advanced-pdes

[bronze gate]

Doesn't that slow you down more than just studying it very deeply?

[quaint herald]

it's annoying at times ofc but try to see the upside

[quaint herald]

It can be easy to gloss over things when reading more passively, and it is hard to stay active when reading in general. Books written badly in this way can force you to be active to catch/repair the authors mistakes lol.

[verbal nebula]

The other main roadblock I'm hitting are things like "interior sphere condition" "exterior sphere condition" and the Alexandroff Principles

Oh and contact sets

This definitely feels more like a PDE approach than what I learned out of Evans or Eskin, which seemed to ues a lot of functional/fourier to get around things

[junior bloom]

Does anyone remember an example of using Riesz interpolation to solve a PDE (strongly or weakly or whatever)

[astral vine]

To purely solve, there is not in my knowledge, but to check out at nice properties/estimates for operators/functions spaces there is some

mostly variation of it than Riesz itself

Use of Haussdorf-Young inequalities to solve some nonlinear diffusion reaction system via Bloch Transform (which is somewhat a twisted Fourier Transform, for which we start from space variable to a space-frequency variables and keep some space information).

[junior bloom]

Hm. Appreciated. I know there are some PDEs for whom existence is dependent on energy controls but the energy controls given by Riesz interpolation are too specific it's definitely unclear to me where it's used.

I recall those properties/estimates being done in a PDEs course for minimal surfaces but the stuff escapes me right now.

[astral vine]

See the appendix for the adapted proof :
M.A.Johnson, K.Zumbrun - Nonlinear stability of periodic traveling wave solutions

Other powerful related applications :
M.A.Johnson, K.Zumbrun - Nonlinear stability of spatially-periodic traveling-wave solutions of systems
M.A.Johnson, P.Noble, L.M.Rodrigues, K.Zumbrun - Nonlocalized modulation of periodic reaction diffusion

[junior bloom]

[slender fulcrum]

are there any microlocal treatments for the maxwell equations?

[astral vine]

@quaint herald I summon you.

[solid flint]

Hello
Are there some smooth functions that are dense in spaces like $V={ u \in H^1(\Omega); u=0 \text{ on } \Gamma_0; \frac{\partial u}{\partial \nu}=0 \text{ on } \Gamma_1}$ such taht $\Gamma_0$ and $\Gamma_1$ are two parts that constitute the boundary of $\Omega$ that's an open subset of $\mathbb{R}^n$ and $\nu$ is the exterior normal vector?

[untold delta]

Mikahopff

[tranquil steppe]

What do you mean by these boundary conditions? For u in H^1 you can make sense of the first one (using the trace theorem), but as far as I know, not the second.

[astral vine]

adter discussing with him in DM, he forgot to mention about the fact that the laplacian lies in L²

[tranquil steppe]

OK, that makes sense.

[junior bloom]

Do we have any characterization of the projection operator $P:L^2(\mathbf{R}^3)\to{u\in L^2(\mathbf{R}^3):\nabla\cdot u=0}$?

[untold delta]

teafortwo

[astral vine]

Yes

thanks to the fourier side

[untold delta]

Functionanatolysis

[junior bloom]

the curl of a riesz potential

do you have a reference

[astral vine]

curl and div on the Fourier side are just

still works for S' by duality of course

Everyone knows it, and it is nowhere explicitly written in my knowledge

Maybe in Sohr

lemme few minutes

[junior bloom]

preciate it

[astral vine]

it is not in Sohr's book

lemme sketch you the proof

use the following identity 3-dimensional vector Identity

$$ i\xi \times(i\xi\times v)-i\xi (i\xi \cdot v) = |\xi|^2 , v$$

[untold delta]

Functionanatolysis

[astral vine]

which says nothing but

$$ \mathrm{curl}, \mathrm{curl} - \nabla \mathrm{div} = -\Delta$$

[untold delta]

Functionanatolysis

[astral vine]

this was a +

first lines are not - signs

but + signs

since I kept the - next to the Laplacian

[tired hollow]

I can’t believe ur speaking a coherent language

[astral vine]

?

No prove above operators I introduce to you above (with the nice corrections) are orthogonal projectors on L² with value in free divergence vector fields

In particular, shows that it restrict to the Indetity on the subspace of Free divergence vector fields

Then by uniqueness of the orthogonal projector on L² free divergence vector fields

above operators are necessarily Hopf-Leray's projection

[astral vine]

Due to this identity you just have to check that the following holds

all the other ones would be direct corollaries

[astral vine]

What do you mean exactly ?

Did I made a mistake somewhere ?

[tired hollow]

Out of context people the average person would say he’s babbling in tongues

That’s what I mean

[astral vine]

Like any other field of mathematics

[tired hollow]

No just a person who doesn’t study mathematics

[astral vine]

Well, in this case that's just even worse

[tired hollow]

huh

it means very smart words

[lilac barn]

Is <u,v> = ∫ D²u D²v an inner product on H²₀(U)? I am trying to prove |D²u| ≥ C |u|_H²₀(U) and I am wondering if I can just use the norm equivalency theorem for Banach spaces?

