6
votes
1answer
92 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 $$ ...
1
vote
1answer
26 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 ...
2
votes
1answer
76 views

Dual of $div$ on spaces where the tangential value is fixed

Say $\Omega$ is a domain in $\mathbb R^3$ with a smooth boundary $\Gamma$. Consider the spaces $$ H_{n,0}=\{v\in H^1(\Omega):n\cdot v \bigr |_{\Gamma} = 0\} $$ and $$ H_{t,0}=\{v\in ...
2
votes
1answer
63 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega:=\mathbb{R}^2\setminus\{0\}$. Consider the $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$ H=-\partial_x^2-\partial_y^2+ ...
1
vote
1answer
54 views

Expression for orthogonal projection onto Hilbert space (is related to Galerkin method)

Let $H=L^2(\Omega)$ and $V=H^1(\Omega)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily orthogonal). Let $V_m = \text{span}(v_1, ..., v_m)$. Define a projection operator $P_m:H ...
2
votes
0answers
65 views

What's this standard duality argument?

I'm reading a proof of the Strichartz inequalities. It shows that $$ \| \int_\mathbb{R} e^{-is\Delta}F(s) \, ds \|_{L^2_x} \lesssim \|F\|_{L^{q'}_t L^{r'}_x}, $$ and then says that by duality, $$ ...
1
vote
0answers
27 views

Regularity theory for $H^k$ space

Lat $\Omega$ be a bounded domain in $\mathbf{R}^n$ with smooth boundary. Let $a(u,v)=\int_\Omega \Sigma a_{ij}\partial_iu\partial_jv+cuv$ where $a_{ij}$ and $c$ are smooth functions on $\bar{\Omega}$ ...
2
votes
0answers
29 views

Completeness of separable solutions to PDEs

Under what conditions will the solutions of a PDE obtained using separation of variables form a complete set for the solution space?
4
votes
1answer
120 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
0
votes
0answers
34 views

estimation in elliptic forms

I obtained before the in equality $$\left\|u\right\|_{H^{1}}\left\|\phi\right\|_{L^{2}}\leq \left\|F\right\|_{L^{2}}\left\|\phi\right\|_{L^{2}} \\ \left\|u\right\|_{H^{1}}\leq ...
1
vote
1answer
33 views

Looking for a basis of $L^2$ with this special property

The setup. Let $\mathbb{T^2}$ denote the two-dimensional torus, i.e. $$ \mathbb{T}^2 \simeq [-\pi,\pi)^2 $$ induced by identifying opposing faces of $[-\pi,\pi)^2$. Note that $$ L^2(\mathbb{T^2}) ...
12
votes
4answers
314 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
2
votes
0answers
44 views

Yosida approximation, solution is $C^\infty$? (Explanation of passage needed)

I'm reading Brezis' book Functional Analysis, Sobolev Spaces and PDEs, Lemma 7.1 page 186. Let $w \in C^1([0,\infty);H)$ satisfy $$w' + A_{\lambda}w = 0$$ where $A_\lambda$ is the Yosida ...
3
votes
1answer
125 views

Is my proof that a function is measurable correct?

Let $V$ be separable and Hilbert. Let $\mathcal V = L^2(0,T;V)$. Assume for each $t \in [0,T]$, $$a(t;\cdot,\cdot):V \times V \to \mathbb{R}$$ is continuous and bilinear. Or equivalently, we have ...
2
votes
1answer
112 views

Want to show an operator is compact

With $V=L^2(0,T;H^1(\Omega))$, let $A:V \to V^*$ with $$\langle Au,v \rangle = \int_0^T \int_{\Omega} \nabla u(t) \cdot \nabla v(t).$$ I want to show that $A$ is a compact operator. So, one way to ...
3
votes
2answers
232 views

Eigenfunctions of Laplacian and orthonormal basis (with different inner products)

Suppose I have $L^2(\Omega)$ which has two inner products that are both norm-equivalent. The eigenfunctions of the Laplacian $\Delta$ we know forms an orthonormal basis of $L^2(\Omega)$ -- with ...
4
votes
2answers
152 views

about weak convergence in $L^{2}(0,T;H)$

I am trying to do an exercise and if the affirmation below is true, my exercise is done . This is the affirmation : Affirmation : Let $H$ a Hilbert space and suppose $u_k$ converges weakly to $u$ ...
4
votes
1answer
65 views

How can we pick $f \in C(0,T;H)$ with $f(T) =0$ and $f(0) = h$, where $h$ is arbitrary?

