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

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Surjectivity of the trace operator in Sobolev spaces

Suppose $U$ is an open bounded set with $C^1$ boundary. It is a well-known result in the theory of Sobolev spaces $W^{1,p}$ that there is a continuous linear operator $T:W^{1,p}(U)\rightarrow L^p(\...
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234 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 ...
7
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2answers
175 views

What is the idea behind Green's function? What does it do?

I have an exam on ordinary and partial differential equations in a couple of days and there is one concept that I am really struggling with: Green's function. I have basically read every PDF-file on ...
7
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1answer
343 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ \|Bx\|...
7
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1answer
250 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 ...
7
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362 views

How to argue this consequence?

Suppose that $\Omega=\mathbf{R}^n_+$ and consider a function $0<u<\sup\limits_\Omega u=M<\infty$ such that: $$\Delta u+u-1=0 \ \ \text{in} \ \ \Omega,$$ $$u=0 \ \ \text{on} \ \ \partial\Omega....
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325 views

Function in $H^1(\Omega)$ which cannot be extended to a greater Sobolev Space

The problem is like this: Consider the open set $\Omega \in \Bbb{R}^2$ by $\Omega=\{(x,y) : 0<x<1, 0<y<x^2 \}$ Is $\Omega$ with Lipschitz boundary? (i.e. the boundary is ...
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665 views

How to find an integral kernel for poisson's equation in the upper half plane

In our lecture we have shown that $\forall f \in L^2(\mathbb{R}^n_+) $ there is a unique $ u $ in the Sobolev space $ H^2(\mathbb{R}^n_+) $ satisfying $ -\Delta u = f. $ Now in our exercise sheet we ...
7
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1answer
169 views

$\nabla \cdot f + w \cdot f = 0$

Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation $$ \nabla \cdot f + w \cdot f = 0 \, ? $$ Note that if $w = \nabla \phi $, then the above equation is ...
7
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2answers
191 views

PDE: What is the most general solution of $F_xF_y=1$ for a real function, $F(x,y)$?

WolframAlpha gives the simple solution, $F(x,y)=cx+\dfrac{y}{c}+c'$ with two constants $c$ and $c'$ . Is this the most general solution?
7
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456 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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207 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 L^2(\mathcal{C}_{\...
7
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1answer
335 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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1answer
444 views

Parabolic linear PDE with continuous coefficients; how to solve and explanation of text needed

I have the following PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u - f(x,t)$$ $$u|_{t=0} = u_0$$ over the domain $S^1 \times [0,T)$. The coefficients and $f$ are in $C^{k,\alpha}$ for some $k$ (and ...
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471 views

Separation of variables for partial differential equations

What class of Partial Differential Equations can be solved using the method of separation of variables?
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96 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 $u \in C_c^\infty(U)$?
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152 views

Surveys: problems, conjectures, and questions in some areas of nonlinear analysis

I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and ...
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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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166 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
113 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 ...
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2answers
190 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 ...
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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 $x$...
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277 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. (...
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182 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 \begin{equation}S(t_0)f=S(t_1)...
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1answer
784 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). ...
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1answer
319 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 ...
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163 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 ...
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2answers
441 views

Change variables into Fokker-Planck PDE

I have a question regarding a PDE (Fokker-Planck) and change of variables. I have a problem deciding what route to take after I use the chain route. I have an expression $$\frac{\partial u}{\partial t}...
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1answer
626 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 - ...
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701 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 C\...
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1answer
170 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(W_t)_{t\...
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317 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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291 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 e^{-s^...
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1answer
176 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
440 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
304 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
316 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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234 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^{*} = (x_1-...
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466 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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51 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}udy. $$ But for the mean-value property of the Heat equation, ...
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190 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 R^2\times\...
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1answer
157 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 \big(G(y)...
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240 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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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 ...
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102 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 $D$}, \tag{1}\\...
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118 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 $u \in H^{s+l}$ (...
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260 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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138 views

When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates ...
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217 views

Importance of Schwartz kernel theorem

I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that ...