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

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1answer
100 views

Rall cable theory equation

I'm currently studying Rall cable theory which is a concept from neuroscience and I've come across this differential equation that governs change of axial current depending of time and distance: ...
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0answers
62 views

Do PDEs arise in the context of Varieties?

My work involves designing numerical methods to solve PDEs on manifolds. This situation often arises in many applications in biology, chemistry, physics etc. I've recently been reading about ...
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1answer
66 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
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1answer
98 views

Sobolev spaces inclusion

I'm having trouble finding an answer related to Sobolev spaces that does not relate to duality. I'm looking for an answer to the following question: When (i.e. for what domains $\Omega$ or such) can ...
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1answer
84 views

Transforming partial differential equations

$13.$ Consider the change of variables $$x = e^{−s} \sin t,\space y = e^{−s} \cos t, \space \text{such that} \space u(x,y) = v(s,t)$$ (i) Use the chain rule to express $∂v/∂s$ and $∂v/∂t$ in terms ...
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2answers
82 views

Does the following have a solution for f(x,y)?

I have the following equations: \begin{equation} {1\over f(x,y)} {\partial f(x,y) \over \partial x} \alpha(x,y) + {1\over f(x,y)} {\partial f(x,y) \over \partial y} \beta(x,y) = \gamma(x,y) ...
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0answers
31 views

About Cauchy Kovalevskaia Theorem

I'm reading an article that says we can garantee a solution u=u(x,t) with $(x,t)$ in a neighboor of origin of $(0,0) \in \mathbb R^{m+n}$ for the problem $$ L_j u = 0, j=1, \ldots n, $$ where $$L_j = ...
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2answers
282 views

General solution of $(u.u_{xy}) - (u_x.u_y) = 0$

$(u.u_{xy}) - (u_x.u_y) = 0$ I'm in bad need to answer this question. Please help. I considered $D= (d/dx)$ & $D'= (d/dy)$ so I came to $(U^2)DD'-(U^2)DD'=0$ but its already obvious that these ...
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1answer
77 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
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1answer
138 views

Fundamental solution of nonlinear PDE

A fundamental solution of a linear PDE (in sense of Schwartz), $Lu=0$ is defined as a distribution $E$ such that $LE=\delta$. Now I wish to find fundamental solution of nonlinear PDE, such as the ...
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1answer
46 views

a question about generalized derivatives in Sobolev space

Let $\delta>-\frac{1}{2}$, $s\in (0,\delta+\frac{1}{2})$, $X_+(x)=x , x>0$ or $X_+(x)=0, x\le 0$. How to prove that$(1-\lambda^2)_+^{\delta}\in W^{s,2}(R)$? Where $W^{s,2}(R)$ is a Sobolev ...
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1answer
35 views

Trace inequality for a subset

Let $\omega$ be a proper subset of an open ball $B$, and $\omega$ has a Lipschitz boundary. Is there a some kind of trace inequality which says that $\|u\|_{L^2(\partial B)} \le C ...
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1answer
58 views

Green's function of an operator

Given a differential operator $L$, how is its Green's function defined? I know that for a an initial condition problem it is the function so that the solution is defined by $u = G*f$, but I couldn't ...
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1answer
57 views

Solving $\dfrac{\partial u}{\partial x}+\dfrac{x}{y}\dfrac{\partial u}{\partial y}=0$

I have solved $\dfrac{\partial u}{\partial x}+\dfrac{x}{y}\dfrac{\partial u}{\partial y}=0$ using characteristics, to obtain $u(x,y)=C$ (for $y=\pm x$) for $C=0$ and $u(x,y)=f(x^2-y^2)$ for $C$ ...
2
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1answer
37 views

Property of a solution of a PDE without dependence in the $x$ variable

Assume $u_0\in L^1\cap L^\infty (\mathbb{R}^d)$ and consider the equation without diffusion $$\frac{\partial u}{\partial t}=-u^p,\;\;\;t\geq 0, x\in \mathbb{R}^d.$$ Show that ...
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1answer
53 views

Don't understand this proof of nonhomogenous Poisson problem (Sobolev space trace and inequality)

See this proof of existence of solution to nonhomogenous Dirichlet problem (from http://www.ann.jussieu.fr/~frey/cours/UdC/ma691/ma691_ch4.pdf page 7): How is the inequality marked with the red ...
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1answer
145 views

$C^1$ domains vs. Lipschitz domains in PDEs, do we need transformation of coordinates?

I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within ...
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1answer
36 views

Show the example belong to the Bessel potentials space (fractional order sobolev space), where $p=2$

If $\delta>-\frac12$, show that $(1-x^2)_+^\delta\in W^{s,2}$, where $s\in (0,\delta+\frac12)$. Thanks in advanced.
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1answer
98 views

Sobolev trace operator bounded from below??

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary $\partial\Omega.$ Is the trace operator $$T:H^1(\Omega) \to H^{\frac 1 2}(\partial\Omega)$$ bounded from below: $$|Tu|_{H^{\frac 1 2}} ...
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1answer
52 views

supersolution of Laplace operator

The problem is that I want to find a solution $u$ s.t. $\Delta u>0, u=0$ on $\partial\Omega$,where $\Omega$ is the domain $[0,1]\times[0,1]$
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0answers
51 views

General integral of an PDE

Consider the PDE $$ \frac{\partial u}{\partial t}+y\frac{\partial u}{\partial x}-a^2x\frac{\partial u}{\partial y}=0 $$ To find the general integral by the method of characteristics, I construct the ...
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1answer
106 views

On pointwise bounded subsequences of a convergent sequence in $L^p$

When trying to rigorously formulate a proof presented to me during a PDE seminar I came across the following difficulty: Let $(u_n) \subset H_0^1 (\Omega)$ be a bounded sequence such that $u_n \to u$ ...
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0answers
48 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
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0answers
120 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
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4answers
172 views

