The laplace-equation tag has no usage guidance.

**3**

votes

**1**answer

94 views

### Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...

**5**

votes

**2**answers

260 views

### Separable coordinate systems for the Laplace and Helmholtz equations?

According to Mathworld, in three dimensions there are 13 coordinate systems in which Laplace's equation is separable, and 11 for the Helmholtz equation. I've read the relevant chapters of the book by ...

**1**

vote

**1**answer

118 views

### W^{2,∞} regularity of solutions of Poisson's equation if the right hand side is in L^{∞}

Let $u$ be solution of $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$.
Is it true that if $f \in L^{\infty}(\Omega)$ then $u \in W^{2,\infty}(\Omega)$?
...

**4**

votes

**0**answers

93 views

### explicit solution for Laplace equation on punctured cylinder

I need an explicit formula for the rotationally invariant solution of $\Delta u=0$ in cylindrical coordinate $(r,\theta,z)$ for a domain like $D=[0,2]\times [0,2\pi]\times [-2,2] - [0,1]\times ...

**1**

vote

**0**answers

32 views

### 2-d laplace equation with corrugated isothermal boundary [closed]

Consider a 2-d laplace equation $\Delta\Theta(x,z)=0$ with a corrugated boundary $ \Theta(x,f(x))=\Theta_0$. You can assume $f(x)$ to be a sinusoidal function.
1.My idea is to set $p=z-f(x)$. But ...

**1**

vote

**0**answers

102 views

### Basic doubt in a free boundary problem for the Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf
In this article the authors considers $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...

**0**

votes

**2**answers

380 views

### Poisson inequality for subharmonic functions

This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads
$$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} ...

**2**

votes

**1**answer

176 views

### A series question related to solution of Laplace equation

Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition:
$$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$
The series solution is ...