0
votes
0answers
24 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
1
vote
1answer
26 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
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
49 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 ...
0
votes
2answers
52 views

Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
0
votes
0answers
25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
0
votes
0answers
33 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
0
votes
1answer
19 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
1
vote
0answers
23 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
4
votes
1answer
61 views

If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?

Let $$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is ...
0
votes
1answer
33 views

composition and commutators of Fourier multiplier operators

I am working with some Fourier multiplier operators arising in study of a PDE. I have a general question: Suppose $S$ and $T$ be two Fourier multiplier operators (on some space) with multipliers $m_1$ ...
0
votes
0answers
17 views

partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
1
vote
0answers
103 views

Partial differential equations and semigroups: explanation of an example.

The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem $$\left\{\begin{align*} ...
2
votes
0answers
56 views

Complex Power of a differential operator

Let $(X,\|\cdot\|)$ be a Banach space and consider a sequence $B_n \colon X \to X$ of bounded operators. I remember from my course in operator theory that the partial sum $$ S_N = \sum^N_{n = 1} B_n ...
0
votes
1answer
59 views

Interpretation of Fredholm Alternative with respect to PDEs

I have studied the Fredholm Alternative, which states the following: Theorem: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator on $H$. Then: 1.$N(I-K)$ is ...
2
votes
1answer
60 views

questions about norm of integral operator

The following is a question I came up with when I was studying the same problem in dimension 1 (for which also I have the questions that follows) but I put in generality. Let $U_1, U_2 \subset ...
5
votes
1answer
160 views

Integral kernel of the resolvent operator

Suppose we have an explicit formula for the integral kernel $k(x,y)$ of an operator $D$ acting on smooth $\mathbb{C}^n$-valued functions defined on an interval $[0,\beta]$, that is $$ Df(x) = ...
1
vote
2answers
581 views

is bounded linear operator necessarily continuous?

Let $U, V$ be separable Banach spaces. Suppose we have a bounded, linear operator $C : U\to V$. Questions are the following *) Shall $C$ be continuous since $V$ is a Banach space? *) In general, ...
0
votes
0answers
84 views

Prove Fredholm's theorem

I'm trying to show the Fredholm alternative, is one of Fredholm's theorems. Let $T$ is a compact operator and $T:E \to E$. where $E$ is a Banach space. We consider the equation: $$u-Tu=f ...
1
vote
1answer
56 views

differentiability/holomorphicity of family of bounded operators

Edit: It seems I made a mistake in the statements on differentiability. I will replace weak differentiable implies strong differentiable with weak continuously differentiable implies strongly ...
1
vote
0answers
95 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
4
votes
1answer
114 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
5
votes
2answers
277 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
3
votes
1answer
57 views

The classification of possible singular supports

I need to find the solutions of $D_{x_1}u=0$ on $\mathbb{R}^{n}$ and to classify the possible singular supports. Any one have an idea how to solve this kind of question? Thanks!
3
votes
2answers
167 views

Core for the Laplace operator in a bounded domain

Let $X$ be a bounded connected open subset of the $n$-dimensional real Euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
4
votes
1answer
47 views

Is there an asymmetric positive definite second-order linear differential operator?

The second-order differential operator is $Lu=-\sum_{i,j=1}^n (a^{ij}(x)u_{x_i})_{x_j} +\sum_{i=1}^n b^i(x) u_{x_i} +c(x) u$. We say it's positive definite if there exsits constant $\beta>0$ such ...
2
votes
1answer
123 views

Elliptic Operators

I'm studying Elliptics Operators like this: $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ for $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I want to know what the difference when: $L$ is elliptic in ...
3
votes
0answers
69 views

Adjoints of an operator

In a paper I'm reading the discussion centres on operators of the form \begin{equation} \mathbf{B} = (-1)^{m+1} \Delta^m_y + \frac{1}{2m}y\cdot\nabla_y \end{equation} Apparently this is symmetric ...
3
votes
0answers
45 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
4
votes
2answers
164 views

Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| ...
3
votes
0answers
112 views

invertible operator Sobolev space

Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
2
votes
1answer
91 views

Evolution operator

We call a function that assigns a starting value of a time-dependent differentialfunction to a solution of a later timevalue as the evolution operator $E(t)$. Look at the thermal equation $$ ...
0
votes
1answer
91 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
0answers
68 views

Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators". The motivating problem for this is to find an approximate kernel ...
9
votes
2answers
228 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
2
votes
0answers
356 views

Adjoint of the infinitesimal generator of a stochastic process

I need help seeing that $$ \mathcal{L}^* g = -\frac{\partial (bg)}{\partial x} + \frac{1}{2}\frac{\partial^2(\sigma^2g)}{\partial x^2} $$ is the adjoint operator of $$ \mathcal{L} = b\frac{\partial ...
2
votes
0answers
269 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 + ...
3
votes
1answer
275 views

Questions about partial differential operators

In Folland's Introduction to Partial Differential Equations (Chapter 2 The Laplace Operator, p.67, Theorem 2.1) Suppose $L$ is a partial differential operator on ${\mathbb R}^n$. Then $L$ commutes ...
6
votes
1answer
252 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 ...
2
votes
0answers
115 views

Principal eigenvalue

How is the principal eigenvalue of elliptic differential operator defined? Is it just a spectral radius?
4
votes
2answers
703 views

How can one relate inverse of a differential operator to an integral operator?

Informally speaking, the integral operator can be regarded as the inverse of some differential operator. In some very special case, finding the inverse of the differential operator is equivalent to ...
6
votes
1answer
173 views

Questions about a PDE: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$

Consider the BBM equation: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$. One may rewrite this equation as following $u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ ...