Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".
0
votes
0answers
5 views
Green formula in Lebesgue weighted space
Working on PDE, I was wondering if Green formula were dependent of the chosen coordinate system ? If you consider your weak form in cylindrical coordinate $(R, \Phi, Z)$ you need to use weighted ...
0
votes
0answers
18 views
Solutions of a system of second order PDE using ray method
I find a solution of the following PDE system:
$\epsilon^2\left(\Phi_{xx}+\Phi_{yy}\right)+\Phi_{zz}=0$ for $z\lt0$ and $g\Phi_z+\epsilon^2V^2\Phi_{xx}=0$ for $z=0$ with $\epsilon\ll1$ with $V=const$. ...
1
vote
0answers
33 views
Advection Diffusion Equation on Semi-Infinite Domain
Regarding the BVP
$$u_t(x,t) - v u_x(x,t)=k u_{xx}(x,t),\qquad x\geq0$$
with BC $u_x(0, t)=0$ for $t\geq 0$, and parameters $v,k>0$, I have some questions.
Does an expression for the Green's ...
1
vote
1answer
22 views
What is elliptic bootstrapping?
While reading about elliptic differential operators, I have seen the phrase elliptic bootstrapping in several places, but none of them explain exactly what it means. I know it has something to do with ...
1
vote
1answer
9 views
requirement of ellipticity property in definning viscosity solution
Why it is important to require ellipticity condition in defining the notion of viscosity solution of degenerate elliptic or parabolic pde.
2
votes
1answer
29 views
Simplicity and isolation of the first eigenvalue associated with some differential operators
Consider the operator $\Delta$ or more generally a second order differential operator $L$, in which the principal part is symmetric and positive definite. It can be proved (see here page 336) that the ...
1
vote
2answers
90 views
Burger's equation
I couldn't solve this problem, can you help me please?
The Burger equation
$$ u_y + uu_x = 0 $$ $ - \infty < x < \infty $ , $ y > 0 $ , $ u(x,0)=f(x) $
My question;
is there any solution ...
5
votes
1answer
85 views
How to prove that the spectrum of the Laplacian over $\Omega\subset \mathbb{R}^n$ is negative?
I am looking for a proof of this well known fact and I guess it has to do with integration by parts (Green's identity). Unfortunately, I only know about 1-d integration by parts( I am just 3rd ...
1
vote
0answers
13 views
Power-sums analogue of Burgers equations
Let $D$ be a domain in $\mathbb C^2$. Let $f_1$, $\ldots$, $f_n$ be a family of holomorphic functions in $D$, satisfying Burgers-type equations:
$$
\frac{\partial f_k}{\partial z_2} - f_k ...
2
votes
1answer
83 views
Laplace equation solution for electrical potential
I'm trying to simulate a lightning in a programming language and I started to read how it can be done, and I found that it could be done by using the Laplacian Growing Model. My experience with ...
0
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0answers
73 views
EDIT variational formulation-exercice [duplicate]
i have this problem.
Let $\Omega$ a bounded domain, connexe and regular, and let $f \in L^2(\Omega).$ Let the variational problem: Find $u \in H^1(\Omega)$ such
$$\int_{\Omega} \nabla u \nabla v dx + ...
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votes
0answers
41 views
Second order PDE
Following: $X_{i}=Kx, u(X_{i}) = \Psi(X_{i}/K)\implies \Psi(x)=u(Kx)$ we get the PDE:
$$
\frac{d^2 \Psi}{dx^2}=K^2\frac{d^2 u}{dX_{i}^2}
$$
How to obtain this expression - Can you help me Please ? I ...
0
votes
0answers
7 views
Uniqueness of Quasilinear Monge Cone
(This page & the page before it are a source for my question if required)
When solving a nonlinear first order pde $F(x,y,z,p,q) = 0$ one can use the implicit function theorem to solve this for, ...
2
votes
0answers
25 views
Non-Integrability of a Pfaffian - Geometric Interpretation?
The question of Solving a Pfaffian ODE can be interpreted as the question of finding the family of surfaces $U = c$ perpendicular to a surface $f$ generated by the vector field
$$F(x,y,z) = ...
2
votes
1answer
44 views
Minimization problem as PDE
In the article "An Image Interpolation Scheme for Repetitive Structures" Luong, Ledda and Philips propose the following approach to denoising digital image.
They consider that regularized total ...
11
votes
4answers
179 views
Correct spaces for quantum mechanics
The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
2
votes
1answer
23 views
Numerical methods for inverting non positive definite matrices
I'm working on a PDE solver and need to invert the following matrix written in block form
$\left(
\begin{array}{cc}
kM & -S \\
-S & M
\end{array}\right)
$
where M and S are the usual mass and ...
2
votes
1answer
36 views
On the proof of deformation lemma “boundedness”
Book- Evans partial differential equation. In the proof of deformation lemma how to say that $V(u)=-g(u)h(\lVert I'(u)\rVert)I'(u)$ is bounded. And how to say that the mapping $u \to ...
