All Questions
Tagged with real-analysis ap.analysis-of-pdes
569 questions
1
vote
0
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Examples of reaction-diffusion systems with analytical solutions
I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:
$$\...
- 1,759
0
votes
1
answer
711
views
Lipschitz domains ambiguous definitions
I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, ...
- 645
0
votes
0
answers
148
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A Grönwall-type inequality for $u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s$?
Quick question: Are we able to show a Gronwall-type inequality assuming that $$u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s,$$ where $\alpha$ is nondecreasing (or constant) and $\beta$ is ...
- 167
3
votes
1
answer
374
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Positive part of Cauchy sequence of Sobolev functions is again Cauchy
Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
- 6,367
1
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0
answers
72
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Compute surface Sobolev norm using local coordinate
For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using ...
- 503
8
votes
1
answer
380
views
Lavrentiev phenomenon between $C^1$ and Lipschitz
Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf_{y\in Lip([a,b])}F(y)<\inf_{...
- 6,367
6
votes
1
answer
376
views
Lavrentiev phenomenon between $C^1$ and $C^2$
Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is
$$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
- 6,367
1
vote
1
answer
675
views
First derivative of cut off function
I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in ...
- 159
1
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0
answers
72
views
Elliptic systems with two dimensions
Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\...
- 1,219
1
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0
answers
85
views
Boundary estimates for elliptic systems
Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\...
- 1,219
5
votes
1
answer
339
views
Finding vector fields on $S^2$ with equal divergence
Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal ...
- 7,197
3
votes
1
answer
146
views
Dirichlet to Neumann operator for a nonlocal ODE
Consider the following nonlocal ODEs on $[1,\infty)$.
#1)
$$\begin{align}
r^2 f''(r) + 2rf'(r)-l(l+1) f(r) &= -\frac{(f'(1) + f(1))}{r^2}\\
f(1) &= \alpha \\
\lim_{r\to \infty} f(r) &= 0
\...
- 39k
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
- 11
4
votes
1
answer
317
views
Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
- 17.2k
4
votes
1
answer
201
views
Spectrum Cauchy-Euler operator
A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations
We consider the operator
$$(Lf)(x) = \...
- 52.3k
0
votes
1
answer
1k
views
Euler-Lagrange equation for a functional
What does it mean that the equation:
$$ \text{div}_{x,y}(y^a\nabla_{x,y}u)=0,\quad \text{in }\mathbb{R}^n\times(0,\infty),$$
is the Euler-Lagrange equation for the functional:
$$ J(u)=\int_{\mathbb{R}^...
CommunityBot
- 1
3
votes
1
answer
1k
views
Continuation (extension) of harmonic functions
Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
- 63.9k
8
votes
2
answers
773
views
Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
- 4,143
4
votes
1
answer
346
views
Energy estimates for nonlinear wave type equation
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
- 39k
1
vote
1
answer
141
views
Averaging and fractional Laplacian
Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is ...
CommunityBot
- 1
2
votes
2
answers
132
views
Density of traces of solutions to an elliptic equation
Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...
- 684
2
votes
0
answers
229
views
Weighted Sobolev norm in terms of Spherical harmonics coefficients
Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
- 969
3
votes
3
answers
219
views
Asymptotic behavior of an integral transform
Given $g\in L^2(\mathbb{R}^3)$, consider the following function ( defined for $r>0$ ):
$$c(r):=\int_{\mathbb{R}^3}\frac{g(x)}{|x|^2+r}dx$$
I'm interested in the behavior of $c(r)$ for large $r$. A ...
- 6,367
4
votes
1
answer
636
views
Existence of a smooth compactly supported function
Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>...
- 91.8k
1
vote
1
answer
150
views
Is the Poisson formula valid when the boundary condition is $ L^2 $?
Dirichlet problem for Laplace equation as follows
\begin{eqnarray}
\Delta{u}&=&0\text{ in }B_r(0)\\
u&=&g\text{ on }\partial B_{r}(0),
\end{eqnarray}
where $ g $ is continuous.
