Tagged Questions
0
votes
2answers
42 views
Relationship between sobolev spaces
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
1
vote
1answer
16 views
Convergence of Discrete Poisson equation
Are there any sources that show the convergence of the discrete poisson equation?
To be clear, by convergence I mean: given the poisson equation in a domain $ M \subset R^2 $, $\Delta \psi = f $, one ...
2
votes
0answers
52 views
(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq ...
5
votes
0answers
122 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
1
vote
0answers
28 views
Can we say approximation by smooth function is an equivalent form of “Weierstrass approximation” theorem in Sobolove?
As I came to know most of properties characterized by approximation by smooth functions in Sobolev space looks equivalent to that of Weierstrass approximation theorem in the space of continuous ...
3
votes
1answer
57 views
Weak solution of a non-linear problem with Lipschitz functions
I'm trying to solve the problem 9.5 in Evans PDE book. The statement goes as follows:
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a $\Lambda$-lipschitz bounded function with $f(0)=0$ and ...
2
votes
1answer
22 views
The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$
I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
0
votes
2answers
39 views
Continuation of smooth functions on the bounded domain
Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
1
vote
0answers
34 views
Smooth Approximation of $L^p$ function
Given a bounded domain $\Omega \subset \mathbb{R}^n$, is it possible to approximate every $L^p(\Omega)$ function (where $1\leq p < \infty$) by smooth functions $\mathit{C}^{\infty}$ ?
2
votes
1answer
52 views
Concerning the noncompactness of the map, $I: W^{1, 2}(\mathbb{R}^n)\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$
Consider the identity map $I:W^{1,2}(\mathbb{R^n})\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$ where $n\geq 3$. Suppose that this map is not compact that is given some bounded sequence of functions ...
1
vote
1answer
29 views
Laplacian $\Delta u$ in spherical coordinates
The Laplacian $\Delta u$ in spherical coordinates is $$\Delta u=\frac{\partial^2u}{\partial\rho^2}+\frac{2}{\rho}\frac{\partial ...
0
votes
0answers
20 views
Sup and lim sup of a function defined by double series
It is unlikely that the following function has a closed form expression:
$$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$
...
2
votes
0answers
39 views
Closed form formula for a double series related to wave equation
Does anyone have a closed form formula for the double series
$$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$
This is related ...
1
vote
1answer
22 views
The existence of a subsequence of harmonic functions that converges pointwise
Let $u_{n}$ be a family of harmonic functions on $\mathbb{R}^n$, and there exists a point $x_{0}$ such that $\{u_{n}(x_{0})\}$ is bounded. Then does it exist a subsequence of $u_{n}$ that converges ...
2
votes
1answer
43 views
What is the *standard duality argument?
What is the standard duality argument? I saw this foor exemplo in the following statement. The case $p < 2$ follows from the standard duality argument. To prove
Theorem: [Calderón Zigmund] If ...
1
vote
1answer
38 views
Proof of Lemma 4.2 in [G-T] pg 55
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n (n\geq 3)$ and let $\Omega_0$ be any domain containing $\Omega$ for which the divergence theorem is true. Let $f$ be bounded and locally Holder ...
4
votes
1answer
135 views
Question about convergence and integration.
I am studying the paper "symmetry and non-uniformly elliptic operators - jean dolbeault, patricio felmer and regis monneau" and in the demonstration of the lemma 8 page: 5, we have the problem:
...
0
votes
1answer
53 views
Differentiable but not Absolutely continuous
Please give an example (if it exists) for a function which is differentiable everywhere but not absolutely continuous.
2
votes
0answers
54 views
Construction of Monotone function which is differentiable on the given set
Given a set $A \subset \mathbb{R}$ of measure 0, is it possible to construct a monotone function whose set of non differentiable points is $A$ ?
1
vote
1answer
63 views
Show that this equation is true.
Consider the following function in $\mathbb{R}^n (n\geq 3)$:
\begin{equation}
H(y)=2b_n\int_{0}^{\infty}e^{\ as}D_n\Phi(y-\tilde{x}+bs)\text{ d} s,\quad (x, y\in\mathbb{R}_{+}^{n}, x\neq y),
...
