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

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Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
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31 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
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45 views

Newton boundary condition for second order pde

I have a few questions about Newton boundary conditions for a second-order partial differential equation: $$-\text{div}(a(x,u,\nabla u)) + c(x,u,\nabla u)$$ considered on a bounded connected ...
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20 views

non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
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30 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
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37 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
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21 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
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43 views

differential equation looks like Bessel but isn't

I have this question What I did is: $U=X(x)*T(t)$ after putting it back into the function I got $-x^2*T''/T= x^2X''-2xX'+2X $ after deviding by $x^2$ remembering to check $x=0$ I get $-T''/T= ...
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49 views

Solving $8v_{\xi\eta} - v_\xi - v_\eta = 0$

While solving a Goursat problem I stumbled upon this PDE. Can't solve it: $$8v_{\xi\eta} - v_\xi - v_\eta = 0$$ This is what I tried: $v \rightarrow V e^{\alpha x + \beta x}$ $v_\xi = V_\xi ...
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51 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
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52 views

Solve the third order PDE

Is it possible to solve this differential equation analytically?
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33 views

Prove that $L[X] = \frac{(pX')' - qX}{r}$ is a formally self-adjoint operator for continuous $p, q,r$ functions.

$$\Large\textbf{Given Problem}$$ Let $p,q,r$ be continuous functions on $[0,L]$ such that $p'$ is also continuous, and $p$ and $r$ are positive. Define $$L[X] = \dfrac{(pX')' - qX}{r}$$ Show, using ...
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104 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
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68 views

aubin-lion lemma

Let $\Omega \subset \mathbb{R}$ and we have $$u_n \rightarrow u \mbox{ in } L^{\infty}(0,T;H^2(\Omega)) \mbox{ weak star }$$ and $$\frac{\partial u_n}{\partial t} \rightarrow \frac{\partial ...
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40 views

Prove hyperbolic operator L is simply the wave operator in a coordinate system moving with velocity $-B/2A$

This exercise comes from A First Course in Partial Differential Equations - H.F. Weinberger 8.2) Show that if the operator $L[u] = A(\frac{d^2u}{dt^2})+B(\frac{d^2u}{dxdt})+C(\frac{d^2u}{dx^2})$ is ...
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29 views

The space of bounded mean oscillation $BMO(B_R)$, live in the Campanato space $\mathcal{L}^{1, n}(B_R)$

Let $B_R$ be an open bounded ball in $\mathbb{R}^n$. I am trying to show that if $u\in BMO(B_R)$ then $u\in \mathcal{L}^{1, n}(B_R)$ and that \begin{equation} \|u\|_{\mathcal{L}^{1, n}(B_R)}\leq ...
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264 views

Find all of the equilibrium points and describe the behavior of the $x' = sin(x), y' = cos(y) $.

Find all of the equilibrium points and describe the behavior of the $$x' = \sin(x), \quad y' = \cos(y) .$$ It has been a while since I took DE...Do we first need to set $x' = y'$ to solve for their ...
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34 views

On the relation between PDEs and Distributions on Manifolds

Given a distribution $\Delta$ of dimension $n$ (continuous or smooth) in a $n+m$ dimensional manifold, can one always find a basis $\{X_j\}$ such that in local coordinates $(x^1,...x^m,y^1,...y^n)$: ...
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66 views

What's this standard duality argument?

I'm reading a proof of the Strichartz inequalities. It shows that $$ \| \int_\mathbb{R} e^{-is\Delta}F(s) \, ds \|_{L^2_x} \lesssim \|F\|_{L^{q'}_t L^{r'}_x}, $$ and then says that by duality, $$ ...
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26 views

Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
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51 views

Hopf Bifurcation of Reaction-Diffusion System

I'm considering the following reaction-diffusion system: $ \frac{\partial u}{\partial t} = f(u,v)+ D_1 \frac{d^2 u}{dx^2} $ $ \frac{\partial v}{\partial t} = g(u,v)+ D_2 \frac{d^2 v}{dx^2} $ where ...
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70 views

Change of variables for linear differential operators

I am trying to find an expression for a change of variables (invertible and $C^{\infty}$) of a linear differential operator in $\mathbb R ^d \newcommand{\t}{\tilde} \newcommand{\p}{\partial}$. i) ...
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74 views

Two Problems about second-order elliptic equtions

First, Assume U is connected. A function $u\in H^{1}(U)$ is a weak solution of Neumann's problem(*) $$ \left\{\begin{matrix} -\Delta u=f \: \:in\: U\\ \frac{\partial u}{\partial \nu }=0\: \: ...
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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 ...
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66 views

Solve the following PDE with fourier transform

Solve the following PDE $$u_t=u_xx+δ(x)δ(t)$$ BC s: $$\lim_{|x|\to \infty}u(x,t)=0$$ IC s: $$u(x,0)= δ(x).$$ solve pde with fourier tranform
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Eigenvalue problem from the PDE point of view

