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

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2
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1answer
171 views

A counter example to this statement

I am struggling with Evans PDE problem 5.10. #12. Here is the question: Let $V\subset \subset U$ be open sets. Show by example that if we have $\|D^hu\|_{L^1(V)}\le C$ for all $0< ...
5
votes
2answers
145 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 ...
1
vote
2answers
184 views

degenerate equation and its fundamental solution

There is a well established theory for the uniformly parabolic equations, i.e. for equations of the form $$u_t=a(x,t)u_{xx}, x \in D, t\in (0,T], u(x,0)=u_0$$ when $a(x,t)\geq a_0 >0$. In fact, if ...
2
votes
1answer
104 views

Heat equation with initial value

I have a 1-dimensional homogeneous heat equation: $$ u''(x, t) = \dot u(x, t)$$ The initial value is $u(x, 0) = \exp\left(-x^2\right)$. I plugged this into the solution formula: $$ u(x, t) = ...
4
votes
1answer
159 views

An inequality for $W^{k,p}$ norms

Let $u \in W_0^{2,p}(\Omega)$, for $\Omega$ a bounded subset of $\mathbb R^n$. I am trying to obtain the bound $$\|Du\|_p \leq \epsilon \|D^2 u\|_p + C_\epsilon \|u\|_p$$ for any $\epsilon > 0$ ...
2
votes
0answers
214 views

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
1
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0answers
184 views

Special solution of Helmholtz equation separated Phase

I'm searching for some special solutions for the Helmholtz-equation in 2D: $(\partial_x^2 + \partial_y^2 + a) f(x,y) = 0$ where $f(x,y) \in \mathbb{C}$ (boundary condition: $\lim_{x,y\to \infty} ...
1
vote
1answer
58 views

Is there a way to solve this without using laplace

Is there a way to solve this without using laplace transform?: $$\frac{\partial y}{\partial x}= \frac{\partial^2 y}{\partial z^2 }-2ik^2 y$$
2
votes
2answers
114 views

existence of solution of a degenerate pde with change of variables

I am looking at the pde $$u_t=x^2u_{xx},\; x\in [0,\infty) ,\; t\in (0,T], \; u(0,x)=u_0(x)$$ This is a degenerate pde with a diffusion coefficient which is not bounded from 0, so I can't apply the ...
2
votes
1answer
122 views

Solve the given Cauchy problem on the bounded interval

$$u_{tt}-16u_{xx}=0, \quad 0<x<3, \quad 0 < t < \infty,$$ $$u(x,0)=x(3-x), \quad u_t(x,0)=\cos(\pi x), \quad 0<x<3,$$ $$u(0,t)=u(3,t)=t, \quad 0 < t < \infty.$$ Determine ...
2
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1answer
136 views

Is there any Elliptic Operator of first order in $U\subset \mathbb R^n$?

Suppose $P=\sum_i a_i \partial/\partial x^i$. It seems for me that there always exists a nonzero $\xi=\{\xi_1, \cdots, \xi_n\}\in \mathbb R^n\setminus 0$ such that the principle symbol $\sigma(P)(x, ...
2
votes
1answer
153 views

Projection onto the nullspace of the Laplacian on a surface

Consider a closed Riemannian manifold $(M,g)$ of dimension 2, so we're talking about a surface. To its Laplacian $\Delta_g$ one can consider the trace of the heat kernel given by \begin{equation} ...
0
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1answer
123 views

What is the Brezis-Kato theorem? Does the space $C_0(\Omega)$ equal to $K(\Omega)$?

What is the Brezis-Kato theorem?Where can I find the details? $\Omega$ is an open subset of $R^N$ $$K(\Omega):=\{u\in C(\Omega):supp\ u \text{ is a compact subset of }\Omega\},$$ $$BC(\Omega):=\{u\in ...
2
votes
2answers
99 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
2
votes
2answers
81 views

How to prove $\Phi \in C^1(L^p(R^N),R)$ under certain condtions?