2
votes
0answers
39 views
Solution to this Poisson equation
I am struggeling with the following PDE. Does somebody here know a solution on the whole $\mathbb{R}^2$ that goes to zero for r approaching infinity?
$\Delta ...
1
vote
1answer
42 views
Derivation of Wave equation
Is there a reference you know of that derives the (dimension $> 1$) wave equation starting from the Navier-Stokes equations?
2
votes
0answers
26 views
theorem about hausdorff dimension involving capacity theory (theorem in the classic book of Heinonen)
I am studying the proof of this theorem :
Theorem: Suppose that $1<p \leq n$ and $E$ is a set in $R^n$ of $p -$ capacity zero . Then the Hausdorff dimension is at most $n-p$.
Proof:
"Since ...
0
votes
0answers
36 views
singular perturbation
I'm looking for literature pertaining to singulary perturbed fourth order parabolic equations. ex:$$u_{t}=-\epsilon u_{xxxx}+f(x,u)u_{xx}+g(x,u,u_x)$$ for sufficiently smooth functions $f,g$.
I ...
6
votes
1answer
47 views
Are a weak derivatives and distributional derivatives are different?
For simplicity, given a real function $f\in L^1_{loc}(\Omega)$, we define both weak or distributional derivatives by
$\int f'\phi = - \int f \phi'$ for all test functions $\phi$.
Now, take $\Omega = ...
0
votes
0answers
48 views
question 5- chap 6 evans PDE
Let $u$ be a smooth solution of $Lu=-\sum_{i,j=1}^n a^{ij}u_{x_ix_j}=0$ in U. Set $v:=|Du|^2+\lambda u^2$. Show that
$$Lv\leq0~~in~U,~ if~\lambda \mbox{ is large enough.}$$
Deduce
...
1
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0answers
33 views
Approximative solution to PDE with additional term.
I am currently struggeling with the following problem:
If I have a solution to the partial differential equation $ \Delta \Psi(r,\theta) = \rho(r,\theta)$ on $\mathbb{R}^3\backslash B(0,R)$(so the ...
3
votes
0answers
23 views
Attach term to solution of PDE(perturbation theory)
Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
2
votes
1answer
60 views
Prove this PDE has a unique (weak) real valued solution
Let $\Omega = (0,1) \times (0,1)$ and let $k \in [0,\frac{3}{2}]$.
Prove the boundary value problem $$\frac{f_{xx}}{2y}+f_{yy}+2k^2\frac{f}{y} = 2k^2$$ subject to ...
1
vote
2answers
33 views
particle models with network interaction - simultaneous estimation
I am working on an application in behavioral ecology that combines network and particle interaction models. I have not been able to find any articles that simultaneously estimate these types of models ...
2
votes
1answer
27 views
Weak maximum principle for the p-Laplacian
For the equation $\Delta_p u = 0 $ in $U$ ($U$ open and bounded), does a weak maximum principle hold? (The maximum and minimum occur on $\partial U$)? If yes, someone can indicate a book with the ...
3
votes
0answers
64 views
Is there any theorem that tells us how many ICs or BCs are needed for getting the determine solution of a PDE or a set of PDEs?
It's a shame that, though I've taken the "Equations of Mathematical Physics" class for one semester and solved numbers of PDEs with Mathematica, I'm still unclear about how many initial ...
3
votes
2answers
66 views
Laplace equation Fourier transform
I tried to solve but I didn't. Please help me.
$$u_{xx} + u_{yy} = 0$$
$$-\infty < x < \infty$$
$y>0$ , $u_{y}(x,0)=f(x)$.
Show that
$$u(x,y)= \frac{1}{2 \pi} ...
1
vote
1answer
41 views
About Mean Value Property of Harmonic Function
I know the question may seem foolish to you but I am not quite sure how to show it in a decent way. My problem is to show that for bounded Borel measurable $f:\mathbb{D}^2\to\mathbb{R}$, (D1) is ...
1
vote
1answer
34 views
Question on boundedness
Page number 479 in partial differential equation by Evans book how to say that the derivative of I is bounded on bounded sets
4
votes
0answers
28 views
3d-diffusion equation in spherical coordinates (numerical), boundary problem
There is one boundary problem
$$\frac{\partial u}{\partial t}= \operatorname{div}\left(a^2 E \nabla u\left(r,\varphi,\psi \right) \right) $$ in a ball
$$ B_{1}(0)=\left\{x \in \mathbb{R^3}: \left\| ...
7
votes
1answer
133 views
+50
Solving Poisson's equation for $\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) \, \delta(y)$
Problem statement
I took an exam, where I had the following task: Determine the electrostatic potential for the charge distribution
$$\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) ...
5
votes
1answer
63 views
The proof of the Helmholtz decomposition theorem through Neumann boundary value problem
As you know, the Helmholtz decomposition theorem is as follows.
