Thread: harmonic functions

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 u_xxx + u_xyy = 0. Can we know if this is the case given we know u_xx and u_yy = 0?

Any help would be appreciated this is a very confusing topic for me. Thanks

Originally Posted by linalg123

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 u_xxx + u_xyy = 0.

What does 'it' refer to here? You have three functions, u, v, and uv. Which are you refering to?

Can we know if this is the case given we know u_xx and u_yy = 0?

Only if you mean "identically equal to 0" which you do not here. You do not even know that u_xx and u_yy, individually, at any specific point. What definition of "harmonic" are you using?

Any help would be appreciated this is a very confusing topic for me. Thanks

ok i guess i didn't state my questions clearly enough, i'm sorry.
PART A
What i know: If i let m=uv, m_xx + m_yy = (u_xx + u_yy)v + 2(u_xv_x + u_yv_y) + u(v_xx+v_yy)
Okay. Now we know that u and v are harmonic, so the first and third terms are both 0.
However, I believe the middle term will only disappear if u and v satisfy the cauchy-riemann equations?

Question: Can the functions u and v both be harmonic in D, but NOT satisfy the cauchy riemann equations?

PART B
we know u is harmonic in D.
i.e u_xx+u_yy= 0

we need to know if this means u_x is also harmonic in D, i.e (u_x)_xx + (u_x)_yy = 0
we are given that harmonic functions have continuous partial derivatives of all orders.

Question: Do we have enough here to say whether u_x is harmonic in D?