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

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391 views

Does Uniform Boundedness in the Sobolev Space $W^{1,2}$ and Convergence in $L^p$ $(1 \leq p < 6)$ Imply Convergence in $L^6$?

Let $B$ denote the open unit ball in $\mathbb{R}^3$. I want to either prove or disprove that a sequence of functions $u_m$ in the Sobolev space $W^{1,2}(B)$ which is uniformly bounded in the ...
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1answer
298 views

Partial differential equation & Lax-Milgram Theory?

Consider the elliptic problem $Lu = \exp(x)$ on $[0,1]$ with $Lu = -\frac{d^2u}{dx^2} + \frac{du}{dx}$ and boundary conditions $u(0) = 5$, $\frac{du}{dx}(1) + u(1) = 2$. Answer the following ...
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2answers
97 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
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177 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: ...
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1answer
126 views

PDE : Mixture of Wave and Heat equations

Today I was given the following equation : $$\frac{1}{c^2}u_{tt} + \frac{1}{D}u_t = u_{xx}$$ with initial conditions : $u(x,0) = 1$ if $|x|<L$ and $0$ otherwise, $u_t(x,0) = 0$. So fairly simple ...
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1answer
136 views

Approximate Holder continuous functions by smooth functions

Let $g \in C^{\alpha} (B_1)$ be given. Can we find a sequence $(f_n) \subset C^{\infty} (B_1)$ such that $f_n \rightarrow g$ in $C^{\alpha}(\overline{B_1})$? If so, how can it be done? I have tried ...
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1answer
156 views

How to prove that the spectrum of the Laplacian over $\Omega\subset \mathbb{R}^n$ is negative?

I am looking for a proof of this well known fact and I guess it has to do with integration by parts (Green's identity). Unfortunately, I only know about 1-d integration by parts( I am just 3rd ...
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2answers
395 views

Product rule of weak derivatives

I am working on proving the following proposition: If $u,v\in {W^1(\Omega)}$ and $uv,uDv+vDu\in L^1_{\operatorname{loc}}(\Omega)$, then we have the product formula $$D(uv)=uDv+vDu.$$ The definition I ...
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1answer
124 views

fundamental solution of the laplace Beltrami operator

It is well know that $s(z,z_0)=\frac{1}{2\pi} \ln \vert z-z_0\vert$ satisfies $\Delta_z s=\delta_{z_0}$ in dimension $2$. Does any one have a reference when we consider a general metric $g$ on an open ...
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1answer
128 views

Self adjointness of an elliptic differential operator

Let $A:D(a)\to L^2(\mathbb{R}^n)$ be an elliptic partial differential operator $$ A(f)=\sum_{i,j=1}^{\infty}\partial_{x_j}(a_{ij}(x)\partial_{x_i}f) $$ where $a_{ij}\in C^{\infty}_b(\mathbb{R}^n)$, ...
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1answer
223 views

How should a numerical solver treat conserved quantities?

Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers? On the one hand, one can observe ...
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1answer
122 views

d'alembert's formula

I'm studying the Cauchy problem for the wave equation $n=2$; $$\begin{cases}u_{tt}=\alpha^{2} u_{xx}, x \in\mathbb{R}, t>0\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} ...
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1answer
297 views

Finding the general solution of a quasilinear PDE

This is a homework that I'm having a bit of trouble with: Find a general solution of: $(x^2+3y^2+3u^2)u_x-2xyuu_y+2xu=0~.$ Of course this should be done using the method of characteristics but I'm ...
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1answer
453 views

How to solve this Initial boundary value PDE problem? [SOLVED]

Today I came across a question on PDE which makes me really frustrating. The question is to solve this initial boundary value problem using method of separation variables: $$u_{tt}=9u_{xx}\text{ for ...
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1answer
104 views

Specific solution to the $1D$ wave equation

So my current solution to the $1D$ wave equation is (with my given boundary and initial conditions): $$y(x,t) = \sum_{n=1}^\infty C_n\cdot \sin\frac{n \pi x}{2 l}\cdot\cos\frac{n \pi c t}{2 l}$$ ...
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1answer
506 views

