Is it permissible to give zakat to one's brother to help him with the expenses of a newborn, if his income is limited and he lives with his father who will cover the necessities, with the aim of alleviating the burden on the father?

If the son is provided for by his father's sustenance, then he is wealthy, and zakat is not permissible for him. However, if the father is unable to provide sustenance, or refuses to do so, or what he provides is insufficient for the son's needs, then it is permissible to give zakat to him.