Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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2
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1answer
24 views

Maximum condition for second order differential operators

Let $A$ be a second order differential operator such that $$Af(x) = \sum_{ij} a_{ij}(x) \big(\partial_i \partial_j f(x)\big) + \sum_j b_j(x) \partial_j f(x) $$ Assume that $x \in B(0,r)\Rightarrow ...
2
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0answers
56 views

How to use Fourier transform to solve Fisher-Kolmogorov?

How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D? \begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation} \begin{equation} u(0,x) = \phi(x) \end{equation} with ...
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0answers
22 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$\begin{align} \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C ...
2
votes
1answer
55 views

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ...
2
votes
1answer
15 views

Maximum principle, quantitative version

Is it true that $$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?
1
vote
1answer
77 views

a vector calculus problem when coping with problem 9 chapter 5 evans

Hi I was trying to understand the solution given by Ray Yang in the post question 9 - chap 5 evans PDE It gets to the sort of things I am quite bad at... When I get to the point ...
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2answers
30 views

Is the parabolic heat equation with pure neumann conditions well posed?

The parabolic heat equation is a partial differential equation given by $\frac{du}{dt}=\nabla^2u+f$. If i impose an initial condition u(x,0) and pure homogeneous neumann boundary conditions that ...
2
votes
1answer
41 views

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in ...
1
vote
1answer
14 views

Maximum principle of p-Laplacian operator

Suppose $\Delta_p u \geq \Delta_p v$ then can it be said that $u \geq v$? The domain considered is a bounded subset of $\mathbb{R}^n$.
1
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1answer
64 views

For which $1\le p\le\infty$ does $u$ belong to $W^{1,p}$(\Omega)$?

Hi could anyone help with a solution for problem 7 Evans PDE chapter 5? I think it is basically about checking which $p$ allows $$\int_{\Omega} |u|^p dx+\int_{\Omega}|Du|^p dx<\infty$$ ? But I ...
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0answers
27 views

non existence of weak derivative evans pde chapter 5 example 2

Hi Im looking at the this very basic example, in proving the non existence of weak derivative. I am confused in the last line $$\cdots\lim_{m\to\infty}\left(\int_0^2 v\phi_m dx-\int_0^1\phi_m ...
0
votes
1answer
22 views

Bound on the size of the solution to $u_{t} - \Delta u - u \leq 0$

Let $B$ denote the open unit ball in $\mathbb{R}^{d}$ and let $u$ be a smooth function such that $u_{t} - \Delta u - u \leq 0$ in $U_{T}:= B \times (0, T]$ and $u = 1$ on the parabolic boundary (that ...
0
votes
1answer
13 views

Regularity for non-homogenous elliptic PDE

Assume $L$ is an elliptic differential operator (second order, with coercive associated bilinear form) with smooth coefficients, and that $\Omega$ has smooth boundary. Does there exist a result of ...
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0answers
25 views

How to apply Runge-Kutta to an implicit scheme?

I see there are some differences in the solution as I increase the resolution of my grid. I'm using Operator Splitting to solve Diffusion Reaction equation \begin{equation} \frac{\partial ...
1
vote
1answer
44 views

Quadrature formula on triangle

I am looking for a quadrature formula on the triangle, with points at the vertices and at the mid-edges, so 6 points, and that is exact for polynomials of degree at least 2, with weights strictly ...
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0answers
20 views

non-linear PDE finite difference approach

How to approach this equation using finite difference method ...
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0answers
21 views

Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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0answers
26 views

Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j ...
1
vote
1answer
24 views

Which 2D domain with fixed area has the lowest laplacian eigenvalue?

I know that a disc has the lowest laplacian eigenvalue among domains with fixed area. But how do I prove it?
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1answer
24 views

Control of $L^{\infty}$ Norm of 3d Heat Equation Solution for $L^{3}$ Initial Data

Let $w_{t}$ denote the 3-dimensional heat kernel $$w_{t}(x)=(4\pi t)^{-3/2}e^{-\left|y\right|^{2}/(4t)},\qquad y\in\mathbb{R}^{3}, \ t > 0$$ Suppose $f\in L^{3}(\mathbb{R}^{3})$, and let ...
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0answers
28 views

Why L belongs to the dual space $H^{-1}$

I'm studying pde using Evans book. In chapter 6 he introduces second order partial differential operators for example : $L= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}$ I can't understand why $L \in ...
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0answers
18 views

2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
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0answers
76 views

{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

I have a question that it maybe simple but I can not understand why we have : {$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ ...
1
vote
1answer
36 views

Help with method of characteristics question.