[astral vine]

To prove your inequality

just apply Poincaré twice

[lilac barn]

But would my argument work too?

[astral vine]

Assuming it is a scalar product how do you prove the coercive inequality ?

[lilac barn]

So, then H²₀(U) is Banach with the induced norm from <-,-> and it is Banach under its usual norm. We have that |D²u| ≤  |u|_H²₀(U) trivially so by the norm equivalency theorem, there exists some C such that |D²u| ≥ C |u|_H²₀(U)

( I haven't thought about why completeness would work out, I was just concerned whether the positive-definiteness would be an issue for the inner product for now)

[astral vine]

that's the difficulty here

and that's why your argument may fail

because your argument does not need the boundedness assumption on U

Think about what happens on U=Rn, in this case the result CANNOT be true

[lilac barn]

The positive definiteness or completeness? If completeness, is it true that under this norm <u,u> = 0 implies u = 0 (I am not sure because here u is a limit of compactly supported smooth functions)

[astral vine]

Depends how you define H^2_0

but your approach fails for many reasons

again think about Rn

[lilac barn]

Ok so can you specify what exactly you mean is failing? The completeness or positive definiteness?

[astral vine]

completness/well posedness of your space

[lilac barn]

Oh yes

I see what you mean

It even fails for <u,v> = \int_U DuDv

[astral vine]

It may fails

again it depends on your deifnition

and the way you deal with objects

[lilac barn]

H^2_0(U) for me means the limit of compactly supported smooth functions in W^2,2(U)

[astral vine]

Okay

Hence the closure of Cc infty in H²

[lilac barn]

Yes

[astral vine]

Okay in this case,

H²_0 is a compelte space under the norm

[untold delta]

Functionanatolysis

[astral vine]

up to there we just said crystal clear stuff

[lilac barn]

Yes. whereas my proposed alternative norm is the third quantity only. (So my question is if H²₀(U) is Banach under |D²u|₂?)

[astral vine]

Yes

usually we denote

[untold delta]

Functionanatolysis

[astral vine]

Now by density take

[untold delta]

Functionanatolysis

[astral vine]

By Poincaré inequality

[untold delta]

Functionanatolysis

[lilac barn]

Wait so u in H²₀(U) implies Du is in H¹₀(U)?

[astral vine]

everything is done for smooth compactly supported function

so yes

obviously

[lilac barn]

Oh okay okay, so I do understand this, but for my other question about whether H²₀(U) is Banach under |D^2u|, you contend that it is a no, where H²₀(U) is defined as stated above?

[astral vine]

No I said it is maybe a no

depending on how you define the space

Here we assumed U is bounded smooth open set

[lilac barn]

What about In the case of limit of compactly supported smooth functions in W^2,2(U) (and U is bounded smooth open)?

[astral vine]

Above inequality still holds by density

thus, here we have completness and equivalence of norms

but for U bounded (and smooth)

which is the very important part

You need poincaré to prove

it

[lilac barn]

Oh okay okay, then this makes sense, I guess no shortcut is being discovered by trying the equivalnecy of norm route since we still need the poincare anyways

Thanks for the help!

[quaint herald]

This shouldn't surprise you, note that if you prove the equivalence of norms (for some class of U), then an immediate corollary is Poincare for U (drop a few terms in the inequality). So you cannot expect to find some miraculous way to make your original strategy work that is not at least the same complexity as Poincare.

[twilit rover]

Can someone explain this

first equation

its from page 53 of Taylor pde volume 1.

how did d/ds change to V

Ok I understand it now, at least formally

[junior bloom]

Hey @astral vine so remember I asked about the projection $\mathcal{P}:L^2([0,T]\times\mathbf{R}^3)\to{u:\nabla_{\mathbf{x}}\cdot u=0}$.

I want to characterize an orthornormal eigenbasis generated by the operator $-\mathcal{P}\Delta$, restricted to the obvious subspace of divergence-free functions. I'm just taking this to be the Fourier kernels $\mathcal{P}\exp(-i\langle k,\cdot\rangle)$ since I'm pretty sure $L^2([0,T]\times\mathbf{R}^3)=\mathcal{P}L^2([0,T]\times\mathbf{R}^3)\oplus H$ for some subspace $H$ and $\Delta$ is linear.

This seems pretty obvious and direct to me and I just need a sanity check because it's a little too good to be true for my work.

[untold delta]

teafortwo

[astral vine]

There is no orthonormal eigen basis

[junior bloom]

I can take the subspace to be smaller, I don't even need it to be a basis.

Oh wait, I see what you're saying. I'm getting at the wrong thing.

Hmm.

[astral vine]

Orthonormal eigen basis for the Stokes Operators, only happens on (bounded) domains

But If you want an exact description of the complementary orthogonal

[junior bloom]

Nah I don't really care about that :berk:

I'm just trying to get a Galerkin system for $-\mathcal{P}\Delta$ lol

[untold delta]

teafortwo

[astral vine]

What do want to know exactly, what is the underlying goal

[astral vine]

On the whole space ?

[junior bloom]

We can take it to be periodic for simplicity.

Bounded and periodic domain.

