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

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2
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1answer
24 views

Convergence in $L^p(0,T;L^q(\Omega))$

If $\Omega\subset\mathbb{R}^3$ is bounded, $$f_n\to f\mbox{ in }L^q(0,T;L^p(\Omega)),\,1\leq q<\infty,\,1\leq p<2 $$ and $$f_n\to g\mbox{ weak-star in } L^\infty(0,T;L^2(\Omega)),$$ then $f=g$ ...
1
vote
1answer
23 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
0
votes
1answer
45 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...
1
vote
1answer
36 views

Linear algebra - as applied to the weak maximum principle

Currently I have little background in theory-based linear algebra (I never took an upper-division linear algebra course), so I am asking this question here. Although this is from page 345 of PDE by ...
3
votes
0answers
86 views

quasilinear partial differential equation

Given a PDE $ f e^2 \frac{\partial f}{\partial x}- e f^2 \frac{\partial f}{\partial y} + M_1 f^4 + M_2 f^2 + M_3=0 $ Note that $M_1$ , $M_2$ and $M_3$ are functions of $\cos (x-y)$ and $\sin (x-y)$ ...
2
votes
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}) ...
0
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0answers
12 views

Examples of quasilinear wave equations

Consider a quasilinear wave equation equation of the form $\sum g^{ij}(u, Du)\partial_i\partial_j u = F(u, Du)$ on $R \times R^n$ subject to initial data $u(0,x)=g, \; \partial_t u(0,x)=h.$ Given ...
1
vote
1answer
39 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 ...
4
votes
1answer
36 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
1
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0answers
37 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
-1
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0answers
29 views

PDE using Laplace transform

! Can anyone please explain how to solve this question?
2
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1answer
28 views

In what sense is the Ricci-Flow equation a “distant relative” of the Black-Scholes equation?

In the book "The Poincare Conjecture: In Search of the Shape of the Universe" by Donal O'Shea, the author states that, "The Ricci-flow equation Perelman wrote, a type of heat equation, is a distant ...
3
votes
3answers
139 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
1
vote
1answer
17 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
2
votes
1answer
12 views

Definition of $H^{-1}$ space in Evans' PDE book

Let $U$ be an open, bounded subset of $R^n.$ Evans' well known PDE book defines the spaces: -$H_0^1(U)$:= $\{f\in H^1(U): \text{there exists a sequence} \; \phi_n \to f \; \text{in the} \; H^1(U) ...
4
votes
3answers
115 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
0
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0answers
32 views

Parabolic PDEs and Gradient Systems

Apologize in advance for the length of this question, I need some help in clearing some things up that I haven't quite got my head around yet. It seems to be easy to find things out about finite ...
1
vote
1answer
25 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
2
votes
0answers
20 views

harmonic functions: comparison of gradients

Consider $\Omega$ a open, bounded, convex domain in $R^n.$ I am trying to justify this: Let $u, v$ non negative harmonic functions in $\Omega$ (in the Sobolev sense). Suppose that $ u = v =0$ in ...
0
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0answers
15 views

Heat semigroup on Morrey spaces

I have a few questions concerning the heat semigroup $e^{\Delta t}$ on Morrey spaces. (1) I read that the heat semigroup on $ M^{p}_{q}(\mathbb{R}^n):= \left\{ f \in L^{q}_{loc}(\mathbb{R}^n): ...
2
votes
1answer
25 views

Does separation of variables in PDEs give a general solution?

When a partial differential equation is solved using the separation of variables method, is the produced solution the most general one that satisfies the equation or have we lost some forms of the ...
0
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0answers
13 views

Method of Characteristics (Change of Co-ordinates)

Here below is the notes about the change of co-ordinates from $xy$-plane to $\xi\eta$-plane. I wanna ask for why dot product works for the change, i.e. $\xi=(x,y) \cdot (a,b)$ and $\eta=(x,y) \cdot ...
2
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0answers
30 views

Method of characteristics for systems of PDE (vs. Lewy's example)

Main question: How does the method of characteristics generalize for systems of first order PDE, as opposed to scalar PDE? Namely, is there such a generalization at all, and if so what information ...
2
votes
1answer
69 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
0
votes
0answers
24 views

Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H(x,y)$$ where the known function $H=x^2+xy+y^2+\frac{1}{x-\alpha}+\frac{1}{\beta-x}$. It is clear that $H$ ...
2
votes
1answer
96 views

Solution to wave equations with predefined phase

I want to construct special solutions to wave equations, specifically to Helmholtz-equation and paraxial wave equation (PWE). Let us consider the PWE $$(\partial_x^2 + \partial_y^2 - ...
3
votes
2answers
80 views

On the solution of constant coefficients PDEs (exponential method)

Having a look to my old PDE notes, I have come across with the following problem: Consider the 2nd order PDE: $$ \varphi_{xx} - \varphi_{xy} = 0, \quad (x,y)\in \mathbb{R}^2, \quad \varphi = ...
0
votes
2answers
15 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
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2answers
22 views

$\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $ holds?

