Tagged Questions

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

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3
votes
0answers
12 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
2
votes
0answers
19 views

Trace operator counterexample

This is homework so no answers please Let $U$ be bounded with a $C^1$ boundary. Show that a ''typical'' function $u \in L^p(U) \ (1 \leq p < \infty)$ does not have a trace on $\partial U$. ...
1
vote
1answer
14 views

Asymptotic Estimate

Consider the following Sturm–Liouville problem $$u''+\lambda u=0, \ 0<x<1$$ $$u(0)-u'(0)=0, \ u(1)+u'(1)=0.$$ Obtain an asymptotic estimate for large eigenvalues. I solved the problem and ...
1
vote
1answer
36 views

Show that f solves the so called wave equation

Task $\text{Let } \; c \in \mathbb{R} \; \text{ be a given parameter, with } \; c > 0$ $\text{ Show that } \; f: (\mathbb{R}^3 \setminus \{ \vec{0} \}) \times \mathbb{R} \to \mathbb{R} \; ...
0
votes
1answer
12 views

Local Estimates for higher order homogeneous elliptic operators

For $u\in W^{2k}_2(\mathbb{R^n})$, $k\geq 1$, it is well known (see, for example, Exercise 12.9.4 in Krylov, N. "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces") that the following ...
1
vote
0answers
25 views

Differential operators

I have a few very simple questions I think. First and foremost what exactly is $|D^2u|^2$? I know $D^2u$ is the Hessian matrix. Secondly, how do I show that $\sum_{i=1}^n u_{x_i}\cdot Du \cdot ...
1
vote
2answers
25 views

Estimate in Sobolev Spaces

Let $u\in H^0(U)\cap H^1_0(U)$ and $v_k\in C^\infty_c$(U) such that $v_k\rightarrow u$ in $H^1_0(U)$ and $w_k\in C^\infty (U)$ such that $w_k \rightarrow u $ in $H^2(U)$. I want to show that $ \int_U ...
1
vote
0answers
7 views

PDE - Neumann Boundary Conditions

Consider $$u_{tt}-4u_{xx}=(1-x)\cos t$$ $$u_x(0,t)=\cos t -1,\ u_x(\pi,t)=\cos t$$ $$u(x,0)=\frac{x^2}{2\pi}$$ $$u_t(x,0)=\cos3x.$$ We use $u(x,t)=v(x,t)+w(x,t)$ and since we have Neumann ...
0
votes
0answers
10 views

Equivalence of lens shaped domain and the existence of a smooth time function

A lens shape doimain is defined here as: Defn A lens-shaped domain $D\subset M$ based on $\Sigma$ is the image of a smooth map $\Phi: \Sigma\times (-1,1) \to M$ where $\Sigma\subset M$ is a compact, ...
0
votes
0answers
8 views

Sharp Gagliardo-Nirenberg inequality: Proof

I have been looking for a detailed proof of Sharp Gagliardo-Nirenberg inequality which states that the following inequality: $\|u\|_{L^r(\Re^n)}\le ...
0
votes
0answers
11 views

Legendre Polynomials Recursion Problem

Using the recurrence equation for Legendre Polynomials: $$(k+1)P_{k+1}(x)=(2k+1)xP_k(x)-k P_{k-1}(x) \text{ , } k \in \mathbb{N}$$ Compute the Integral: $$ \int_{-1}^1xP_k(x)P_{k+1}(x)dx $$ I am ...
0
votes
0answers
18 views

Solving a PDE via method of characteristics

I'm interested in solving the following PDE via the method of characteristics: $$\frac{\partial f}{\partial t} - ax\frac{\partial f}{\partial p}+ bp \frac{\partial f}{\partial x} = 0,$$ with ...
0
votes
1answer
28 views

A quick question regarding Green's Functions

I've been given the following conditions for some 3D function, $\phi$; $$\nabla^2 \phi(x,y,x) = 0$$ $$\phi(x,y,0) = f(x,y)$$ My question is, would the equivalent Green's function problem be; ...
0
votes
1answer
24 views

Solving second order partial differential equation

I' trying to solve this differential equation: $$y^2 \frac{\partial ^2 u}{\partial y^2} - 2xy \frac{\partial ^2 u}{\partial x \partial y} + x^2 \frac{\partial ^2 u}{\partial x^2} + 2y \frac{\partial ...
0
votes
0answers
17 views

Removable singularities for Dirichlet problems of Laplace equaions?

