0
votes
2answers
48 views

Improve Liouville's Theorem in Evans ' PDE

Here is Liouville's Theorem Suppose that $u \colon \mathbb{R}^n \to \mathbb{R}$ is harmonic and $u \geq 0$. Prove that $u$ is constant. (In this problem , instead of $u$ is bounded now $u \geq 0$ ...
0
votes
1answer
42 views

What can you say about f if g is harmonic?

Suppose that f : R → R is such that, whenever g : $R^n$→ R is harmonic, so is f(g(x)). What can you say about f? This is my attempt , and I think f is a linear function.
1
vote
1answer
50 views

Mean value proof in Evans PDE

Here is the proof I don't really understand about the part beginning using Green's formula. How can Du(y) become du/dv . Is is using the directional derivative formula ? Aslo how can you get/pull ...
1
vote
1answer
40 views

Solving the Laplace partial differential equation with particular boundary conditions [closed]

How this Laplace partial differential equation $$ u_{xx}+u_{yy} =0 $$ with initial conditions on $y=0 $ as $$ u(x,0)=0 $$ $$ u_{y}(x,0)=n^{−1} \sin{nx} $$ has solution $$u(x,y)=n^{−2} \sin({hny}) ...
1
vote
1answer
46 views

To show unique solution for the Laplace equation

The problem is in the top while its weak form at the end, source $\hskip 1in$ I know that the solution is unique because the boundary condition is Dirichlet. But I want to show this. How can you ...
0
votes
2answers
17 views

Prove $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$

Prove that for some $C$, $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$. Recall that $$ \Delta^2u=u_{rr}+r^{-1}u_r+r^{-2}u_{\theta\theta}$$ My answer is below.
0
votes
1answer
39 views

Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
0
votes
1answer
51 views

Comparison of the gradients of two harmonic functions near the boundary

Let $\Omega$ an open bounded domain in $R^n$. Let $u,v$ be nonconstant smooth functions in the interior of $\Omega$ and harmonic in $\Omega$. Suppose that $u,v \in C(\overline{\Omega})$ and $u \geq ...
0
votes
0answers
17 views

Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
2
votes
1answer
53 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
0
votes
2answers
83 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
0
votes
1answer
22 views

Laplace Equation with non-const Dirichlet Boundary Conditions

I'm struggling to get a Laplace problem with inhomogeneous boundary conditions solved. My memories are very rusty, and it almost works out, but I've got my brain twisted in some way. So I'm kindly ...
0
votes
1answer
53 views

Applications of PDE and laplace equation

The edge r = a of a circular plate is kept at temperature f(theta). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state. I'm ...
0
votes
1answer
23 views

Dirichlet boundary value problem in convex domains with discontinuous boundary values

Consider $\Omega$ an open, bounded, and convex domain in $\mathbb{R}^n$. Let $g \in L^{2}(\partial \Omega)$ such that the problem $$ \left\{ \begin{array}{ccccccc} \Delta u = 0, \ \text{in} \ ...
8
votes
1answer
70 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
4
votes
2answers
81 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 ...
0
votes
0answers
24 views

Harmonic function in circle - exercise from Partial Differential Equations book by Y. Pinchover

Could I please ask about help with the following exercise: Let $u(x, y)$ be the harmonic function in $D = \{ (x, y) : x^2 + y^2 < 36\}$ which satisfes on $D$ the Dirichlet boundary condition: $$ ...
5
votes
2answers
68 views

Natural question about harmonic functions

Let $U$ an open, bounded and convex set in $R^n$. Let $(u_k)_{k \in N}$ a sequence of functions defined in $\overline{U}$. Suposse that each $u_k \in C(\overline{U}) \cap C^{2}(U)$ and harmonic in ...
2
votes
4answers
90 views

Solving Laplace's Equation - weird boundary conditions?

The potential is given by: $$V = \sum_{n=0}^{\infty} \left[a_n r^n +b_nr^{-(n+1)}\right] P_n(cos \theta) $$ I want to find potential for $r \geq a$ using th definition $I_n = \int_0^1 P_n(x) \space ...
0
votes
1answer
39 views

Inequality for a harmonic function with gradient bounded from below

Consider $K \subset \mathbb R^n$ a compact set . Let $R > 0 $ such that $B(0,R) \supset K$ and $\partial B(0,R) \cap \partial K = \emptyset .$ Let $u : \overline{B(0,R)} \rightarrow \mathbb R$ a ...
0
votes
2answers
44 views

Questions about the Laplace's equation in polar coordinates

The Laplace's equation in polar coordinates at a cyclic disk: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}, \ \ \ 0 \leq r \leq a, \ \ \ 0 \leq \theta \leq 2 \pi$$ $$u(a,\theta)=h(\theta), \ ...
2
votes
1answer
45 views

Estimates for harmonic functions

Assume $u$ is harmonic in $U$. Then $$ |D^{\alpha}u(x_{0})|\leq \frac{C_{k}}{r^{n+k}}\|u\|_{L^{1}(B(x_{0},r))} $$ for each ball $B(x_{0},r)\subset U$ and each multi-index $\alpha$ of order $|\alpha| ...
6
votes
1answer
74 views

counterexample for ill posedness of the laplace equation

Consider the wave equation with initial data: $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$u(0,x) = u_0(x)$$ $$u_t(0,x) = u_1(x)$$ Hadamard showed that this problem is ill-posed: there exist large solutions ...
1
vote
1answer
21 views

Is $w =\text{ max}(v_1, v_2)$ subarmonic if $v_1$ and $v_2$ are?

