According to the Catholic Church (and other Nicene churches), does the divine name YHWH (יַהְוֶה) apply equally to all Persons of the Trinity?
For the Catholic Church and other Nicene churches (the Eastern Orthodox, the Oriental Orthodox, the Armenian Orthodox, the Assyrians; as well as the majority of Protestants), the divine name YHWH (which is closely linked to the expression “I am” or “I Am Who I Am” (see Ex. 3:14), applies to God in His divine nature—hence to all three Persons of the Trinity.
The attribution of YHWH begins with Jesus himself, who multiple times attributes to himself the divine name “I am.” The most straightforward example is John 8:58:
Jesus said to them, “Truly, truly, I say to you, before Abraham was, I am.”
Hence, Jesus, the Son, attributes the divine name YHWH to himself. It is less immediately clear from the Scriptures that the Holy Spirit can also be called YHWH, but it follows from our knowledge of Trinitiarian theology.
Each of the Persons of the Trinity is perfectly identical with the Divine Nature or Essence—in orthodox Trinitarian doctrine, the Persons are not “parts” or “manifestations” of the Trinity, but strictly relations of opposition. (See for instance, St. Augustine, De trinitate I, 6, and V, 5.) In the words of the Nicene-Constantinopolitan Creed, the Son and Spirit are both consubstantial (homoousios) with the Father. (The Creed does not explicitly say so with respect to the Holy Spirit, but that is the fully intentional implication of saying that the Holy Spirit is “worshiped and glorified” together with the Father and Son.)
The name YHWH, however, refers to the very essence of God: clearly, it pertains to all three Persons to “be” in the absolute sense. Hence, YHWH can be applied equally well to all three Persons, although it is frequently attributed to the Father, because He is the ultimate origin or monarch of the whole Trinity.
(The Creed actually implies this: in liturgical celebrations the divine name YHWH is traditionally not used—a holdover from the ancient Jewish reluctance to utter that name. That is why in most translations of the Old Testament, the name YHWH is generally translated as “the LORD”—translating the Hebrew “Adonai.” The Creed, however, calls the Holy Spirit the “Lord and giver of life,” hence the liturgical substitute for the divine name YHWH is used for the Holy Spirit, just as it is for the Son and the Father.)
Yes. The divine name applies to each of the persons of the Trinity.
Matthew 28:19 is explicit that the name of the Trinity is one and the same.
Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,
Matthew 28:19 (ESV)
The name of God tells us about God's very being (Exodus 3:14). Thus, the Trinity, by having a common name, is of same nature.