I'm trying to solve the partial differential equation, $$ \frac{\partial F}{\partial t} = (z-1)\left(kF - d \frac{\partial F}{\partial z}\right) $$ where $k$ and $d$ are constants greater then $0$, ...
0
votes
1answer
30 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
3
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0answers
49 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
0
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0answers
35 views

Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business. In the ...
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0answers
37 views

Clarification on the validity of separable solutions to PDEs

So we know that various famous PDEs have solutions that can be found by assuming the function is a product of functions each in one variable, e.g. 'Let $V(a,b,c)=A(a)B(b)C(c)$' etc. What I'd like to ...
0
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1answer
14 views

Calculating Laplacian after substituting polar coordinates in derivation of fundamental solution to Laplace's Equation

I'm following the derivation of the fundamental solution to Laplace's equation in section 2.2.1 of Evans's PDE book. It's the standard approach. We assume a radially symmetric solution $v(r)$ and do ...
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0answers
19 views

Nonlinear-Variation of Helmholtz Equation

I was wondering on the solution of the equation $$\nabla^2P(\vec r)=v(\vec r)P(\vec r)^2\phantom{.......}(1)$$ Or more simply, if there exists a coordinate system where: $$\nabla^2P(\vec r)=P^2(\vec ...
0
votes
1answer
59 views

Analytical solution to complex Heat Equation with Neumann boundary conditions and lateral heat loss

I have solved a PDE in this from numerically on Mathematica, but does anyone know if there is a way to solve the following PDE analytically, an analytical solution would really help me. This is an ...
1
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0answers
44 views

A system of partial differential equations

I have 6 partial differential equations that in the first look they don't seem very difficult, but all my efforts for solving them were unsuccessful. $$\frac{\partial f(x,y,z)}{\partial ...
0
votes
1answer
50 views

What is the difference between single and double layer potential

I want to know the difference between single layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...
0
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0answers
15 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
2
votes
1answer
59 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
1
vote
1answer
30 views

Relationship between Faedo-Galerkin Method and Semigroup Method

I have multiple questions relating to Galerkin Method and the Semigroup Method of proving existence of solutions to PDEs. In the Galerkin Method, we decide on a function space, find eigenfunctions ...
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0answers
33 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
0
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1answer
28 views

Stepping backwards with Forward Euler?

Let us say I want to use Forward Euler scheme to solve the heat equation $$ \frac{\partial u}{\partial t} = -\frac{\partial^2 u}{\partial x^2} $$ in the domain $t \in (0, 1)$, $x \in (0,1)$ but ...
1
vote
1answer
31 views

Maximum principle for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega),$ we denote by $S(t)f$ the solution of the equation $$ ...
2
votes
1answer
25 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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1answer
45 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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1answer
51 views

Intuition behind eigenfunctions of the Laplacian operator

I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the ...
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0answers
32 views

Adi-method for Diffusion-reaction equation in 2d

i'm trying to solve this pde using an adi-method (alternating-direction-implicit). $\frac{d f}{d t}=D\nabla^2_{x,y} f+Q(x,y)f+C$ After discretizing, the equation looks like this. Implicit in ...
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0answers
17 views

Non-linear hyperbolic systems

This question is about a naive approach to non-linear hyperbolic systems, thinking in the context of elasticity. To set up the problem suppose $\Omega\subset \mathbb{R}^n$ is open and bounded. ...
0
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0answers
20 views

Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
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vote
1answer
28 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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1answer
35 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
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vote
1answer
27 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
0
votes
1answer
17 views

about convection term in the NSE

Let $u=(u_1(x, y), u_2(x, y))$ be two dimensional vector field and consider the convection term $(u\cdot\nabla) u$ in the NSE. In some books it is usually written in this form, but in some other books ...
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1answer
60 views

Operator theory curiosity

I'm not an expert in operator theory... but i was wandering if there's some practical applications. For example (the first one i came up with) compared to normal calculus techniques that usually the ...