In Evan's textbook "Partial Differential Equation", question 5 in section 2.5 says "$|Du|^2$ is subharmonic if $u$ is harmonic.". This can be easily proven, but do we really need the derivative $D$? I don't see any point of taking the derivative of $u$.
2026-04-05 18:24:58.1775413498
$|Du|^2$ is subharmonic if $u$ is harmonic.
215 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in HARMONIC-FUNCTIONS
- Harmonicity is a local property?
- Harmonic functions satisfying given inequality
- Is there Phragmen-Lindelof for $\mathbb{C}_+$ where $f$ is not bounded on $i\mathbb{R}$ but has polynomial growth?
- Solution of a non homogeneous Laplace equation on the unit disk.
- Complex Analysis - Harmonic function as real part of holomorphic function
- Show that u is harmonic
- Physicists construct their potentials starting from the Laplace equation, why they do not use another differential operator, like theta Θ?
- Prove a family of harmonic functions is locally bounded
- Why is $ u=\log(\sqrt{x^2+y^2})$ not harmonic for $x^2 + y^2 <1$?
- Modulus and argument of a holomorphic function.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?