Green's functions of Stokes flow

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A fundamental solution for a ...
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1answer
420 views

Eigenvalues For the Laplacian Operator

How do I show that the asymptotic speed of the eigenvalues $\lambda$ of the Laplacian Operator is $O(m^{2/n})$ where $m$ is the index of the eigenvalues and $n$ is the dimension of the space?
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1answer
44 views

holonomic D-modules

I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables, ...
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1answer
140 views

Spherical means (Kirchoff's formula) for variable speed wave equation

Suppose $$ \begin{cases}u_{tt} - \Delta u = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\ u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases} $$ then, depending on the dimension $n$, we have a ...
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1answer
178 views

Folland PDE chapter 6.C problem 1

Problem 6.C.1: Suppose $0 \neq \phi \in C^\infty_c(\mathbb{R}^n)$ and $\{ a_j \}$ is a sequence in $\mathbb{R}^n$ with $|a_j| \to \infty$, and let $\phi_j(x) = \phi(x - a_j)$. Show that $\{\phi_j\}$ ...
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1answer
96 views

Lie-brackets and solution space of PDE

I have a linear, first-order homogeneous PDE system with polynomial coefficients $$L_j\, f =0,\text{ for } j=1,..,J\quad \text{ where } L_j \text{ is a first order, diff. operator with polynomial ...
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2answers
225 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
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1answer
103 views

Dirichlet Problem with piecewise smooth boundary

Suppose a domain $ \Omega \subset \mathbb{R^2} $ with $ \partial \Omega $. For $ f \in C^{\infty}(\mathbb{R^2}) $, the dirichlet problem is to find $ u $ with $ \Delta u = 0 $ in $ \Omega $, and $ f = ...
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1answer
129 views

Estimation on elliptic operator

Assume the strongly elliptic property, i.e. $$\sum_{|\alpha|= m}a_\alpha \xi^\alpha\neq0,\ \forall \xi \in \mathbb{R}^d\backslash\{0\},$$ and $$\sum_{|\alpha|\leq m}a_\alpha ...
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1answer
190 views

Conditions for Unique Solution for this PDE

$$ U_{xy}+\frac{2}{x+y}\left(U_{x}-U_{y}\right)=0 $$ with the boundary conditions $$ U(x_{0},y)=k(x_{0}-y)^{3}\\ U(x,y_{0})=k(x-y_{0})^{3} $$ where $k$ is a constant given by ...
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1answer
55 views

Is there a version of mean value property for nonharmonic funcions?

We know by mean value property that harmonic functions satisfies the equalities \begin{equation} u(x) = \dfrac{1}{|B_r|}\int_{B_r}f dx = \dfrac{1}{|\partial B_r|}\int_{ \partial B_r}udS. ...
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1answer
108 views

How does the Hahn-Banach theorem implies the existence of weak solution?

I came across the following question when I read chapter 17 of Hormander's book "Tha Analysis of Linear Partial Differential Operators", and the theorem is Let $a_{jk}(x)$ be Lipschitz continuous in ...
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1answer
123 views

Calculus on surfaces and chain rule

Define the surface gradient operator on any surface $S$ as $$\nabla_S f = \nabla f - \nabla f \cdot \nu_S \nu_S$$ where $\nu_S$ is the outward unit normal on $S$. Let $T:S_1 \to S_2$ be a $C^2$ ...
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1answer
70 views

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
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3answers
240 views

A question related to Wave Equation

Let $L>0$. Suppose $f, g$ are $C^2$ functions on $\mathbb{R}$ such that $$f(t)+f(-t)+\int_{-t}^t g(s)\,ds=0$$ and $$f(L+t)+f(L-t)+\int_{L-t}^{L+t} g(s)\,ds=0$$ for all $t\in \mathbb{R}.$ Does it ...
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2answers
106 views

