If the landlord has a new tenant who wants to take possession, don't they have a duty to mitigate my brother's lease obligations? Is there California law that addresses this situation?
Yes, the doctrine of mitigation of damages. See Valle de Oro Bank v. Gamboa, 26 Cal.App.4th 1686, 1691 (1994):
The doctrine of mitigation of damages holds that "[a] plaintiff who
suffers damage as a result of either a breach of contract or a tort
has a duty to take reasonable steps to mitigate those damages and will
not be able to recover for any losses which could have been thus
The application of this doctrine in landlord-tenant disputes is also palpable in CA Civil Code §1951.2(a). The statutory language is needlessly convoluted, but Danner v. Jarrett, 144 Cal.App.3d 164, 166-7 (1983) points out that "section 1951.2 [...] encourages the lessor to mitigate damages [...]. Its formula for damages permits the lessee to prove what rental loss could have been avoided" (citations omitted).
Consistent with the previous statement, Powerhouse Motorsports Group, Inc. v. Yamaha Motor Corp., USA, 221 Cal.App.4th 867, 884 (2013) mentions that "[t]he burden of proving a plaintiff failed to mitigate damages [...] is on the defendant".