I am reading a paper(Thomas Bartsch, Zhi-Qiang Wang, Commun. in Partial Differential Equations, 20,1725-1741(1995)), and encounter the following problem in page 1731: ($f1$) $f \in C(R^N \times R, ...
4
votes
2answers
99 views

Symmetries in a nonlinear heat equation

I have to solve the following nonlinear PDE: $$\partial_t u(x,t)=ku(x,t)^2 \partial_{xx}u(x,t)$$ where $k$ is a constant with $k>0$. Is it possible to find some symmetry in this equation which ...
8
votes
2answers
209 views

Black-Scholes PDE with non-standard boundary conditions

I have the PDE $$ -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$ with initial and boundary conditions: ...
2
votes
3answers
98 views

wave equation with initial values and boundary condititon

I have a homogenious 2-dimensional wave equation: $$ - \frac{\partial^2 u}{\partial x^2} (x, y, t) - \frac{\partial^2 u}{\partial y^2} (x, y, t) + \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} (x, ...
0
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1answer
68 views

Does formulating a PDE's analytical solution (if possible) imply its existence?

Or are there counterexamples? Especially some that are not quite artificial by e.g. taking the limit of a PDE series?
2
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1answer
118 views

How to solve this partial differential equation?

I need to solve this partial differential equation, $$Z(\frac{\partial Z}{\partial x}-\frac{\partial Z}{\partial y})=(x+y)^2+Z^2$$ Wolframalpha gave the last solution, $$Z = \pm ...
2
votes
1answer
147 views

Compactness in $C([0,1])$

I read in a paper (pp8) that the set $$A=\left\{f\in W^{1,1}(0,1):\sup |f|\leq C,\ \int_0^1|f'(t)|dt\leq M \right\}$$ is compact in $C([0,1])$, where $C$ and $M$ are fixed constants. I understand that ...
3
votes
1answer
233 views

Harmonic function with condition on part of its boundary

Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
2
votes
1answer
142 views

A Gronwall-type inequality

I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that. $$ f^2(t) \leqslant g^2(t) ...
1
vote
0answers
59 views

Differentiation of the norm

Let $f = f(t,x) = f(t, x_1, \cdots , x_n) . $ If $f \in C^1 ([0,M],W^{s,2}(\Bbb R^n))$ (which means that $g(t) :=\| f \|_{W^{s,2}(\Bbb R^n)}$ is differentiable on $[0,M]$ and $\frac{d}{dt} g(t)$ is ...
0
votes
1answer
183 views

Analysis of the fundamental solution to the heat equation

It is well known that there is a heat kernel (or fundamental solution) of the Cauchy problem for the heat equation on $\mathbb{R}^{n}$. I have a simple question. How do I show that the fundamental ...
2
votes
1answer
246 views

How to solve second order PDE with first order terms.

I know we can transform a second order PDE into three standard forms. But how to deal with the remaining first order terms? Particularly, how to solve the following PDE: $$ ...
-2
votes
1answer
95 views

How to solve PDE?

Solve $\Delta u=0$ on $\Omega$, $u=1$ on the boundary of $\Omega$, and $\lim_{x\to\infty}u(x)=0$, where $\Omega=\{x \mid ||x||>1\}$.
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0answers
63 views

What is an affine function?

Consider a functional, what is meant by a minimal sequence consistent of 'piecewise affine functions'?
2
votes
2answers
170 views

How to use the dense theorem to prove this exercise?

Let $\frac{2n}{n+1}\leqslant p<n,\quad q=\frac{np}{2n-p},\quad u\in L^1(R^n)\cap W^{1,p}(R^n)$, then prove $u^2\in W^{1,q}(R^n).$ I hope someone can show me how to prove it by dense theorem ...
0
votes
1answer
117 views

EDP problem in a ball

How to solve the exterior problem on a ball with radius $r$ in the 3d space? I have to found u such that: $\Delta u = 0$ in $B(0,r)^C$ Thanks!
3
votes
1answer
329 views

Density of space in a Sobolev space

An exercise from Gilbarg-Trudinger Elliptic Partial Differential Equations states the following : "Using Lemma 9.12, show that for a $C^{1,1}$ domain $\Omega$ the subspace $$\{u \in ...
2
votes
1answer
127 views

Removal of singularities for harmonic functions with finite energy

Denote by $B = B(0,1) \subset \mathbb{C}$ the open unit disc and by $B' = B \setminus \{ 0 \} \subset \mathbb{C}$ the punctured unit disc. Assume that $u : B' \rightarrow \mathbb{R}$ is a harmonic ...
10
votes
1answer
271 views

Regularizing effect of the heat equation

Consider the heat equation on $\mathbb{R}_+\times\mathbb{R}^d$ \begin{align*} \partial_t u -\Delta_x u &= f, \\ u(0,x)&=u_0(x). \end{align*} In the case where $u_0\in L^2(\mathbb{R}^d)$ ...
0
votes
1answer
612 views