Solving PDE on finite domain

Given the PDE, $\Delta u=x$ in the region $x^2+y^2<1$. And $\frac{\partial u}{\partial r}=y$ on $x^2+y^2=1$. I am supposed to find all solutions. The only machinery I know for finding solutions on ...
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2answers
265 views

Regularity of an infinite series arising with the heat equation

Let $(t,y)\in(0,\infty)\times\mathbf{R}$, and $\displaystyle f(t,y) \equiv \sum_{k=-\infty}^{\infty}\frac{\exp(-(y-2\pi k)^2/2t)}{\sqrt{2\pi t}}$. This infinite series arises if one attempts to solve ...
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1answer
240 views

What is the counter example?

The Liouville's theorem states that if $u$ is a non negative, subharmonic function, $L^\infty(\mathbf{R}^n)$, then $u$ is constant ($n\leq2$). Someone knows a counter example if $n>2$, or where can ...
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1answer
63 views

An estimate of $C^2$

Let $u$ be a function on a domain $\Omega\subset R^n$, and $D^2u=\left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{n\times n}$ be the Hessian of $u$. If $D^2u$ is positively definite and ...
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2answers
167 views

Does the PDE $\frac{1}{t}\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0$ have a name?

$$\frac{1}{t}\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0$$ Does this PDE have a specific name? Is it a wave equation? Can we transform it into a wave equation?
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1answer
287 views

Heat kernel on a noncompact manifold

I'd like to understand some issues about the heat problem related to the Laplacian of a Riemannian manifold especially when the manifold is noncompact. So first recall the heat equation on a ...
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1answer
79 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
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1answer
74 views

For a system of PDEs, how many equations are needed generally for the system to have unique solution?

For an algebraic system of equations or a system of ordinary differential equations the following rule holds:(right?) the total number of unknown variables must be equal to the number of equations ...
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1answer
94 views

Uniformly quasi-isometric patches over classes of Riemannian manifolds

Suppose $(M^d,g)$ is a closed, connected Riemannian manifold. Is there a constant $R > 0$ such that for all $z \in (M,g)$, for all $x \in B_R(z)$, \begin{equation} \frac{1}{2} \lVert \xi ...
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1answer
366 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
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2answers
164 views

Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable (or differentiable only on a set of measure 0) ? Thanks
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1answer
283 views

Lax Pairs for Linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
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2answers
139 views

$ W_0^{1,p}$ norm bounded by norm of Laplacian

Let $f\in W_0^{1,p}(U)$, for $U$ a bounded domain and $p < n/(n-1)$. I am trying to prove that there is an inequality of the form $$\|f\|_{W^{1,p}} \leq C \int_{\Omega} |\Delta f| $$ where ...
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1answer
203 views

Claims in Pinchover's textbook's proof of existence and uniqueness theorem for first order PDEs

The reference here is Pinchover & Rubinstein's An introduction to partial differential equations, pages 36-37. It's about the existence and uniqueness of a solution to the equation $a(x,y,u)u_x + ...
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1answer
224 views

$L^p$ norms of Fourier transform of solutions of hyperbolic Burger's equation at the time of first blow-up

I am struggling to understand the behavior of the Fourier transform (in the $x$ variable) of initially smooth solutions of the hyperbolic Burger's equation in 1-D, $ \partial_t u + u~ \partial_x u ...
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2answers
294 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...
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2answers
350 views

minimizing the norm of a curl over a domain

According to my computations: The function which minimizes $$\int_\Omega \|\operatorname{curl} f\|^2\,dx$$ should satisfy $$\operatorname{curl}(\operatorname{curl}f) = 0$$ everywhere on $\Omega$, ...
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1answer
369 views

is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?

I asked this on mathoverflow, and was suggested to ask it here. Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have ...
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3answers
94 views

Laplace's equation on a square domain with a central point reservoire

Could someone please tell me the solution to this problem. I have $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ on the square domain $-L<x<L, -L<y<L$ with ...
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0answers
600 views

How prove this hypersingular operator $(T\psi)((z(t))$

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...