I study the eigenvalue problem for the Laplacian $-\Delta$ on a bounded domain $\Omega\subset \mathbb{R}^n$ with Dirichlet boundary conditions, but unfortunately my knowledgue about solving a boundary ...
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27 views

$L^2$ method to solve the PDE $\bar{\partial}u=f$, where $f\in L^{ 2 }_{ (0,p) }(\Omega )$

Suppose that $\Omega$ is a pseudoconvex domain, for $f\in L^{ 2 }_{ (0,p) }(\Omega )$,and ${ \bar { \partial } f }=0$. Use $L^2$ method to show that there exists solution $u\in L^{ 2 }_{ (0,p-1) ...
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58 views

Complex Power of a differential operator

Let $(X,\|\cdot\|)$ be a Banach space and consider a sequence $B_n \colon X \to X$ of bounded operators. I remember from my course in operator theory that the partial sum $$ S_N = \sum^N_{n = 1} B_n ...
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Is numerical stability preserved under basis transformation for pde

I'm using a Backward Time Central Space method to solve the heat equation in polar coordinates. In Cartesian coordinates, it is easy to show this is unconditionally stable (by assuming solution of the ...
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67 views

transformation of variables in a fokker planck PDE

I am trying to solve the following Fokker Planck PDE: $$ \dfrac{\partial u(t,x)}{\partial t} = -\dfrac{\partial u(t,x)}{\partial x} + \dfrac{1}{2}\dfrac{\partial^2 }{\partial x^2}[ 3 x^2 u(t,x)]. $$ ...
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Parabolic PDEs Maximum Principle

Consider diffusion equation for t>0 and $ \boldsymbol{x} $ in a bounded doman $ D$ in $\mathbb{R}^n$, and a given scalar field $a(\boldsymbol{x}) >0$ that is uniformly bounded and continuously ...
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30 views

Change of variables in PDE's

How the change of variables in PDE effects the initial and boundary condition? will the condition be accordingly changed with the substitution use for change of variables? any example will be ...
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25 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
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33 views

The help for Max Principle for the heat equation

Give a direct proof that if U is bounded and $ u \in C_{1}^2(U_{T}) \cap C(\overline U_{T})$ solves heat equation, then $\max_{\overline U_{T}} u$= $\max_{\tau_{T}}u$ I appreciate it Thanks
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112 views

Show that convolution satisfies partial differential equation

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
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60 views

Need help on computing odd, even extensions of a function

OK I am going over d'Alembert solutions. And I came across the following example. $$ f(x) = \begin{cases} \frac{3}{10}x &0 \le x \le \frac{1}{3} \\ \frac{3(x-1)}{20} & \frac{1}{3} \le x \le ...
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127 views

Poisson Equation in a Rectangle

The problem is to solve $$\Delta\phi=\frac \lambda {\varepsilon_0}\delta(x-x',y-y')\quad;\quad \phi(0,y)=\phi(a,y)=\phi(x,0)=\phi(x,b)=0.$$ My idea was to try and represent the RHS as a series of ...
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Completeness of separable solutions to PDEs

Under what conditions will the solutions of a PDE obtained using separation of variables form a complete set for the solution space?
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51 views

Analytical solution nonlinear partial differential equation

Do you know how I can solve nonlinear PDEs analytically i.e does the perturbation method work? e.g $$a^2u_{tt} - u_{xx}+ f(u)=0$$ where $f$ is nonlinear in $u$, with boundary condition. what the ...
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Find a maximum principle for elliptic PDE of degree 2 in divergence form

In our reading we had the following maximum principle for elliptic PDE of degree 2: Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of the linear Dirichlet task $$ ...
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110 views

Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
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25 views

Time continuity of a solution to the generalized porous medium equation

Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain. Now assume there is a weak ...
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64 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
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reference request: a graduate level textbook on viscosity solutions of IPDEs

I am particularly interested in IPDE which arises from optimal stopping/control problems so I would like some references on integral-partial differential equations, which are related this area. I have ...
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118 views

Temperature Distribution for a Finite Rod

The question: The temperatures at ends x = 0 and x= L of a rod length L with insulating sides are held at temperatures $T_1$ and $T_2$ until steady-state ...
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44 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
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81 views

Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
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29 views

subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...
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66 views

How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?

Let $U\subset\mathbb{R}^n$ be a open set and $f:U\to\mathbb{R}$ a function in $C^\infty_c(U)$. Evans PDE book uses the following result $$\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial ...
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94 views

Don't understand proof of global solution to parabolic PDE

Let $u \in L^p(0,T;V)$ denote the solution to the parabolic PDE $$u_t + \Delta u = f\qquad\text{a.e $t \in [0,T]$}$$ where $u_t \in L^q(0,T;V^*).$ We have the usual assumptions on $V \subset H \subset ...