1
vote
1answer
67 views
General solution for the Eikonal equation $| \nabla u|^2=1$
Does there exist a formula for the general solution of the Eikonal equation? $| \nabla u|^2=1$. I'm looking for something similar to "the general solution of $\dfrac{\partial u}{\partial x}(x,y)=0$ is ...
1
vote
2answers
57 views
Maximum Principle for Poisson Equation
For a smooth $u(x)$, $x \in \mathbb{R}^n$, satisfying:
$\Delta u = -f$ for $||x||<1$ , $u=g$ on $||x||=1$
I want to show that there exists a constant $C$ such that:
$$\max\{|u|:||x||\leq 1\} ...
5
votes
2answers
93 views
Weakly differentiable but classically nowhere differentiable
Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable (or differentiable only on a set of measure 0) ? Thanks
1
vote
1answer
41 views
Weak Differentiability of Holder functions
Is it true that every Holder function is weakly differentiable? If not please give counterexample. Thanks
3
votes
1answer
59 views
Smooth approximation of characteristic function of a bounded open set
Let $U$ be an open bounded set of $\mathbb{R}^n$. Is it possible to approximate $\chi_U$ as almost everywhere limit of increasing sequence of smooth functions?
0
votes
2answers
73 views
Existence of smooth function with given compact support
Let $K$ be a compact set in $\mathbb{R}^n$. Does there exist a smooth function $\phi$ such that $0<\phi\leq 1$ on $K$ and 0 outside of $K$
0
votes
1answer
39 views
theorem in capacity theory
I am trying to understand the proof of a theorem of capacity theory
the theorem can be found in the book "nonlinear potential theory of degenerate elliptic equations" the authors are Juha Heinonen, ...
0
votes
1answer
45 views
Solving the equation $Pu = f$, given that every $\ell \in \text{Ker}(P')$ has $\ell(f) = 0$
(Stanford Real Analysis Qualifying Exam: Spring 2012) (Ideal time: 24 minutes)
5) Let $X$, $Y$ be separable reflexive Banach spaces. Let $P \colon X \to Y$ be a bounded linear map, and $P'\colon Y^* ...
5
votes
3answers
209 views
A question related to Wave Equation
Let $L>0$. Suppose $f, g$ are $C^2$ functions on $\mathbb{R}$ such that
$$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$$
for all $t\in \mathbb{R}.$
Does it ...
2
votes
1answer
70 views
Is this function harmonic? [G-T] page 121
On page 121 of Gilbarg-Trudinger's book (Elliptic PDE of second order) they have the following Green's function in $\mathbb{R}^n (n\geq 3)$:
\begin{equation}
G(x, ...
1
vote
1answer
73 views
Elliptic operators on compact space are Fredholm
I have come across this fact in a reading of mine, but I cannot seem to prove it, and I cannot seem to find a proof of it.
Mostly, I am confused why the range of an elliptic operator between ...
1
vote
1answer
57 views
Behavior of the pointwise norm of the gradient w.r.t. to boundary conditions in elliptic PDEs
Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$.
If I know that
\begin{align}
&\bullet\quad ...
3
votes
1answer
64 views
Confused about Proof of Thm 4.9 Gilbarg Trudinger
Thm 4.9 in Gilbarg-Trudinger's book states that :
if $B$ is a ball in $\mathbb{R}^n$ centred at $x_0$ and
$f\in C^{\alpha}(B): \sup_{x\in B} (\text{dist}(x, \partial B))^{2-\beta}\vert f(x)\vert ...
1
vote
1answer
30 views
Necessary Condition for $C^{2}$ Regularity of this Function
If I define
$$u(x,t):=\frac{1}{4\pi}\int_{B(x,t)}\frac{f(y,t-|x-y|)}{|x-y|}\;dy$$
for $(x,t)\in\mathbb{R}^{3}\times(0,\infty)$, what regularity of $f$ is required so that $u$ is at least $C^{2}$ ...
3
votes
3answers
121 views
Non Uniformly Elliptic Equations page 117 [G-T]
Let $\Omega\subset\mathbb{R}^n$ be open and bounded. Suppose also that $\Omega$ satisfies the exterior sphere condition at $x_0$ and let $B=B_R(y)$ be a ball such that $B\cap\overline{\Omega}=x_0$. ...
