All Questions
Tagged with ap.analysis-of-pdes fa.functional-analysis
1,304 questions
1
vote
1
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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(...
- 1,219
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 ...
- 839
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 ...
- 839
1
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0
answers
298
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Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?
The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ ...
- 597
4
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0
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194
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$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
- 307
0
votes
1
answer
417
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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 ...
- 309
2
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0
answers
65
views
Regularity of solution to first order time dependent variational problem
Consider the following first order evolution problem over some regular bounded domain $\Omega\subset\Bbb R^d$
$$\frac{\partial\phi}{\partial t}(\mathbf{x},t) +\vec V(\mathbf{x},t)\cdot \nabla\phi(\...
- 153
4
votes
0
answers
127
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Algebra properties regarding Gevrey spaces: closed under multiplication
In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
- 517
3
votes
1
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164
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Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?
Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
- 1,245
4
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1
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458
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Contractivity of Neumann Laplacian
I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.
In ...
- 1,759
2
votes
0
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94
views
From some priori estimates can we estimate higher Sobolev norm?
Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that
$$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$
where $u|_{\partial\Omega}=\phi$.
Can we ...
- 309
1
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0
answers
103
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Regularity results for non uniform elliptic equation
I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
14
votes
1
answer
830
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Spectrum of matrix involving quantum harmonic oscillator
The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator
$$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$
Fix two numbers $\alpha,\beta \...
- 192
3
votes
0
answers
96
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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
$$\begin{cases} -\Delta ...
- 969
2
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0
answers
147
views
Dimension of critical set of p-harmonic function
Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical ...
- 91
3
votes
1
answer
552
views
Lax-Milgram and the existence of solution to parabolic equation
I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not ...
- 54
3
votes
0
answers
145
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Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
- 931
1
vote
1
answer
341
views
Plummer and Coulomb kernel for the Poisson equation
Consider the $d$-dimensional Coulomb "kernel" defined by:
\begin{equation}
x \in \mathbb{R}^{d} \mapsto g(x):=\left\{\begin{array}{ll}
\log \frac{1}{|x|} & \text { if } d=2 \\
\frac{1}{|...
- 407
1
vote
1
answer
116
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uniform convergence of $H^r$ projectors on compact sets?
Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
- 5,380
3
votes
1
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216
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Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
- 124
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\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 ...
- 124
1
vote
0
answers
251
views
Regularity of a Fokker-Planck PDE with unbounded coefficient
Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\
- 311
3
votes
1
answer
107
views
Finiteness of Schatten $p$-norm of truncated free resolvent
Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$.
For $p \geq 1$, let ...
- 91
5
votes
0
answers
419
views
Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
- 51
3
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0
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159
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Does the weak formulation of a parabolic PDE applies to a (good) non-test function?
Let $\rho:\mathbb R^d\times[0,\infty)\to(0,\infty)$ such that $\int \rho_t(x)\,dx=1$ for all $t\geq0\,$, $\rho$ is Holder-continuous (in both variables) and $\rho_t\in W^{1,1}(\mathbb R^d)$ for a.e. $...
- 311
2
votes
0
answers
125
views
Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system
...
- 1,245
5
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1
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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 ...
- 324
2
votes
1
answer
171
views
Mean value formula for fractional heat equation
For the solution $u(z) = u(t,x)$ of the heat equation $u_t -\Delta u = 0$ we have
$$u(z_0) = \int_{\Omega_r(z_0)}u(z) K_r(z_0-z) dz,$$
where $$\Omega_r(z_0) = \left\{z \in \mathbb{R}^{N+1}: \Gamma(z_0-...
- 161
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 ...
- 379
4
votes
1
answer
698
views
Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions
Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type
$$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$
to hold for all $u\in X\subset H^1(\Omega)$ and $C$...
- 287
2
votes
0
answers
49
views
Determining a space of differentiability
I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by
$$
P(t,s)f:...
2
votes
0
answers
162
views
$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?
Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
- 61
4
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0
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143
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If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
- 309
2
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1
answer
2k
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Norms in Sobolev space $W^{1,\infty}$
Let $n\in\mathbb{N}$ and consider the Sobolev space $W^{1,\infty}(\mathbb{R}^n)=\lbrace u\in L^{\infty}(\mathbb{R}^n):\partial_iu\in L^{\infty}(\mathbb{R}^n) \rbrace$. A function is in $W^{1,\infty}$ ...
- 237
1
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0
answers
72
views
Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
2
votes
1
answer
494
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Fundamental Theorem of Gamma-Convergence
Let $(X, d)$ be a metric space and $F_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F_\varepsilon$ is an equicoercive sequence of functions on $X$, i.e. for all $t\in\mathbb{R}$ there exists ...
- 155
6
votes
1
answer
182
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Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,135
0
votes
1
answer
98
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Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?
Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$
where the coefficient $a$ are smooth and bounded and $D$ is a bounded
and smooth domain of $\mathbb R^d$
$$
\begin{...
- 446
4
votes
0
answers
146
views
Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
- 1,755
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
$$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
2
votes
0
answers
654
views
Convergence of operator in norm resolvent sense and their eigenvectors
Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
- 21
16
votes
2
answers
1k
views
How to generalize the various vector calculus theorems to distributions?
Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
- 415
1
vote
2
answers
106
views
Green function of symmetric stable process in dimension 1 and 2
Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
- 99
2
votes
1
answer
287
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Reference request for semilinear PDEs in dimension 2
I am interested in the study of the (semi-linear, I suppose) equation
$$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\
u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$
...
- 597
5
votes
0
answers
156
views
Homotopy, contraction mapping and the inverse function theorem on Banach spaces
We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
- 179
1
vote
0
answers
105
views
Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?
Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
- 303
6
votes
0
answers
113
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
0
votes
1
answer
147
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The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]
Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
- 47
0
votes
0
answers
93
views
Regularity of semilinear parabolic PDE in the whole space
I need regularities in Holder space of the following parabolic PDE:
$$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
- 1,399
5
votes
1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 536