I am studying Perron method to prove the existence of solution to \begin{equation} \Delta u = 0 \quad \text{in } \Omega \\ \ u = g \quad \text{in } \partial \Omega \end{equation} In the proof they ...
0
votes
1answer
67 views

Question about the proof of Harnack's inequality in Evans and Gilbarg's PDE book

I have some trouble in understanding the proof of Harnack's inequality. Since I have consulted two books, I explained my three questions one by one. In Evans' book Partial Differential Equations, 2nd ...
0
votes
0answers
31 views

Laplace's equation boundary conditions

I am supposed to determine a solution from the following boundary conditions in a rectangular area $V$. $f(0,y)=f(x,a) = f(x,0)=0$ and $f(b,y) = C\ sin(\frac{\pi}{a}y)$, for $a,b \in \mathbb{R}$. I ...
1
vote
1answer
37 views

Electromagnetic fields and Laplace equations along a square

I'd like to solve Laplace equation satisfying the following BCs: $$\phi(x,y=0)=0$$ $$\phi(x=0,y)=0$$ $$\phi(x,y=1)=9\sin(2\pi x)+3x$$ $$\phi(x=1,y)=10\sin(\pi y)+3x$$ where $0\leq x,y\leq 1$. I have ...
1
vote
0answers
41 views

Question about the maximum principle for the Laplace's equation

Maximum principle for the Laplace's equation: $$\nabla^2 u= \Delta u=u_{xx}+u_{yy}$$ $$\Delta u=f(x,y) \text{ Poisson }$$ Problem with boundary values of the form Dirichlet: $$\left.\begin{matrix} ...
0
votes
0answers
23 views

Prove that a function is harmonic by the use of the mean property

Given $y\in\mathbb{R}^N$, define $v_y:\mathbb{R}^N\setminus\{y\}\to\mathbb{R}$ by $$v_y(x) = \frac{|y|^2 - |x|^2}{|x-y|^N}.$$ By denoting $u(x) = |y|^2 - |x|^2$ and $w(x) = |y-x|^{-N}$, one can ...
0
votes
1answer
27 views

Proof that laplace's equation is rotationally invariant using chain rule

Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) ...
1
vote
0answers
86 views

Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
0
votes
1answer
35 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
0
votes
1answer
43 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 $u$ ...
1
vote
1answer
45 views

Laplace's equation, integral, tends to steady state?

If $v(x,y)$ solves Laplace's equation $v_{xx} + v_{yy} = 0$ on a bounded domain $S$, and $u(x,y,t)$ solves $u_t = u_{xx} + u_{yy}$ on $S$, with $u=v$ on $\partial S$ for all $t$, one can show that ...
0
votes
1answer
38 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, ...
1
vote
1answer
66 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 } ...
5
votes
3answers
156 views

Laplace's equation on a square domain with a central point reservoire

Could someone please tell me the solution to this problem. I have $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ on the square domain $-L<x<L, -L<y<L$ with ...
1
vote
0answers
263 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 $x_0$, ...
3
votes
2answers
114 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 ...
1
vote
0answers
50 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 ...
2
votes
2answers
106 views

Why don't elliptic PDE's have a time coordinate?

Usually second-order linear PDE's are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the ...
5
votes
1answer
198 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, ...
0
votes
1answer
89 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
0
votes
1answer
53 views

polar Laplace equation solution:

Question: $ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial {\theta}^2} = 0,\:\:\:\: 0\leq r \leq 3 \:\:\:\: ...
1
vote
1answer
44 views

One problem about harmonic functions

Problem. Given open, bounded set $\Omega\subset\mathbb R^d$ with smooth boundary $\partial\Omega$ and given smooth function $\varphi$ on $\partial\Omega$. As known, problem $$ \begin{cases} ...
5
votes
1answer
133 views

Estimates on Derivatives

I have trouble in filling in the details of the proof on Estimates on derivates, from page 29 of PDE Evans, 2nd edition. Namely, I am lost at some steps. The book gives: Theorem 7 (Estimates on ...
0
votes
2answers
195 views

Uniqueness of harmonic solution

On page 28, line 1 in PDE Evans, 2nd edition, the theorem and proof state Theorem. Let $g \in C(\partial U), f \in C(U)$. Then there exists at most one solution $u \in C^2(U) \cap C(\bar{U})$ of ...
4
votes
1answer
79 views

Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
1
vote
1answer
112 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
4
votes
1answer
199 views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...