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

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440 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 anywhere (or differentiable only on a set of measure 0) ? Thanks
7
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1answer
205 views

Is $C_c^{\infty}$ dense in $X_0^{\alpha}$?

While reading papers on fractional Laplacian, I always meet space $X_0^{\alpha}(\mathcal{C}_{\Omega})$ which is defined as following: $$X_0^{\alpha}(\mathcal{C}_{\Omega})=\{z\in ...
7
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1answer
326 views

differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
7
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2answers
462 views

Separation of variables for partial differential equations

What class of Partial Differential Equations can be solved using the method of separation of variables?
7
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1answer
107 views

$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}$

Having difficulty in solving the following partial differential equation: $$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}.$$ Will it be easier if we ...
7
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1answer
158 views

Solution of $\frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0$

I've been trying to to solve the following PDE: \begin{equation} \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0 \end{equation} I ...
7
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2answers
109 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
7
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2answers
178 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
7
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2answers
134 views

Very simple partial differential equation

I am solving $$ \frac {\partial f}{\partial x} = \frac y{x^2 + y^2} \\ \frac {\partial f}{\partial y} = \frac {-x}{x^2 +y^2} $$ As $y$ was held constant when the partial derivative with respect to ...
7
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2answers
274 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
7
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2answers
177 views

When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in ...
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2answers
127 views

Can the orbits of a semigroup touch without one being included in the other?

Question. Let $(S(t))_{t \ge 0}$ be a continuous semigroup of linear operators on some Banach space $X$. Might there exist $f, g\in X$ and $0<t_0<t_1$ such that ...
7
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1answer
748 views

Wave Equation, Energy methods.

I am reading the book of Evans, Partial differential Equations ... wave equation section 2.4; subsection 2.4.3: Energy methods. Arriving at the theorem: Theorem 5 (Uniqueness for wave equation). ...
7
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1answer
314 views

Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space ...
7
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1answer
162 views

What is known about the equation $u_{xy}+u_{yx}=0$?

I was thinking a bit about PDEs and realized that I haven't seen any PDEs whose solutions possess non-equal mixed partial derivatives or where this possibility is at least taken seriously. So, I was ...
7
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1answer
618 views

Complex and real forms of the Poisson integral formula

In my complex analysis book there is the expression $$\frac{1 - |z|^2}{|1 - \bar z e^{it}|^2}$$ and it says that when $z = re^{it}$, we can write the above expression as $$P_r(t) = \frac{1 - r^2}{1 - ...
7
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1answer
683 views

Heat equation in bounded domain

Let $\Omega \subseteq \mathbb{R}^n$, $n\geq 2$, be a bounded domain with boundary $\partial \Omega \subseteq C^2,v$ outer unit normal vector on $\partial \Omega$, $h \in L^2(\Omega)$. Let $u \in ...
7
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1answer
92 views

Is it true that $\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over 2}$?

Is it true that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over 2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$?
7
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2answers
298 views

Prerequisits for Gauss-Green theorem

Consider the following theorem from the appendix C from Evans PDE book: I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my ...
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2answers
242 views

Solutions of the heat equation of the form $u(x,t)=v(x/\sqrt{t})$

Assume $n=1$ and $u(x,t)=v(\frac{x}{\sqrt{t}})$. (a) Show that $u_t = u_{xx}$ if and only if $$v''+\frac z2 v' = 0. \tag{$*$}$$ Show that the general solution of $(*)$ is $$v(z)=c \int_0^z ...
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1answer
169 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about ...
7
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1answer
434 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
7
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1answer
301 views

Help me understand this proof (showing that something is a norm).

I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237. I need help to understand the ...
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2answers
300 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 ...
7
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1answer
233 views

Rewriting an integral

This is concerning Poisson's equation with oblique boundary condition (Gilbarg Trudinger p121) We let $\Gamma(|x-y|)$ denote the fundamental solution to Laplace's equation. Also, let $x-y^{*} = ...
7
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2answers
459 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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0answers
184 views

The study of the properties of the solution of Heat equation.

