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Questions tagged [elliptic-equations]

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

For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see .

750 questions
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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
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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
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17
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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
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11
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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
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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
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9
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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…
Eulerian
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9
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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
9
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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
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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
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8
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1 answer

A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ has no $C^2$ solution in any neighbourhood of the…
user294388
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Monotonicity formula for harmonic functions

Let $\Omega$ be an open subset of $\mathbb{R}^N$ and $u:\Omega\to \mathbb{R}$ be a harmonic function. I need to prove that, for every $x_0\in\Omega$, the function $$ \rho\mapsto \frac{1}{\rho^N}\int_{|x-x_0|<\rho} |\nabla u|^2 \; dx $$ is…
Albert
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8
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2 answers

In what conditions a weak solution is a classical solution?

I'm studying elliptic equations in divergence form $$-D_{j}(a_{ij}(x)D_{i}u) + c(x)u = f(x) \text { in a domain } \Omega \subset \mathbb{R}^{n}$$ I call a function $u \in H^{1}(\Omega)$ a weak solution if for every $\phi \in…
Cézar Bezerra
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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
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Have the solutions to semilinear Elliptic PDEs: $-\Delta u=f(x,u)$ with Neumann Boundary conditions been established?

Suppose $U$ is a bounded domain in $\mathbb{R}^n$ ($n \geq 3$) with smooth boundary. Consider the semilinear Elliptic PDE: $$ \begin{cases} -\Delta u=f(x,u) & \text{in } U, \\ \partial_{\nu}u=g & \text{on } \partial U \tag{1}…
miyagi_do
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7
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1 answer

In which sense are the Cauchy-Riemann equations elliptic

I often read that the rigidity and smoothness properties of holomorphic functions can be explained by the fact that the Cauchy-Riemann equations are elliptic. In which sense is that true? Obviously they are not elliptic in the sense of the…
Bananach
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