1
vote
0answers
122 views
Green's function for the Laplace operator $\Delta$ in a rectangle (or square)?
What I'm looking for is an explicit form of the Green's function $G$ for the Laplace operator $\Delta$ in a rectangular (or square) domain $D\subset\mathbb R^2$, e.g. $D=[0,1]\times[0,1]$. Namely, I'm ...
2
votes
1answer
86 views
Uniqueness of PDE BVP/IVP Modified Wave Equation
Let $U\subset\mathbb{R}^{n}$ be open, $q(x)\geq0$ continuous, and suppose $u\in\mathscr{C}^{2}(U\times[0,T])$ solves
$$\left\{\begin{array}{rl}
u_{tt}-\Delta u=q(x)u&\text{in}\;U_{T}\\
...
2
votes
0answers
62 views
Boundedness for Reaction Diffusion BVP with Arbitrary Exponent $\alpha$
Let $U\subset\mathbb{R}^{n}$ be open, $U_{T}$ and $\Gamma_{T}$ be the parabolic cylinder and boundary of $U$ for arbitrary $0\leq t\leq T$, respectively, and suppose $u$ solves
$$
...
2
votes
1answer
88 views
Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$?
Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$? More precisely, I want to prove that
THEOREM. A sequence $\{f_n\}$ is convergent in $C^k(\bar\Omega)$ (or some more ...
3
votes
2answers
45 views
Showing Solution to Some Random PDE Tends to Zero Uniformly
Another qual problem that is causing me some difficulty...
Consider the PDE
$$
\left\{\begin{array}{rl}
u_{xxt}+u_{xx}-u^{3}=0&\text{in}\;[0,1]\times(0,\infty)\\
...
3
votes
2answers
67 views
Cauchy Problem for Heat Equation with Holder Continuous Data
This exercise comes from a past PDE qual problem. Assume $u(x,t)$ solves
$$
\left\{\begin{array}{rl}
u_{t}-\Delta u=0&\text{in}\mathbb{R}^{n}\times(0,\infty)\\
...
0
votes
1answer
49 views
Question about boundness of derivatives.
My doubt is in the paper: Further qualitative properties for elliptic equations in unbounded domains, by Berestycki, Caffarelli and Niremberg (page: 93)
My question is simples. For any direction ...
1
vote
0answers
69 views
Question about uniform convergence.
My doubt is in the paper Monotonicity for Elliptic Equations in Unbounded Lipschitz Domains, by Berestycki, Caffarelli and Nirenberg (page: $1099$).
Consider the set
...
4
votes
2answers
65 views
Elementary application of Brouwer's fixed point Theorem
A professor of mine has suggested to me to look at this theorem and to find a problem related to it to explain in a future class.
I found an understandable proof in "Linear operators" by ...
6
votes
1answer
96 views
Wave Equation, Energy methods.
I am reading the book of Evans, Partial differential Equations ... wave equation section 2.4; subsection 2.4.3: Energy methods. Arriving at the theorem:
Theorem 5 (Uniqueness for wave equation). ...
3
votes
1answer
87 views
A problem about $C^1$-convergence! (Elliptic theory)
Let a function $u:\overline\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies
$$\Delta u+f(u)=0 \ \ \ \mbox{in} \ \ \Omega,$$
and consider
...
5
votes
2answers
130 views
characteristic curves for second-order equations
Reading about characteristic curves for second-order equations, in particular semi-linear equations of second order with two independent variables:
...
5
votes
0answers
74 views
Can we do some scaling argument in the presence of inhomogeneous norms?
Notation:
$B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$.
$\hat{f}$ stands for the Fourier transform of $f$.
Question. The following inequality holds true for all $f\in ...
0
votes
1answer
30 views
Question about moving planes method.
In the paper Inequalities for second-order elliptic equations with applications to unbounded domains I, by Berestycki, Caffarelli and Nirenberg (page $486$), they considered a set ...
5
votes
1answer
91 views
d'alembert's formula
I'm studying the Cauchy problem for the wave equation $n=2$;
$$\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}
...
