Suppose that the functions u and v are harmonic in a domain D:
(a) Is the product uv necessarily harmonic in D?
(b) Is du/dx harmonic in D? (You may use the fact that harmonic functions have continuous
partial derivatives of all orders.)
In the previous question i have shown that if v is harmonic conjugate for u in a domain D, then uv is harmonic in D.
Now i'm assuming that uv is only harmonic in D if v is harmonic conjugate for u, but can u and v both be harmonic without being conjugate of the other? If so, can someone tell me how to explain it in words in a general sense without actually being given specific values for u or v.
For b) i think it will be harmonic if uxxx + uxyy = 0. Can we know if this is the case given we know uxx and uyy = 0?
Any help would be appreciated this is a very confusing topic for me. Thanks