0
votes
0answers
12 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
0
votes
2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
1
vote
1answer
41 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
0
votes
0answers
25 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
0
votes
0answers
31 views

TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
2
votes
1answer
45 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
0
votes
1answer
20 views

About Sobolev-Poincare inequality on compact manifolds

Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$. We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u ...
2
votes
1answer
16 views

Weak convergence in some space.

I have the sequence $\{u_{k}\}_{k}$ weak convergent in the space $L^{2}(0,T; W^{1,2}(\Omega))$. What exactly does it mean? Do it imply weak convergence $\{u_{k}\}_{k}$ in $L^{2}(0,T;\Omega)$ or $\{ ...
1
vote
0answers
26 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
0
votes
1answer
13 views

About the weighted Sobolev norm

I'm wondering that the Sobolev norm with weight $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ is equivalent to the norm $$ ...
1
vote
1answer
34 views

If $\lVert u(t) \rVert \leq \lVert u_0 \rVert$ for solution of PDE, is $\lVert u(t) \rVert \leq \lVert u(s) \rVert$ for all $t \leq s$?

Suppose I have some nonlinear PDE on some time interval $[0,T]$ $$u_t + A(u) = 0$$ $$u(0)=u_0$$ and I have managed to show existence and uniqueness of solution with $\lVert u(t) \rVert \leq \lVert u_0 ...
0
votes
0answers
25 views

A problem of convergence in $L^2(0,T;L^6(\Omega))$

Why $u_m\to u\mbox{ in }L^2(0,T;L^2(\Omega))\Rightarrow u_m 1_{\{|u_m|\leq\lambda\}}\to u 1_{\{|u|\leq\lambda\}}\mbox{ in }L^2(0,T;L^6(\Omega))?$
2
votes
0answers
28 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
1
vote
1answer
32 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
0
votes
1answer
48 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
1
vote
0answers
37 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
0
votes
0answers
36 views

Problems with understanding a proof.

I have some problems with understanding the Div-Curl lemma's proof. More precisely I don't understand the first part of the proof of the Theorem 1.1 in the article below: ...
6
votes
1answer
94 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 $$ ...
2
votes
1answer
46 views

Elliptic partial differential equations and elliptic operators

I'm starting to study elliptic partial differential equations and I just want to know if there are any connections between the following concepts: An elliptic partial differential equation is given ...
1
vote
1answer
39 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
0
votes
2answers
49 views

Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases} $$ doesn't admit weak solutions. I'm proceeding by ...
5
votes
0answers
106 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
1
vote
1answer
28 views

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ make sense?

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ exist, i.e., is $u^+$ weakly differentiable in time? By $u^+$ I mean the positive ...
2
votes
2answers
48 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
1
vote
0answers
13 views

Existence of Galerkin approximations to PDE

Given a weak PDE $$ \langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0 $$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow ...
3
votes
0answers
75 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
0
votes
1answer
25 views

If $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^1)$ and $u$ and $u_t$ are bounded, is $u \in C([0,T];L^\infty$)?

Let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$ so $$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$ holds for all $\varphi \in C_c^\infty(0,T)$. Suppose we know ...
3
votes
1answer
59 views

Characterization of weak solution

5 Nonlinear elliptic variational inequalities Preliminaries In order to explain the importance of elliptic variational inequalities, first consider the weak solution of the linear ...
0
votes
1answer
28 views

Using Gauss's Theorem in weak formulation

Can anyone see how exactly Gauss's theorem used in the following case: In defining the weak solution to the linear elliptic equation we start with $$-\sum_{j,k=1}^{n}D_{j}(a_{jk}D_{k}u) + cu = f ...
1
vote
1answer
27 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
0
votes
1answer
29 views

The spherical mean of $h(x,y) = x$.

The spherical mean of a function $h : \mathbb R^2 \to \mathbb R$ is given by $$ \frac{1}{2\pi r} \int_0^{2\pi} h(x + r \cos(\theta), y + r \sin(\theta)) d \theta $$ Now I want to compute the ...
2
votes
1answer
45 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
10
votes
1answer
109 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
0
votes
0answers
27 views

What are the tempered solutions of $\frac{\partial^2 T}{\partial x\partial y}=0$?

I would like to know what are the solutions of $\frac{\partial^2 T}{\partial x\partial y}=0$ when T is assumed to be a tempered distribution on the plane (or, what is the same, an slowly growing ...
2
votes
3answers
57 views

2D Heat Equation with special initial condition

I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...
0
votes
1answer
26 views

estimate to show bilinear form continouus

Consider the bilinear form $$a(u,v)=2\mu\int_\Omega \mathrm{trace}\left(\varepsilon(u)^T\varepsilon(v)\right)dx + \lambda\int_\Omega \mathrm{div}(u)\mathrm{div}(v)dx$$ with ...
0
votes
1answer
42 views

Show that $(\log |x|)^2\notin \text{BMO}([-1,1])$.

