0
votes
0answers
8 views

Strong maximum principle for weak solutions?

For a general linear parabolic equation, is a strong maximum principle possible when the solutions are merely weak solutions (i.e. they lie in a Bochner space)? Is there some proof possible that does ...
-1
votes
0answers
13 views

$\big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} \le C_{>0}\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$ holds? [duplicate]

I want to know that whether the following inequality holds or not for complex-valued functions $f_1$, $f_2$, $f_3$ on $\mathbb R$: $$ \big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} ...
1
vote
0answers
41 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
0
votes
0answers
33 views

general existence theorem of nonlinear parabolic PDE on a unit circle

I wish to study existence/uniqueness of the solutions to a system (possibly coupled) of nonlinear PARABOLIC PDE arise from biology on a unit circle. Could any one suggest me any references for ...
1
vote
2answers
62 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
3
votes
0answers
32 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
2
votes
1answer
24 views

Convergence in $L^p(0,T;L^q(\Omega))$

If $\Omega\subset\mathbb{R}^3$ is bounded, $$f_n\to f\mbox{ in }L^q(0,T;L^p(\Omega)),\,1\leq q<\infty,\,1\leq p<2 $$ and $$f_n\to g\mbox{ weak-star in } L^\infty(0,T;L^2(\Omega)),$$ then $f=g$ ...
1
vote
0answers
37 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
1
vote
1answer
16 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
2
votes
1answer
12 views

Definition of $H^{-1}$ space in Evans' PDE book

Let $U$ be an open, bounded subset of $R^n.$ Evans' well known PDE book defines the spaces: -$H_0^1(U)$:= $\{f\in H^1(U): \text{there exists a sequence} \; \phi_n \to f \; \text{in the} \; H^1(U) ...
0
votes
0answers
32 views

Parabolic PDEs and Gradient Systems

Apologize in advance for the length of this question, I need some help in clearing some things up that I haven't quite got my head around yet. It seems to be easy to find things out about finite ...
1
vote
1answer
25 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
0
votes
0answers
15 views

Heat semigroup on Morrey spaces

I have a few questions concerning the heat semigroup $e^{\Delta t}$ on Morrey spaces. (1) I read that the heat semigroup on $ M^{p}_{q}(\mathbb{R}^n):= \left\{ f \in L^{q}_{loc}(\mathbb{R}^n): ...
0
votes
2answers
15 views

Prove $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$

Prove that for some $C$, $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$. Recall that $$ \Delta^2u=u_{rr}+r^{-1}u_r+r^{-2}u_{\theta\theta}$$ My answer is below.
0
votes
2answers
22 views

$\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $ holds?

I want to find the relation of $p$ and $k$ such that the inequality $$ \|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $$ holds when r.h.s $<\infty$. Here $f$ ...
0
votes
1answer
28 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
1
vote
2answers
57 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
1
vote
1answer
62 views

estimation of gradient

$$(\mathcal{P}_{\varepsilon}) : \left\{\begin{array}{ll} \displaystyle -div\left(A(x)\nabla u_\varepsilon(x)\right)= \dfrac{a(x)}{|u_\varepsilon(x)|+\varepsilon} &\mbox{ in }\Omega \\\\ ...
0
votes
1answer
32 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
2
votes
0answers
27 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
0
votes
1answer
64 views

Formulas for Schrödinger unitary groups of operators

Let $\Omega$ an open set of $\mathbb{R}^n$. Consider the Hilbert space $X=L^{2}\left(\Omega\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\Omega)$. Is there any ...
1
vote
0answers
28 views

Elliptic regularity on the torus: reference request

Suppose we work on the two dimensional torus $\mathbb T^2$. Let $L_a^2$ be the space of square integrable functions with zero space average and $H_a^m$ be the corresponding Sobolev space. Suppose we ...
2
votes
1answer
41 views

Global bounded solution of $u_{tt}=\Delta u-mu+h$ in the Hilbert space $X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Let $\Omega$ be an open subset of $\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ...
0
votes
1answer
24 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
43 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
26 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
35 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
46 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
23 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
19 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
29 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
16 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
35 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
26 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
29 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
34 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 ...
1
vote
0answers
40 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
103 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
47 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
40 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
50 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
110 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
52 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
14 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
81 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
31 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
64 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 ...