Let $\Omega$ be open and simply connected, and bounded region in $\mathbb R^3$.
Then the smooth vector field $E : \Omega \to ...
1
vote
1answer
26 views
A not smooth distribution
Is exist the not smooth distribution which satisfying:
$\left ( D_{t}^{2}-D_{x}^{2} \right )u(x,t)=0$
I can't find at least one not smooth distribution like this...
Thanks for the help!
1
vote
0answers
27 views
Yosida approximation, solution is $C^\infty$? (Explanation of passage needed)
I'm reading Brezis' book Functional Analysis, Sobolev Spaces and PDEs, Lemma 7.1 page 186.
Let $w \in C^1([0,\infty);H)$ satisfy
$$w' + A_{\lambda}w = 0$$
where $A_\lambda$ is the Yosida ...
2
votes
0answers
38 views
When does a Green's function exist?
If I have a simply-connected compact domain $ \Omega $ in $ \mathbb{R}^2 $, endowed with a Riemannian metric $ g $, does there exist a green's function on $ \Omega $ for the laplace operator induced ...
3
votes
1answer
112 views
Is my proof that a function is measurable correct?
Let $V$ be separable and Hilbert. Let $\mathcal V = L^2(0,T;V)$. Assume for each $t \in [0,T]$,
$$a(t;\cdot,\cdot):V \times V \to \mathbb{R}$$ is continuous and bilinear.
Or equivalently, we have ...
2
votes
1answer
22 views
The classification of possible singular supports
I need to find the solutions of $D_{x_1}u=0$ on $\mathbb{R}^{n}$ and to classify the possible singular supports.
Any one have an idea how to solve this kind of question?
Thanks!
2
votes
2answers
74 views
Prove $\varphi\in\mathcal{S}(\mathbb R^n)$ if and only if the following inequality holds..
I need some help for showing the following result: Let $\varphi\in C^\infty(\mathbb R^n)$. Then $\varphi\in \mathcal{S}(\mathbb R^n)$ if and only if for all $\alpha\in\mathbb N^n$ and $N\geq 0$ there ...
1
vote
0answers
22 views
Projection of fine mesh with low degree polynomial to projection of coarse mesh with higher degree polynomial of a funtion $u$.
One way of improving some finite element approximations is to take the approximate solution and project it (mesh element-wise) using $L^2$-inner product onto a space of piecewise polynomials of higher ...
3
votes
0answers
57 views
A “straightforward” inequality.
In section 5.5 page page 117 of Fanghua, Quing's book, in order to obtain $W^{2,p}$ estimates the idea is to show that for $p \in (1,\infty]$ the condition $\theta \in L^p(\Omega)$ implies $D^2u \in ...
2
votes
1answer
49 views
Can I use inverse inequality for an infinite space, $H^1$?
Let $u \in H^1(\Omega)$ and $Q_0 u$ is a $L^2$ projection of $u$ to a polynomial finite space $P_k(T)$ where $T \in \mathcal{T}_h$ is a finite element and $\mathcal{T}_h$ is a set of all the elements.
...
1
vote
1answer
40 views
How to show this equality?
I need some help for showing the following equality $$\displaystyle \left(\sum_{i=1}^nx_i\right)^m=\sum_{|\alpha|=m}\frac{m!}{\alpha!}x^\alpha$$ for all $x\in\mathbb R^n$. Here $\alpha\in\mathbb N^n$ ...
2
votes
0answers
39 views
Restriction on exponent; Weak Harnack inequality for strong solutions
The weak Harnack inequality for strong solutions goes as follows (Taking $Lu = a^{ij}(x)D_{ij}u + b^i(x)D_iu+c(x)u$ to be elliptic)
Let $u\in W^{2,n}(\Omega)$ satisfy $Lu\leq f$ in $\Omega$ for ...
2
votes
1answer
49 views
Construct harmonic function on noncompact manifold
$M$ is a non-compact Riemannian manifold, $p \in M$. Consider Dirichlet problems: $\Delta u = 0$ in ${B_p}\left( i \right)$ ($i = 1,2, \dots $), $u{|_{\partial {B_p}\left( i \right)}} = {f_i}$, ${f_i} ...
3
votes
1answer
23 views
Density of a subspace in $\mathcal{D}(0,T;V)$ under $H^1$ norm
Let $V$ be Hilbert. Let $\mathcal{D}((0,T);V)$ be space of infinite differentiable functions with values in $V$ with compact support. Are functions of the form
$$\sum_j \psi_n(t)w_n$$
where $\psi_n ...
0
votes
1answer
32 views
Isometric embeddings with prescribed second fundamental form
I'm looking for some non-rigidity result for isometric embeddings in euclidean space (codimension 2).
For example, any isometric embedding of the round $S^2$ into $\mathbb R^3$ is unique up to rigid ...
