What is the ruling of Islamic law regarding two sisters' refusal to sell an inherited apartment, or to purchase their brother's share of the inheritance at a suitable price, under the pretext that the apartment is a trust for them, knowing that the brother needs his share in the apartment to secure suitable housing for himself and his children?

No heir has the right to monopolize the inheritance before its distribution, nor to prevent others from receiving their rightful share. Therefore, your sister is not entitled to live in the inherited apartment before its division without the consent of the other heirs. The questioner has the right to request the division of the apartment, or its sale if division is not possible, and the distribution of the proceeds. Anyone who refuses this can be compelled to either divide or sell through the Shar'i judge. The questioner is advised not to rush to court, but rather to seek mediation from wise relatives or righteous individuals for reconciliation. He must maintain ties with his two sisters and not cut them off, even if they cut him off or treat him poorly, for the one who maintains ties is not merely reciprocating.