Solving $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=c$

Question: let $a,b,c$ be positive constants. Find $u=u(x,y)$ if is satisfies the partial differential equation $$\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=c$$ and the ...
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1answer
181 views

Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
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1answer
222 views

Contour Integration (Choice of Contour)

Let $ \alpha \le 0 $ and $\sigma > 0$ . I want to choose a contour, including $ [\sigma - iR, \sigma+iR] $ , such that i can apply Cauchy's Residue theorem and evaluate: $$ \lim_{R \rightarrow ...
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1answer
686 views

Inhomogenous Heat equation using fourier transform

Is it possible to transform the inhomogenous heat equation: $ u_t = u_{xx} + h(x,t)$ for $ - \infty < x< \infty , t > 0$ and $u(x,0) = 0$ to the integral equation: $$\int_0^t ...
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1answer
57 views

Show that $k \bigtriangleup T+ \nabla k \cdot \nabla T=0$ leads to $\bigtriangleup (k^{1/2} T)- \frac{\bigtriangleup(k^{1/2})}{k^{1/2}}k^{1/2}T=0$

I am trying to see the following derivation given in the book 'Kernel Functions and Elliptic Differential Equation in Mathematical Physics' by S. Bergman and M. Schiffer. We deduced the differential ...
2
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1answer
34 views

If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $|f(x,t)| \leq C$ for all $x$ and $t$?

Here $\Gamma$ is a bounded closed $C^k$ hypersurface. If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $f$ uniformly bounded on $[0,T]\times \Gamma$? Or even does it hold that $|f(t)| \leq C_t$ for ...
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2answers
71 views

Problem with solving PDE

I'm trying to solve this equation: $u_{tt} = u_{x_1x_1} + u_{x_2x_2} + u_{x_3x_3}$ $u(x,0) = x_1^2\sin(x_2+x_3)$ $u_t(x,0) = 0$ In what form to find a solution? I tried in form $u = ...
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1answer
42 views

Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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1answer
61 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
4
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1answer
169 views

Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
2
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1answer
109 views

For which real values of $\alpha$ PDE $\Delta u(x,y)+2u(x,y)=x-\alpha$ has at least one weak solution?

Problem. Consider boundary value problem: \begin{cases} \Delta u(x,y)+2u(x,y)=x-\alpha, & \text{in $\Omega$,} \\ u(x,y)=0, & \text{on $\partial\Omega$,} \\ \end{cases} where $\alpha$ is ...
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1answer
48 views

Lipschitz domain and surface measure

Let $S$ be the boundary of a Lipschitz domain $\Omega$. We know it has a surface measure $\mu$. Can we write $d\mu = f(x)dx$ with $f$ explicity given in terms of the Lipschitz maps that make up the ...
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1answer
75 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
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1answer
55 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
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1answer
311 views

How to find the infinitesimal generator of this semigroup?

Definition 1: Let $X$ be a Banach space. A semigroup is a family $\{T(t)\}_{t\geq 0}$ of continuous linear operators $T(t):X\to X$ such that $(i)\;\;T(0)=I$, where $I$ is the identity operator; ...
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2answers
50 views

Explicit solutions to $-\triangle u = k f(u)$

Does anyone know any examples of $f$'s for which $-\triangle u(x) = k f(u(x))$ has an explicit solution (i.e. a formula for the solution, not a numerical approximation scheme) in terms of $k$? I am ...
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1answer
55 views

Is there something useful for third boundary condition on Poisson equation

There is a following task for a friend of mine to make: $$-\Delta u=f,x\in(0,X),y\in(0,Y)\\-u_x+\alpha(y)u\Bigr|_{x=0}=g(y)\\u\Bigr|_{x=X}=c(y)\\u\Bigr|_{y=0}=a(x)\\u\Bigr|_{y=Y}=b(x)$$ She needs ...
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2answers
86 views

Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...
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0answers
47 views

Properties of the first eigenvalue of the $p$-Laplace operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $p\in (1,\infty)$, $p\neq 2$. Consider the usual Sobolev space $W_0^{1,p}(\Omega)$ and its dual $W^{-1,p'}(\Omega)$, where $1/p+1/p'=1$. Define ...
2
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1answer
64 views

Partial differentials vs normal differential (notation question/clarification only)

In physics, it seems like the use of $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ are used somewhat interchangeably. My understanding is that, technically $\dfrac{dy}{dx}$ is only ...
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1answer
228 views

property of local Sobolev space

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if ...
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1answer
104 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
5
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2answers
223 views

Analytical solution to PDE

I am trying to solve the following linear pde where $u=f(x,y)$ in the domain $y \in (0,\infty)$: $$y\dfrac{\partial{u}}{\partial x} = \dfrac{\partial^2 u}{\partial y^2}$$ with boundary conditions: ...
2
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1answer
66 views

Maximum Principle for Heat Equation Proof (?)

Is the following a correct proof? It is easier than a proof I have been provided with, but I feel like it is wrong. Prop: If $ u$ satisfies $ u_{t} = \sum_{i=1}^{n} u_{x_{i} x_{i} } $ on $\ D ...
3
votes
2answers
204 views

Nodes of eigenfunctions and Courant's nodal domain theorem

I am looking for a reference for properties of eigenfunctions of the Laplacian (on the Euclidean plane, and maybe also Laplace-Beltrami on a general manifold): The discreteness of the set of ...
0
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0answers
50 views

How to solve this system of equations analytically?

$$ \begin{cases} \ddot{r}-r\dot{\theta}^2-r\left[\sin\left(\theta\right)\right]^2\dot{\varphi}^2 &=&0 ...