I am studying the following basic heat equation. (All notations follows Evans book) \begin{align} u_t -\Delta u =0 &\text{ in }\mathbb R^2\times(0,\infty)\\ u=u_0 &\text{ on }\mathbb ...
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1answer
155 views

Density given by variable-coefficient PDE

I am looking for a time-dependent probability density $f(x,y,t)$ solving the equation $$-\frac{\partial f}{\partial t} = \alpha\cdot \big(y - F(x)\big)\frac{\partial f}{\partial x}+\beta\cdot ...
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0answers
215 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
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0answers
76 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f\left(u\right) = 0, \\ u\big\vert_\Gamma = u_0 $$ by Newton’s method when its convergence is global ...
7
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0answers
101 views

Existence of classical solution for equation $\Delta u + v(x) u = 0$

It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, ...
7
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0answers
111 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
7
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0answers
248 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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0answers
322 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
7
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1answer
203 views

How can I find the limiting value of a time-dependent PDE?

I've managed to reduce a question in probability to the following simple looking PDE: $$ u_t = -t u_x + \frac{1}{2} u_{xx}, {\rm ~for~} x>0, \, t \in \mathbb{R} \;, $$ with a limiting initial ...
6
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3answers
3k views

Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
6
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2answers
2k views

What is the intuition behind a function being 'weakly differentiable'?

As part of an optimization paper I am reading now, they are talking about a function $g$ being "weakly differentiable". I looked it up on the wiki but I do not have enough context to start cracking ...
6
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2answers
336 views

Difference between $u_t + \Delta u = f$ and $u_t - \Delta u = f$?

What is the difference between these 2 equations? Instead of $\Delta$ change it to some general elliptic operator. Do they have the same results? Which one is used for which?
6
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4answers
193 views

$2^{nd}$ order PDE: Solution

I am trying to solve the following equation: $$\frac{\partial F}{\partial t} = \alpha^2 \, \frac{\partial^2 F}{\partial x^2}-h \, F$$ subject to these conditions: $$F(x,0) = 0, \hspace{5mm} F(0,t) = ...
6
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3answers
1k views

Elliptic Regularity for solutions in distributional sense

I know that there are a lot of (great) books treating regularity of weak solutions of elliptic pdes (such as Gilbarg-Trudinger), but what about regularity of very weak solutions, that is, solutions in ...
6
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1answer
103 views

Finding a general solution for $u_{xx}-4u_{xy}+3u_{yy}=0$

Let $$u_{xx}-4u_{xy}+3u_{yy}=0.$$ Find the general solution given the solution $u(x,y)=f(\lambda x+y).$ My attempt was as follows: let $u(x,y)=e^{\lambda x+y}$. Then by computing $u_{xx},u_{xy}, ...
6
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2answers
2k views

What does the heat kernel in the heat equation represent $u(x,t)$?

Okay so I am studying for my PDE course and I am convering Fourier transforms. In fact I am using fourier transforms to find a solution to the heat equation on an infinite length rod. After going ...
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2answers
785 views

What is elliptic bootstrapping?

While reading about elliptic differential operators, I have seen the phrase elliptic bootstrapping in several places, but none of them explain exactly what it means. I know it has something to do with ...
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2answers
350 views

Problem about Ricci Flow

On page 12 of "Lectures On Ricci Flow" by Peter Topping is written: In two dimensions, we know that the Ricci curvature can be written in terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working ...
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2answers
127 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
6
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2answers
213 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
6
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2answers
204 views

Decomposition of functionals on sobolev spaces

It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on ...
6
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3answers
148 views

How to prove a PDE preserves mass?

My general question is: suppose you are given a PDE (possibly with boundary conditions). What does it mean to say the PDE "conserves mass"? Specifically, if you are given the PDE $$- \nabla \cdot ...
6
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2answers
2k views

Find the Green's Function and solution of a heat equation on the half line

Consider the heat equation on the half line $$u_t = ku_{xx},\quad x > 0,\, t > 0,\\ u(x,0) = 0, \,x \in\mathbf R,\\ u(0,t) = \alpha(t),\, t > 0. $$ Find Green's function and the solution. ...
6
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2answers
584 views

Intuition and applications for the p-Laplacian

Consider the p-Laplacian of a suitably nice function $u$: $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$ Are there useful ways of thinking about the p-Laplace operator, or of thinking about ...