It is already known that if $u$ is harmonic in $\Omega\backslash\{x_0\}$ where $\Omega$ is a pre-compact domain in $\mathbb{R}^n$, $n\geq2$ and $u=o(|x-x_0|^{2-n})$ when $x\to x_0$, then the singular ...
0
votes
1answer
16 views

Sturm-Liouville problem

Find the eigenvalues and eigenfunctions of the the Sturm-Liouville problem $$(x^2v')'+\lambda v=0, \ 1<x<b$$ $$v(1)=v(b)=0, \ b>1.$$ The general solution is ...
1
vote
1answer
13 views

Solving the non-homogeneous heat equation with homogeneous Dirichlet boundary conditions

The problem: Solve $$ \frac{dT}{dt} + k \frac{d^{2}T}{dx^2} = \exp \Bigl[-\alpha t\Bigr] \sin\left( \frac{2\pi x}{L}\right)\text{ with }T(0,t) = T(L,t) = 0,\ T(x,0) = f(x) $$ with $k$, $\alpha > ...
1
vote
1answer
23 views

Extension theorems in Sobolev spaces: Solving for constants

I saw this problem in PDE book and tried searching for the idea behind solving it which I have not been able to find yet. If we have $n\ge2$, $B=\{x\in\mathbb R^n:|x|<1\}$ and ...
0
votes
0answers
14 views

Product of $n$ $1$-dimensional solutions of heat equation

Suppose $u_{1},\ldots,u_{n}$ are solutions of the one-dimensional heat equation $\partial_{t}u=\partial_{y}^{2}u$. It is easy to verify that $$ v(x,t):=\prod_{j=1}^{n}u_{j}(x_{j},t) $$ solves the ...
1
vote
0answers
35 views

How can I solve this PDE?

$\dfrac{\partial \hat{Q}}{\partial t} - \dfrac{Am}{\rho} \dfrac{\partial ^3Q}{\partial t \partial z^2} = 0$ I really do not know which method could I use to solve it!
1
vote
0answers
18 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
1
vote
0answers
13 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
0
votes
0answers
6 views

Unique solution for PDE?

How can one tell if a solution is existent or unique? For example: $yu_y+uu_x=u-y$ $u(x,1)=x$ I've found the solution to be $u=x+1-y$, but have been told there are infinitely many solutions. Is ...
0
votes
1answer
16 views

Cauchy problem for the inhomogeneous wave equation written as $u_{xy}=1$

I have a question on a PDE assignment that's giving me problems interpreting. Solve the following Cauchy problem for the inhomogeneous wave equation: $u_{xy} = 1$ $u(x,-x)=6$ ...
0
votes
0answers
34 views

Gelfand triples for Product Spaces

For $V = H^1(\Omega)$ and $H=L^2(\Omega)$. If we identify H with it's dual space $H^*$, then we have the following relation: \begin{equation} V \subset H \subset V^* \end{equation} Does this also ...
2
votes
0answers
16 views

Can you give a nonconstant function to show difference between The Weak Maximum Principle and The Strong Maximum Principle

We know that the Weak Maximum Principle and Strong Maximum Principle in every PDE book,such as Theorem 3.1 and Theorem 3.5 in David Dilbarg's book. But I never see a author give a nonconstant ...
0
votes
0answers
8 views

finite subspace of $L_0^2 $- Stokes Problem

Consider $L^2_0(\Omega)$ be the $L^2(\Omega)$ functionspace with mean-value $\int_\Omega u \, dx$. For numerical reasons I search a finite subspace of $L^2_0$ on a triangulation, but I don't know how ...
0
votes
3answers
48 views

PDE - Energy - Wave Equation

I dont know how solve a) and b), I'm read the book of Walter Strauss, but I have a lot of doubts, for the c) first, I tried estimate $(k(t)e^{-2at})'$ and integrate the inequal, but not unwind... :( ...
2
votes
0answers
28 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{(k-1)A_{k-1}(t)-kA_k(t)}{\alpha+2\beta t} + \delta_{k \beta} . ...
0
votes
0answers
39 views

Partial differential equation (heat equation with other terms)?

