# Tagged Questions

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

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### Compute the continuous spectrum of an unbounded operator in $L^2(\mathbb{R}^2)^2$.

In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily ...
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### Spectrum of laplacian in a parallelogram

Is the spectrum of the laplacian on an arbitrary parallelogram with dirichlet boundary conditions known?
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### What is the difference between the domain of influence and the domain of dependence?

When analysing the wave equation $$u_{tt} = c^2 u_{xx}$$ in my PDE's module, I understand the 'domain of dependence' which is where the value $u(x_0,t_0)$ is only depends on the initial value of $x$ (...
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### Complete integral of pde without independent variables

Show that the complete integral of pde $F(u,p,q)=0$ ($p=u_{x}$ and $q=u_{y}$) is $$f(x,y,u,a,b) = x + ay + b - \int\frac{du}{g(u,a)},$$ where the function $p=g(u,a)$ is computed from the ...
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Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation: $$\nabla^T f = - (\nabla^T \phi ) f$$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the ...
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### Exact solution of 2d poisson equation on an annulus

I have to find the exact solution of the following: $$-\nabla^{2}{u}=1$$ on the two dimensional domain $$\Omega=\{z=(x,y):|z|\ge{1} \wedge |z|\le{R}\}$$ On the boundaries of $\Omega$ we impose ...
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### Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
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### Example of an oscillation Young measure

I'm taking a course in which Young measures are introduced for oscillation and concentration. I have understood the examples the lecturer has given us for concentration Young measures, but cannot get ...
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### Is there a version of Friedrich's lemma for Bochner spaces?

Let $u\in H^{1}([0,T], H^{1}_{0})$ that satisfies $u_{tt}+L u=f\in L^{2}(0,T; L^{2})$. where $L$ is a uniformly elliptic operator in divergence form given by $$L=(a^{ij}(t,x)u_{x^{i}})_{x^{j}}$$ ...
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### Solving the wave equation with Neumann conditions(?) using Green's functions.

I either have to solve the wave equation using Green's functions or Fourier transforms. I am given an infinite string which is fixed at $x_0=0$ and at rest for $t<0$. At $t=0$, it receives an ...
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### What is a “standard symbol” in analysis?

I was reading some Wikipedia pages about analysis when I came across this strange "standard symbol" terminology in the "Fourier integral operator" page. It seems to be a function (or a distribution) ...
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### Energy method for one dimensional wave equation with Robin boundary condition

Show that the initial-boundary value problem \begin{align} & {{u}_{tt}}={{u}_{xx}}\text{ }(x,t)\in \left( 0,l \right)\times \left( 0,T \right),\text{ }T,l>0 \\ & u\left( x,0 \...
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### Using characteristics and picard iteration to solve nonlinear wave equation

My question: I want to solve $u_{tt}-u_{xx}+\lambda u=0$ in $0 \leq |t|\leq x \leq 1$ with the conditions that $u(x,x)=u(x,-x)=1$. My attempt: I let $\xi=x+t$ and $\nu=x-t$. This transformed the ...
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### The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is $$u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r}$$ where $F$ and $G$ are arbitrary functions. I want to ...
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### Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...
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### looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
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### Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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### The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E$, where $\mathcal E$ denotes the ...
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### How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $u>0$

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in C^{\infty}(\Omega)$ then how can we prove there is a weak solution \$u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \cap L^{\infty}(\...