It is ...
- 12.9k
1
vote
1
answer
123
views
Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$
Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$.
To prove that, as $\epsilon \to 0$, the weak solution ...
- 52.3k
7
votes
1
answer
735
views
Conserved positive charge for a PDE
Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$:
\begin{equation}
\frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
CommunityBot
- 1
2
votes
1
answer
177
views
Determine the sign (positive or negative) of an integral with the fractional Laplacian
Let $u,v:\mathbb R \to \mathbb R$ and $\phi: \mathbb R \to \mathbb R_+$ be smooth bounded functions. Assume also $\phi' \ge 0$. Assume that $u(0) - v(0) = 0$ and that $0$ is a strict global minimum of ...
- 17.2k
1
vote
1
answer
225
views
How to prove the reverse Hölder inequality for Laplace equations?
Let $ u\in H^1(2B) $ be a weak solution of $ \Delta u=0 $ in $ 2B $, where $ B=B(0,1) $ is a ball with center $ 0 $ and radius $ 1 $. Then there exists some $ p>2 $ such that
\begin{eqnarray}
\left(...
- 27.5k
2
votes
1
answer
195
views
Sufficient conditions for the convexity of the discrete Fourier transforms
Let $f : [0,2\pi] \to \mathbb{R}$ be some function. Then the discrete Fourier transform of $f$ when sampled at $2\pi i/N$ is then given by
$$
X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)...
CommunityBot
- 1
0
votes
1
answer
417
views
Application of Green function for non linear PDE [closed]
In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$.
Is the same thing hold for ...
- 34.7k
3
votes
0
answers
96
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
- 969
2
votes
1
answer
268
views
Existence of the derivative of functionals of Brownian motion
Let $v(x, t) = \mathbb E [f(x + W_t)]$ with a Brownian motion $W$. Then, Malliavin calculus leads to the sensitivity in $x$:
$$\partial_x v(x, t) = \frac{1}{t} \mathbb E [ f(x + W_t) W_t ].$$
I am ...
- 128k
8
votes
2
answers
622
views
Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 17.2k
3
votes
1
answer
301
views
Convergence of a level set when $f^n\to f$ in $C^1$ sense
Let $f^n$ be a family of $C^1$ functions and $f(x)=-|x|^2+1$ such that
$$f^n\to f$$
in $C^1$ sense as $\varepsilon\to 0$. I want to ask that does the level set $\{f^n=0\}$ converges to $\{f=0\}$ in ...
CommunityBot
- 1
3
votes
1
answer
216
views
Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
- 12.6k
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 128k
2
votes
0
answers
58
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,143
0
votes
1
answer
109
views
Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...
- 6,367
1
vote
0
answers
46
views
Boundary estimates for Neumann derivative of solution to Laplacian equation with Dirchlet boundary data
Let $\Omega \subset \mathbb{R}^n$ be a smooth domain. Consider the following Laplacian equation with Dirichlet boundary condition.
\begin{equation}
\begin{cases}
\Delta u=0\quad &\mbox{in $\Omega$}...
- 1,350
0
votes
2
answers
241
views
Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$
Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...
- 39k
5
votes
1
answer
279
views
Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$
Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
- 3,584
2
votes
0
answers
297
views
Examples of harmonic functions
I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...
- 246
4
votes
0
answers
318
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
- 261
2
votes
0
answers
153
views
unique continuation in a disk
Let $D$ be the unit disc in $\mathbb R^2$ centered at the origin. Let $w \in C^{\infty}_c(D)$ satisfy
$$ (1-r^2)^2\Delta w +w =0.$$
Prove that $w \equiv 0$.
- 4,143
2
votes
0
answers
252
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 6,367
4
votes
0
answers
135
views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
- 6,367
5
votes
0
answers
140
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
- 6,367
6
votes
1
answer
182
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 128k
6
votes
0
answers
151
views
Gap between consecutive Dirichlet eigenvalues
Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say
$$ -\Delta \phi_k = ...
- 4,143