How do Christians reconcile Isaiah 42:4 with the death of Jesus? My understanding of Isaiah 42:4 is that it states that the Messiah will not falter before his job is completed. How do Christians reconcile this with the fact that Jesus died?

He will not falter or be discouraged till he establishes justice on earth. In his teaching the islands will put their hope.  Isaiah 42:4 NIV
“He will not be disheartened or crushed until He has established justice in the earth;  And the coastlands will wait expectantly for His law.”  Isaiah 42:4 NASB
This is easily reconciled because here is nothing to reconcile! Christ has never faltered, been disheartened, or crushed. He fulfilled His mission while here on earth, continues to do so as our advocate, and will continue to do so. His death and resurrection demonstrated this, rather than negating it!

A look at Christian commentaries shows that this is the common understanding among Christians.


  Matthew Henry's Concise Commentary
  
  Isaiah  42:1-4 This prophecy was fulfilled in Christ, Mt 12:17. Let
  our souls rely on him, and rejoice in him; then, for his sake, the
  Father will be well-pleased with us. The Holy Spirit not only came,
  but rested upon him, and without measure. He patiently bore the
  contradiction of sinners. His kingdom is spiritual; he was not to
  appear with earthly honours. He is tender of those oppressed with
  doubts and fears, as a bruised reed; those who are as smoking flax, as
  the wick of a lamp newly lighted, which is ready to go out again. He
  will not despise them, nor lay upon them more work or more suffering
  than they can bear. By a long course of miracles and his resurrection,
  he fully showed the truth of his holy religion. By the power of his
  gospel and grace he fixes principles in the minds of men, which tend
  to make them wise and just. The most distant nations wait for his law,
  wait for his gospel, and shall welcome it. If we would make our
  calling and election sure, and have the Father delight over us for
  good, we must behold, hear, believe in, and obey Christ.


Even in His darkest hour, as He prepared to fulfill His mission on the cross, He never faltered.  He did the will of the Father, putting aside His own needs without hesitation.


  Matthew 26:39  (KJV)
  
  And he went a little further, and fell on his face, and prayed,
  saying, O my Father, if it be possible, let this cup pass from me:
  nevertheless not as I will, but as thou wilt.


In my book. that's not faltering.  were it me, I'd have faltered. I'd have run away, lied, done anything to escape, yet He never faltered.  He went ahead, knowing what was to come, because He cannot falter.

Extra credit: Here's a list of sermons that preach on that verse, many of which focus on Christ as the triumphant servant.



Edits to address points missed per the comments below.  Namely, this comment:


  Maybe you know more than me and can show that there has been justice
  on this earth for the past 2000 years.


This was addressed in the very first paragraph on this answer, but I will expand upon it.  Repeating the relevant sentence, with some emphasis added:

He fulfilled His mission while here on earth, continues to do so as our advocate, and will continue to do so. 

The misunderstanding here of commonly accepted Christian doctrine, is that in asking whether there has been justice on this earth in the last 2000 years is that His work was done on the cross.  That is not the case.  Christianity isn't based on His death, it's based on His resurrection, and the work that He continues to do.  He has still not, to this very second, faltered. He is still doing the will of his father.

The idea that the verse is nullified because perfect justice wasn't established on this earth by the time of His death ignores the very basic tenets of the Christian faith.  For there to be a discrepancy, His would would have had to end with His death.  It didn't.

Rather than going deep into detail on all of the doctrinal points that this question ignores, I'll simply summarize them with links for further study.


Christ was resurrected.  So His work didn't end at the cross.  
He then spent a little more time on this earth after his resurrection, and then ascended to heaven.


Lest you wonder about the three days between His death and resurrection and somehow try to interpret that as a "pause", we know little of what He was doing during those three days, but we do know that He was active, "preaching to the spirits in prison".  (Although the exact details on what that means aren't clear.)

Christ continues, right now, even as I type this, to work as our Advocate.  (still hasn't stumbled.)


So, up until the present time, at no point has Christ stumbled, wavered, or strayed from his path.  We have no evidence whatsoever that he was discouraged, even in His darkest hour, or since.  Thus, the fact that perfect justice hasn't been established yet is not a problem.

Scripture does promise that He will establish new Heaven and a new Earth.  This is the point at which perfect justice will be established on earth, but that is a promise for the future.  

Only if he stumbles before that future promise comes to pass could there possibly be any potential discrepancy here.