Can some one help me solve the following PDE with the given intial and boundary conditions? $\gamma t\frac{\partial^{2}f}{\partial x^{2}}=t\frac{\partial f}{\partial t}-\alpha f$ Initial condition: ...
0
votes
1answer
25 views

Dirichlet Problem

I have to solve the following Dirichlet Problem $$\Delta u=0\quad\text{in}\,\,\, D,$$ $$u(\mathrm{e}^{it})=\frac{1}{2}(\mathrm{e}^{it}+\mathrm{e}^{-it}),$$ for $$u \in C^2(D)\cap C(\overline{D}).$$ ...
0
votes
0answers
47 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
0
votes
0answers
7 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...
0
votes
0answers
8 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
2
votes
0answers
25 views

A particular case of Gelfand triple

I am currently working on hyperbolic equations on bounded domains. For this reason, I am considering in particular functions $u$ such that: $u \in L^2(0;T;H^1_0(\Omega)), \quad u' \in ...
0
votes
0answers
23 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
1
vote
1answer
33 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \Delta u$ ...
1
vote
1answer
27 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
0
votes
0answers
14 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
0
votes
1answer
24 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
0
votes
0answers
14 views

find coefficient-functions s.t. PDE has no global solution

Consider the PDE \begin{align*} a_1(x)u_{x_1} + a_2(x)u_{x_2} & = 0 \quad \mathrm{in}\ \mathbb R^2 \\ u & = u_0 \quad\mathrm{on}\ \Gamma \end{align*} with $\Gamma=\{(x_1,0)|x_1\in\mathbb ...
1
vote
1answer
48 views

How to solve parabolic equation via implicit Euler in 2 dimensions?

I have the following parabolic equation: $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} $$ over domain $(x,y)\in [0,10] \times [0,10]$ ...
0
votes
1answer
29 views

Laplace equation - PDE

For the PDE:$$u_{xx}+u_{yy}=0$$ $$u(0,y)=\sin\pi y, \ u(1,y)=0$$ $$u(x,0)=u(x,1)=0.$$ I have that $u(x,y)=X(x)Y(y)$, then $$-Y(y)=\mu Y(y), \ Y(0)=Y(1)=0$$ $$X''(x)=\mu X(x), \ X(1)=0.$$ Thus, ...
1
vote
1answer
30 views

How to deal with $u_{ttt}$ in derivatives estimates of $u_{tt}$ if $u_{ttt}$ is not defined?

Suppose we have proved the equation $$u_{tt}-u_{xx}=0\quad\text{in}\quad(0,T)\times(0,\ell)$$ with some boundary and initial conditions has an unique solution $u\in C^2(0,T,H^2(0,\ell))$ and we need a ...
0
votes
0answers
14 views

Partial differential equation question (using chain rule?)

How do I do this? Find how the equation $\dfrac{\partial^2 w}{\partial x \partial y} +w=0$ will be transformed if we look for the particular solution w=f(u) where $u=(x-x_0)(y-y_0)$. I got $uf_{uu} ...
1
vote
0answers
12 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
1
vote
1answer
35 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
2
votes
0answers
26 views

How can two PDE's have the same classical solution, but different weak solutions?

For example: The implicit solution for the inviscid Burgers' Equation, in two forms: $u_y+(\frac{1}{2} u^2)_x = 0$ and $(u^2)_y+(\frac{2}{3} u^3)_x = 0$ share the same smooth solutions, but they ...
1
vote
0answers
60 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...
2
votes
1answer
22 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...