rejects him, Orsino must be undeterred, since the reader discovers Orsino has sent for Feste—Olivia’s fool—o deliver his message of love again.
In Act 2, Scene 4, Orsino once again asks Cesario to plead with Olivia to return Orsino’s love. This shows that Orsino is unable to accept rejection, and he believes that if he is persistent enough, Olivia will finally give in to him.