Let $C(0,T;H)$ be the space of continuous functions $f:[0,T]\to H$ where $H$ is Hilbert. For every $h \in H$, why is it possible to pick a function $f \in C(0,T;H)$ such that $f(0) = h$ and $f(T) = ...
1
vote
1answer
55 views

$\bigcup_{n}V_n$ is dense in $V$ implies $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$?

Let $V$ be a separable Hilbert space with basis $w_j$ and let $V_n$ denote the linear span of $w_j$ for $j=1,...,n$. Clearly $V_n$ are Hilbert spaces and $V_n \subset V_{n+1}$ for all $n$. We have ...
5
votes
1answer
72 views

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
5
votes
1answer
406 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
2
votes
1answer
175 views

Pde problem with Neumann BC's

Let $U \subset\mathbb{R}^n$ be a bounded open set with smooth boundary $\partial U$. Consider the Neumann boundary problem $$-\Delta u +u=f, \quad \left.\frac{\partial u}{\partial ...
2
votes
2answers
71 views

Inner product? Yes or no?

I define an "inner product" on $H_0^2(U)$ where $U \subset R^n$ is bounded open set: $$\langle u,v\rangle = \int_U \Delta u \Delta v dx.$$ I need this when trying to find a weak solution for my PDE ...
2
votes
1answer
584 views

PDE weak solution problem

My professor grades really strictly (details). I would be very happy if you could help me with this problem: Let $U \subset R^n$ be a bounded set. Consider $ \Delta^2 u = f$ on $U$ and ...
0
votes
1answer
52 views

If $v = v_1 + v_2$ and $\lVert v\rVert \leq 1$ then $\lVert v_1\rVert \leq 1$. Why is it true? (Hilbert spaces and orthogonality)

We have $H^1_0 \subset L^2$ where $w_j$ is an orthogonal basis on $H^1_0$ and orthonormal basis on $L^2$. Let $v= v_1 + v_2$ with $\lVert{v}\rVert_{H^1_0} \leq 1$, where $v_1 \in \text{span}\{w_j\}$ ...
5
votes
1answer
292 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
3
votes
1answer
101 views

Why define a function space in this fashion? (PDE and functional analysis)

Suppose that for $t \in [0,1]$, $X(t)$ is a Hilbert space of functions, eg. $X(t) = L^2(\Omega_t)$ where $\Omega_t$ is a bounded domain. Define a space $$H := \{\overline{v}:[0,1]\to \bigcup_{t ...
2
votes
2answers
155 views

elliptic pdes and associated bilinear forms for Lax-Milgram

I have a simple question on elliptic pdes, actually I can not understand clearly from definitions. Thats why I want to try think on an example. Let us have an elliptic pde $$-A \Delta ...
4
votes
1answer
498 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
0
votes
1answer
90 views

Differential Operator on $L_{2}$ problem

I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated, Consider the partial differential equation, $\frac{\partial ...
2
votes
1answer
46 views

Inequality involving a sequence in Hilbert space

Let $H = L^2(0,T;V)$ where $V$ is separable in $H$. We have $y_n(t) = \sum_{i=1}^n c_{i,n}(t)b_i$ where $b_i$ are basis vectors in $V$. Suppose we have the estimate $$\lVert y_n \rVert_H^2 = \int_0^T ...
3
votes
1answer
200 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
2
votes
0answers
267 views

Can we construct a Hilbert space where the operator $A_u v := -\frac{1}{2} v'' + (vF + v\int_\mathbb{R} Su + u\int_\mathbb{R} Sv )'$ is symmetric?

It seems not to be a easy problem. I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v^{\prime\prime} + (vF + v\int_\mathbb{R} Sp + ...
6
votes
1answer
246 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
1
vote
2answers
533 views

How do the solutions to the wave and heat equations converge in general?

I would like to check my understanding with someone if possible. When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
4
votes
3answers
357 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...
1
vote
3answers
177 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...