I am trying to show that $u(x)\equiv (\log |x|)^2\notin \text{BMO}([-1, 1])$ by showing that it doesn't satisfy the John-Nirenberg inequality. If $u\in\text{BMO}[-1, 1])$ then this inequality says ...
0
votes
1answer
21 views

How to get comparison principle from contraction principle for PDE

let $u$ and $v$ be two solutions to some PDE with initial data $u_0$ and $v_0$. Is $$|u(t)-v(t)|_{L^1} \leq |u_0-v_0|_{L^1}$$ a contraction principle? I read that "contraction principle gives ...
1
vote
1answer
56 views

Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?

Suppose $x_n\rightharpoonup x$ in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ with the weak* topology, in other words, $\forall f\in L^{1}([0,T];L^2(\mathbb{T}^2))$ we have $$\lim_{n \to\infty} \int _0 ^T ...
1
vote
1answer
38 views

Spherical Harmonics completeness relation

The answer to this question is probably negative, but let me ask it anyway. If I have an expression of the form $$\tag{1}I =\int\!\mathrm{d}\Omega\, Y_1^0Y_l^{*m}(\theta,\phi) \times ...
3
votes
2answers
68 views

If $u=v$ on $A \subset \Omega$, then $\nabla u = \nabla v$ on $A$ too

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $A \subset \Omega$ be measure nonzero. For $u, v \in H^1(\Omega)$, if $u=v$ (a.e) on $A$, how to prove that $\nabla u = \nabla v$ on $A$? ...
1
vote
0answers
56 views

Exercise on additional regularity of reaction-diffusion equation problem

I have to make exercise 6.2.6.1 form the notes of Lunardi. Prove the following additional regularity properties of the solution to (6.12): (i) if $u_0 \in BUC(\mathbb{R}^n,\mathbb{R}^m)$, then $u(t, ...
1
vote
0answers
37 views

A bound on $\nabla u$ in $L^\infty(0,T;L^2)$; how to make argument rigorous?

Suppose $u \in L^2(0,T;L^2)$, $u_t \in L^2(0,T;H^{-1})$ and $f \in L^\infty((0,T)\times\Omega)$. I have the weak form $$\langle u_t, \varphi \rangle_{H^{-1}, H^1} + \int_\Omega\nabla u \nabla \varphi ...
0
votes
1answer
42 views

If $u$ has a weak time derivative, does it make sense for $f(u)$ to have a weak derivative where $f$ is a piecewise function?

Let $\Omega$ be a bounded domain. I write $L^2$ instead of $L^2(\Omega)$ etc. Let $u \in L^2(0,T;H^1)$ with weak derivative $u' \in L^2(0,T;H^{-1})$. Consider $$f(u(x,t)) = \begin{cases} -1 &: ...
1
vote
1answer
67 views

A couple of inequalities (explanation needed), how to show $\liminf_{k \to 0}k^{-1}\int(\nabla u - \nabla v)\nabla (T_k(b(u)-b(v))) \geq 0$

Let $T_k(x) = \max\{-k, \min(x,k)\}$, a truncation function at levels $k$ and $-k$. Let $b$ be a Lipschitz increasing function with $b(0)=0$ and $b'$ and $b^{-1}$ also Lipschitz. I have seen it ...
1
vote
0answers
13 views

Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE: $\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 ...
1
vote
1answer
51 views

Explanation of a passage about a smooth approximation to $L^p$ function

I'm reading J.L Vazquez "Porous Medium Equation" book. In it, he says the following: We are given a function $a:\Omega \times (0,T) \to \mathbb{R}$ such that $a \geq 0$. We find a smooth ...
0
votes
0answers
17 views

What about the HUM for a finite-dimensional system?

We consider the finite-dimensional system \begin{equation}\begin{cases}y'(t)=Ay(t)+Bv,\ t\in (0,T)\\ y(0)=y^0\end{cases}\end{equation} Where $v$ is the control, $A\in Mat(N\times N); B\in Mat(N\times ...
0
votes
1answer
40 views

Resolvent of the Laplacian

We know that the Laplacian $-\Delta$ on $\mathbb{R}^n$ is a radial operator. I was wondering if its resolvent $(\lambda - \Delta)^{-1}$ is also a radial operator on $L^2(\mathbb{R}^n)$. I strongly ...
0
votes
1answer
31 views

If $u \in L^2(0,T;L^2)$ and $u' \in L^2(0,T;H^{-1})$, is $u \in C^0([0,T];V)$ for some space $V$?

If $u \in L^2(0,T;L^2)$ has weak derivative $u' \in L^2(0,T;H^{-1})$, is $u \in C^0([0,T];V)$ for some Banach space $V$? For what $V$ Lebesgue spaces does this hold? I cannot find any results.