For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).

# Questions tagged [elliptic-equations]

525 questions

**27**

votes

**7**answers

### Why is it useful to show the existence and uniqueness of solution for a PDE?

Don't get me wrong, I understand that it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps, for example, to actually solve the problem.
I am reading books that deal with…

D1X

- 1,982
- 14
- 24

**14**

votes

**5**answers

### Unique weak solution to the biharmonic equation

I am attempting to solve some problems from Evans, I need some help with the following question.
Suppose $u\in H^2_0(\Omega)$, where $\Omega$ is open, bounded subset of $\mathbb{R}^n$.
How can I solve the biharmonic equation
…

Theorem

- 7,367
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- 52
- 93

**12**

votes

**1**answer

### What is the parametric equation of a rotated Ellipse (given the angle of rotation)

The Formula of a ROTATED Ellipse is:
$$\dfrac {((X-C_x)\cos(\theta)+(Y-C_y)\sin(\theta))^2}{(R_x)^2}+\dfrac{((X-C_x) \sin(\theta)-(Y-C_y) \cos(\theta))^2}{(R_y)^2}=1$$
There:
- $(C_x, C_y)$ is the center of the Ellipse.
- $R_x$ is the Major-Radius,…

Gil Epshtain

- 411
- 1
- 3
- 11

**9**

votes

**1**answer

### Is there analytical solution to this heat equation?

I have a PDE of the following form:
$$\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2f}{\partial\phi^2} = A\cos\theta\,\max(\cos\phi, 0) +…

titanium

- 514
- 4
- 13

**9**

votes

**0**answers

### Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator.
However, I learn also from the conversations in the same post that the hypoelliptic operators can be Fredholm but…

Ali Taghavi

- 1,740
- 1
- 11
- 34

**8**

votes

**0**answers

### Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta u dx $$ where $n$ is the unit outward normal ,…

user490539

**8**

votes

**0**answers

### Elliptic regularity on the Hypercube

Assume
$$
Lu=f\quad \text{in } [0,1]^d\\
u=0 \quad\text{ on } \partial[0,1]^d
$$
for some strongly-elliptic operator $L$, and $f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the result for smooth domains.
I am pretty sure the…

Bananach

- 7,304
- 1
- 24
- 41

**8**

votes

**0**answers

### Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type
$$
-a\Delta u + f\left(u\right) = 0,
\\
u\big\vert_\Gamma = u_0
$$
by Newton’s method when its convergence is global and monotonic.
Could you advice some references…

jokersobak

- 224
- 1
- 13

**7**

votes

**2**answers

### Evans' PDE Problem 6 Chapter 6 - Existence and uniqueness of weak solutions of Poisson's equation with mixed Dirichlet-Neumann boundary conditions

Suppose $U \subset \mathbb R^n$ is an open, bounded and connected set, with smooth boundary $\partial U$ consisting of two disjoint, closed sets $\Gamma 1$ and $\Gamma 2$. Define what it means for $u$ to be a weak solution of Poisson's equation…

sound wave

- 795
- 1
- 6
- 14

**7**

votes

**3**answers

### How to solve a second order partial differential equation involving a delta Dirac function?

In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function:
$$
a \, \frac{\partial^2 w}{\partial x^2}
+ b \, \frac{\partial^2 w}{\partial y^2}
+ \delta^2(x,y) = 0 \, ,
$$
subject…

Hohenstoffen

- 191
- 10
- 29

**7**

votes

**2**answers

### Is it possible to solve a hyperbolic moving boundary problem?

J. L. Davies says in his book,
"The basic principle in PDEs is that boundary value problems are associated with elliptic equations while initial value problems, mixed
problems, and problems with radiation effects at boundaries are
associated…

Nanashi No Gombe

- 1,102
- 8
- 24

**6**

votes

**1**answer

### Solving the 2D Poisson equation with variable boundary location

I am trying to find $z(r,\phi)$ from the 2D Poisson equation in polar coordinates:
$$\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2z}{\partial \phi^2}=C \tag{1}$$
where $C$ is a…

Michiel

- 394
- 3
- 24

**6**

votes

**1**answer

### Solutions of $u_{xx} + 2 \mathrm i u_{xy} + u_{yy} = 0$

A problem in A. Friedman, Partial Differential Equations, 1969, chapter 1, section 14, is concerned with the elliptic, but not strongly elliptic PDE on the open unit disk $\mathbb D = \{z= x+ \mathrm iy \mid |z| < 1\}$ given…

Kehrwert

- 587
- 2
- 11

**6**

votes

**1**answer

### Equivalence of two harmonic problems on different domains

I want to solve
\begin{cases}
\Delta u = 0,&\text{ in }\mathbb{R}^3\setminus B_1(0) \\
u=0,&\text{ as }\Vert x\Vert\rightarrow +\infty \\
u=1,&\text{ on }\partial B_1(0).
\end{cases}
I know that the solution to this problem is
$$
\bar{u}(x) = \Vert…

Dadeslam

- 644
- 4
- 19

**6**

votes

**0**answers

### Existence theorems for self adjoint elliptic systems

Let's consider an elliptic (vectorial, homogeneous, constant coefficient) system of order 2m
$$
\begin{cases}
Lu=f &\text{in }\Omega\\
Bu=0 &\text{on }\partial\Omega
\end{cases}
$$
which is self adjoint, meaning that
$$
\int_{\Omega}\langle Lu,v…

Sobolev

- 195
- 8