# Tagged Questions

19 views

73 views

### limit of harmonic functions Gilbarg and Trudinger page 27

Here is a passage from Gilbarg and Trudinger page 27 Let $\Sigma$ be a bounded domain in $\mathbb{R}^n$ ($n\geq 3$) with smooth boundary $\partial\Omega$, and let $u$ be the harmonic function (often ...
22 views

41 views

### Local regularity for harmonic functions (Laplace's equation)

I need a local Sobolev regularity result for a smooth solution $u$ of $$-\Delta u=0$$ with the equation satisfied in an open set $U$ (I have no boundary conditions). I know that such a smooth $u$ ...
37 views

### Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
59 views

### Uniqueness for second order PDE $-\Delta u = c(u)$

I'm trying to solve the following problem: Let $\Omega$ be an open set in $\mathbb R^n$ and consider the equation \begin{cases} -\Delta u = c(u) & \text{ on } \Omega \\ u = 0 & \text{ on } ...
192 views

### Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
107 views

### If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set ...
45 views

### Biharmonic boundary condition

I try to solve $$\Delta^2u=f$$ on unit square. with $f=4sin(\pi x)sin(\pi y)$ Using $v=-\Delta u,$ leads to $$v+\Delta u=0,$$ $$-\Delta v=f.$$ By Dirichlet boundary condition on $u$. What is ...
177 views

### Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
44 views

115 views

### Laplace equation on unbounded set

I have gotten stuck with a problem for PDEs class for a few days. I did not figure out how to start a solution for it. Problem: Let $g \in C(\partial B(0, R))$, $n > 2$. Find a formula for a ...
168 views

### Reflection principle for harmonic functions

$U^+ \colon= \left\{x\in \mathbb{R}^n\mid |x| < 1, x_n>0\right\}$ is an open half-ball. Assume $u \in C^2 (\overline{U^+}$) is harmonic in $U^+$ with $u=0$ on $\partial U^+ \cap \{x_n=0\}$. ...
115 views

### poisson's equation with robin's boundary, boundary value problem

Consider the Poissonâ€™s equation with Robinâ€™s boundary conditions as follows \begin{array}{ll} âˆ’\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} ...
57 views

### Stationary heat equation problem

We are given the following BVP: $u_{xx}+u_{yy}=0,\ x\in\mathbb{R},\ y>0\\ u(x,0)=f(x),\ x\in\mathbb{R}\\ u(x,y)\rightarrow 0\ \text{as}\ x^2+y^2\rightarrow\infty$ where ...
54 views

### $\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
116 views

### Poisson Equation in a Rectangle

The problem is to solve $$\Delta\phi=\frac \lambda {\varepsilon_0}\delta(x-x',y-y')\quad;\quad \phi(0,y)=\phi(a,y)=\phi(x,0)=\phi(x,b)=0.$$ My idea was to try and represent the RHS as a series of ...
353 views

### Show that the Poisson kernel is harmonic as a function in x over $B_1(0)\setminus\left\{0\right\}$

Show that the Poisson-kernel $$P(x,\xi):=\frac{1-\lVert x\rVert^2}{\lVert x-\xi\rVert^n}\text{ for }x\in B_1(0)\subset\mathbb{R}^n, \xi\in S_1(0)$$ is harmonic as a function in $x$ on ...
103 views

### Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
142 views

### Determining if the product of two particular harmonic functions is a harmonic function

Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$. I would like to show that: $v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$ is also ...
119 views

### Harmonic function vanishes with its normal derivative on a part of boundary; can Green's formula be applied to broken boundary? [duplicate]

Let $\Omega$ be an open domain, and let $\Sigma$ be a smooth and nonempty portion of the boundary. Let $u$ be a harmonic function in $\Omega$ and $u=D_\nu u=0$ on $\Sigma$. ($D_\nu$ is the derivative ...
114 views

### Harmonic function (PDE) - Orthogonal matrix

Let $u\in C^2(\mathbb{R}^n)$ be harmonic in $\mathbb{R}^n$, i.e. $$\Delta u:=\sum\limits_{k=1}^{n}\frac{\partial^2 u}{\partial x_k^2}=0\mbox{ in }\mathbb{R}^n.$$ Let ...
I am trying to solve this exercise : Consider $U$ open , bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U)$ a harmonic function. Suppose that for each $x \in \partial U$ there exists a ...