I want to find the relation of $p$ and $k$ such that the inequality $$ \|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $$ holds when r.h.s $<\infty$. Here $f$ ...
5
votes
0answers
52 views

Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume ...
2
votes
0answers
27 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
0
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0answers
28 views

Interior $H^2$ regularity - using a textbook's identity to show an important estimate

This is yet another continuation of my previous question, and concerns the same long textbook proof which I asked in those two questions as well. This is now from page 331 of PDE by Evans, 2nd ...
0
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2answers
28 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
2
votes
1answer
46 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
2
votes
1answer
31 views

$1/r^2\int_{\mathbb{S}_r}u-u(x)$ converging to $\Delta u(x)$?a

When reading some papers on PDEs, the following shows up several times: For a $C^{\infty}$ function $u$, $\frac{\int_{\mathbb{S}_r(x)}u-u(x)}{r^2}$ converges to $1/2n\Delta u(x)$ uniformly on ...
0
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0answers
9 views

Paper about Global Hypoellipticity Doubt..

I'm tying to read a paper about global hypoellipticity and I came across the following statement: Let $E$ be an elliptic, normal differential operator of order $e$ on a smooth manifold $M$... The ...
0
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1answer
28 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
5
votes
2answers
87 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
1
vote
2answers
57 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
5
votes
2answers
75 views

Solving the PDE $u_{tt}+2u_{tx}+u_{xx}=2c$

Consider the second-order parabolic inhomogeneous second-order PDE $$ u_{tt}+2u_{tx}+u_{xx}=2c $$ I have seen two ways to solve this problem. I would like to know (1) if Solution 1 is correct (2) if ...
0
votes
1answer
35 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 ...
3
votes
4answers
85 views

Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$ y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
1
vote
1answer
62 views

estimation of gradient

$$(\mathcal{P}_{\varepsilon}) : \left\{\begin{array}{ll} \displaystyle -div\left(A(x)\nabla u_\varepsilon(x)\right)= \dfrac{a(x)}{|u_\varepsilon(x)|+\varepsilon} &\mbox{ in }\Omega \\\\ ...
2
votes
2answers
65 views

Solving second order non-homogenous PDE

$$ d_1u_{xx} +d_2 u_{yy} = -2 $$ I need to solve this given PDE. I tried to solve it using change of variables. The variables are $$ \xi = y-ax \ , \ \ \ \eta = y+ax$$ where $$a = \sqrt{-d_1d_2} $$ ...
1
vote
0answers
12 views

order of pseudo - differential operator

We consider the pseudo - differential operator $$Jv=(1+i\,a(x)\,(1-\Delta)^{-1}\,a(x))\,v$$Studying the work of Camille Laurent,"Global controllability and stabilization for the nonlinear Schrodinger ...
2
votes
1answer
35 views

Solve 2 connected ODEs describing a domain

This problem confused me for a long time. I have 2 ODEs which describe part of our domain. They are connected at middle: $$ \frac{d^2}{dx^2} u = -a, x<x_0 $$ $$ \frac{d^2}{dx^2} u - \frac{u}{b^2}= ...
3
votes
1answer
91 views

Checking Boundary Conditions for Candidate Solutions to PDE

Consider the one-dimensional heat equation $ u_{xx}=u_t$ with the boundary data $$u(x, 0)=f(x), \quad u(0, t)=u(L, t)=0.$$ The standard method of solving this equation is by finding a candidate ...
1
vote
4answers
67 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
0
votes
1answer
32 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
0
votes
1answer
19 views

To show surface orthogonal to each other

How do we show that the surfaces represented by $Pp + Qq = R$ where $p= \frac{\partial z}{\partial x}$ and $q= \frac{\partial z}{\partial y}$ are orthogonal to the surfaces represented by $Pdx + Qdy + ...