Using d'Alembert's solution to solve the 1-D wave equation

Solve $$u_{tt}=7u_{xx},-\infty< x <\infty$$ $$u(x,0)=x^2, u_t(x,0)=\cos(3x),-\infty< x <\infty$$ If you use d'Alembert's solution for this problem after doing change of variables and ...
0
votes
1answer
187 views

Fundamental solution of a particular differential operator

Show that he distribution given by the locally integrable function $\dfrac{1}{2} e^{|x|}$ is a fundamental solution of the differential operator $\begin{equation} -\dfrac{\partial^{2}}{\partial ...
1
vote
1answer
94 views

Proving a sequence of distributions converge in $C^{-\infty}(\mathbb{R})$

Question: I have a sequence $(T_n)$, where $T_n$ is given by the locally integrable function $ne^{-\dfrac{n^{2}x^{2}}{2}}$, converges in $C^{-\infty}(\mathbb{R})$ and compute its limit. I suspect ...
2
votes
1answer
293 views

parallelogram identity in the wave equation

Using the parallelogram identity, I need to solve the following initial boundary value problem for a vibrating semi-infinite string with a nonhomogeneous boundary condition: $ u_{tt} − u_{xx} = 0 $ ...
7
votes
1answer
175 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 ...
1
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1answer
79 views

how to convert a multivariate polynomial to a list in Maple

Motivation: I need to take higher derivatives with respect to many variables. The particular derivatives I take are based on a homogeneous multivariate polynomial. This polynomial is 12 pages in Maple ...
0
votes
1answer
489 views

Method of Eigenfunction Expansion

The solution of a PDE can be represented by a Fourier cosine series $$ u(x,t)=\sum_{n=1}^\infty A_n(t)\cos\frac{n\pi x}L. $$ Applying a given initial condition $$ u(x,0)=100, $$ lets us solve for ...
3
votes
3answers
465 views

Heat equation, separation of variables and Fourier transform

I have a question about the heat equation $\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}$ with the conditions that $\varphi(x,t=0) = f_0(x)$ and $\lim_{x \rightarrow\pm ...
1
vote
1answer
121 views

Singular support of a tempered distribution is compact?

I am reading Introduction to the Theory of Distributions by Friedlander and Joshi. As definition 8.6.1, they define the singular support of a tempered distribution $u$ to be the complement of {$x$: ...
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0answers
58 views

The Maximum Principle and Some Notation

I am working on this question: Suppose that $\Delta u\leqslant0$ in $\Omega$ and $u=f$ on $\partial\Omega$, and $\Delta v\leqslant0$ in $\Omega$ and $v=f$ on $\partial\Omega$. Prove that ...
4
votes
1answer
650 views

Wave equation with variable speed coefficient

Consider the wave equation initial value problem in $\mathbb R^3$ with spatially variable wave speed, denoted by \begin{align*} \frac{\partial^2}{\partial t^2}u(x,t)-c(x)\Delta ...
0
votes
1answer
80 views

Diffusion equation with dirichlet condition by seperation of variables

The problem is this: $$\begin{cases} U_t = 3U_{xx}, \quad 0 < x < 2\pi, \\ U(0,t) = U(2\pi,t) = 0, & \\ U(x,0) = 2 \sin x + 5 \sin 3x \end{cases}$$ I want to express this as an infinite ...
0
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0answers
109 views

Chain rules for differential forms

I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z ...
9
votes
1answer
250 views

Regularity of elliptic PDE with coefficients in some Sobolev space

Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$? By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ ...
0
votes
1answer
95 views

How to prove this change of variables?

Consider the sets: $$Q_0=\{x\in\mathbf{R}^n:0<x_1<2, \ |x_\alpha|<1, \ for \ 1<\alpha\leq n\},$$ $$Q_l=\{x\in\mathbf{R}^n:0<x_1<2l, \ |x_\alpha|<l \ for \ 1<\alpha\leq n\}.$$ ...
1
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0answers
25 views

How to prove this equivalence?

Consider the general elliptic operator $$M=\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j},$$ where $a_{ij}$ are continous functions. The function $u$ satisfies $$|Mu|\leq A(|\nabla ...
1
vote
0answers
64 views

Proving $ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) $ by using the Sobolev inequality.

Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ...