# Tagged Questions

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

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### Fundamental solution of Laplacian in 2d doesn't agree with 3d version physically?

The fundamental solution of the Laplacian, with $(x, y) \in \mathbb{R}^3$, is $$\Phi(x, y) = -\frac{1}{4\pi|x - y|}$$ So, say, the further away $x$ is from some point source $y$ the lesser the ...
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### Constructing a PDE given solution with arbitrary differential function?

I have the following multivariable function: $u = x + y + F(xy)$ where F is an arbitrary differentiable function. I would like to construct a first order PDE with u(x,y) as the following solution. ...
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### Complicated exponential function inverse laplace transform

The problem is from wave equations for describing flexural vibration of Euler-bernoulli beam. The equations are listed in the following pics. I. equations and 16 elements to be analyzed II.cF:wave ...
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### Poincare's inequality with difference quotient

For the classical Poincare's inequality, if $u \in H^1_0(\Omega)$, then $$\int_\Omega u^2 \,dx \le C \int_\Omega |\nabla u|^2 \,dx.$$ Do we have something similar with the difference quotient? That ...
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### Product of two $W^{1,p}_0$ functions

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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### Bounded solution to the heat equation on a strange domain.

Find a solution to the heat equation $u_{xx} = u_t$ that is bounded for $0\leq x < \infty$, $-\infty < t < \infty$. Would $u(x,t) = f(x)$ since if there was any time dependence the solution ...
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### What does it mean for a boundary to be analytic in the context of a PDE?

I am reading a paper where they assume the boundary of a domain is "Analytic". They never define it. Is this a standard definition, and, if so, what is it?
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### Can we show that each element of the Sobolev space $H^k(D)$, with $D\subseteq\mathbb R^d$ being a bounded domain, has a continuous representative?

Let $d\in\left\{2,3\right\}$ $D\subseteq\mathbb R^d$ be a bounded domain $\lambda$ be the Lebesgue measure on $D$ $H^k(D)$ be the Sobolev space Can we show that each element of $H^k(D)$ has a ...
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### Test space for weak formulation

From an applied functional analysis course where the weak formulation was covered, I think I have a high-level understanding of the concepts that are involved. To derive a weak formulation for a PDE, ...
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### well posedness of 1D linear Schrodinger

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm ...
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I want to use the method of manufacturing solution MMS to find the exact solution of the following linear system of two equations $$\begin{cases} u_t -(u_{xx} + u_{yy}) = u + v\\ v_t -(v_{xx} + ... 2answers 41 views ### Construction of Sobolev space I am reading about the construction of Sobolev spaces from L^2. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ... 1answer 20 views ### Regularity of solution to an elliptic PDE Let R be the shifted open unit cube $$\Big(-\frac{1}{2}, \frac{1}{2} \Big)^3 \subset \mathbb{R}^3,$$ and let k \in \mathbb{C} be a constant with \textrm{Re}(k) ... 0answers 10 views ### Proof of A\ne \mathrm{conv}~ \mathrm{relbd} A\Rightarrow A is a flat or a half-space Picture below is from the 16 page of Schneider R. "Convex Bodies--The Brunn-Minkowski Theory". I am fuzzy with the red line, what is the translate of H^- ? And what is the effects of convexity ... 1answer 63 views ### Why is a differential equation a submanifold of a jet bundle? I'm reading The geometry of jet bundles by D.J Saunders and struggle with the definition of a differential equation 6.2.23. on page 203. First of all, Saunders introduces a differential operator ... 1answer 24 views ### Solve the PDE using characteristics method Consider the PDE$$3 \frac{\partial{u}}{\partial{t}} + t^2\frac{\partial{u}}{\partial{x}} = 0: t \lt 0 $$with the initial condition$$u(x,0) = f(x): 0 \lt x \lt 1$$Determine the characteristics ... 0answers 13 views ### Sobolev space membership of logarithmic function Determine the largest s\in(0,1) for which the following integral converges$$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy
I am considering the following equation $\partial_t f+v\cdot\partial_x f+\partial_x\psi\cdot\partial_xf =0$ with initial data: $f(0,x,v)=f_0(x,v)$. There are some analysis like the uniqueness and ...