Actrually no periodic

[astral vine]

In this case there is lot of troubles I don't mention

[junior bloom]

Hm. Well if it's simpler take it for the whole domain.

[astral vine]

the Stokes Operator involves boundary conditions

[junior bloom]

R^4.

Except there are no bases 🙀

[astral vine]

the problem is being in R³ in space

yes

There are basis on free divergence L² vector fields

but none well adapted to the Stokes operator

[junior bloom]

Now I wonder what people building finite-element methods for the basic Laplace operator have been doing.

Alright back to the drawing board, very confused.

[astral vine]

Usually they didnt mention they use some boundary conditions

but there are necessarily underlying BC

Just people are so used to it, they do not mention it anymore

Maybe @river path knows better about this kind of stuff

[junior bloom]

You know what I'm just going to cite a paper I haven't read, they build some orthonormal basis for the Galerkin method for the Laplacian

I can work out what the Galerkin method is doing on the boundary later

[astral vine]

But on domains

the Stokes operator is no longer

[untold delta]

Functionanatolysis

[astral vine]

This is no longer true on domains

[junior bloom]

I'm taking this operator as a primitive, I can deform the domain however I want.

TO match the operator.

This might sound weird but basically this operator appears in some reduction of an SPDE and it's not important the domain on which the SPDE exists, since I'm just trying to get local controls.

So I can pick any generally non-moronic domain and get something I want.

[astral vine]

local control on an operator which is absolutely non-local

taht sounds very weird

[junior bloom]

Local not in spacetime but in spacetime x probability measure on some limit.

[astral vine]

I'm not into SPDEs/Numerics so I can tell that much

[junior bloom]

I don't do numerical analysis either, I'm just pretty sure there's a convergent Galerkin system because I saw it used in a similar problem.

I don't actually know what it is in generality.

[junior bloom]

Oh to be clear I'm trying to control an object being acted on by these operators.

So if u solves P\Delta u = f for some f I want to know stuff about u, and the approach I've seen is a Galerkin decomposition of \Delta to control solutions to the Laplacian.

Then a limit to a solution of the real Laplacian.

Using an appropriate eigenbasis.

But whatever, shelving this for now.

[astral vine]

where f lies ?

in L² ?

L² free divergence vector fields ?

[junior bloom]

We can take it to be L^2_loc, Holder alpha for alpha less than 1/2.

[astral vine]

L² free divergence vector fields with 0 tangential BC ?

[junior bloom]

We can project as needed.

Yeah, let's project it onto divergence free fields.

[astral vine]

Not really

[junior bloom]

No for my problem I'm projecting as needed.

[astral vine]

because due to possible boundary conditions a pressure gradient may appear

[junior bloom]

Hm.

[astral vine]

(and except for exceptionnal boundary conditions or the whole space, it ALWAYS appear)

On domains the Projection does not commutes with the Laplacian

[junior bloom]

I don't get it. But I can choose where I project for this problem because there's a theorem that says there's some weak convergence of a type I care about.

And existence of a non-projected solution.

So I actually have P\Delta u + SomeEllipticOperator u = a very badly behaved f, if I know u solves this what bounds on energy can I get and what do they look like on a limit. Ideally the limit is on u itself as I increase the resolution of my eigenfunctionspace by adding more eigenfunctions.

[astral vine]

On what domain ?

[junior bloom]

I can choose, doesn't really matter as long as it can be increased, or it can start as infinite.

[astral vine]

It matters

because your equation maybe very illposed

(on the whole space I buy it is somewhat well defined)

[junior bloom]

I'm sure it matters for this problem as a PDEs problem but for the general context I'm working in I don't care, I can pick out the domain that makes it easiest for me, so long as it's something totally connected.

And the boundaries aren't fractal or anything.

[astral vine]

Assuming smoothness

[junior bloom]

We can do that.

[astral vine]

i buy the fact you don't care

[junior bloom]

For the boundary

Not for $f$

[untold delta]

teafortwo

[astral vine]

In fact you have to know because changing the domain/BC really change the whole structure of the equation

The behavior could be totally different

Not having smoothness on f is not really important

as long as it lies in a Lebesgue/Sobolev space

even locally in some cases

[junior bloom]

The reason boundaries don't matter for me is because I'm just trying to show (roughly speaking) that the limit of u (using an increasing resolution in a Galerkin approximation to assist) converges to a solution of something else where I can pick the boundaries too. It's actually not important what the boundary is or even if I can only show it for a particular boundary, I'm just trying to get a solution to make sense in the limit.

[astral vine]

but the eigenbasis should depend on the domain right ?

That's why I don't get it

[junior bloom]

Yep, so I need to pick out a domain and eigenbasis that work together.

[astral vine]

and choose appropriate boundary conditions

[junior bloom]

Sure.

Yeah.

[astral vine]

notice vvarious as : derivatives does not commutes with the Leray Projection

[junior bloom]

So what I'm dealing with is more compllicated than P\Delta but the claim I'm hearing from chats IRL is we can just build an eigenbasis from \Delta.

And I'm trying to work that out for the Galerkin system.

But anyways, shelved for now until I understand numerical PDEs.

[astral vine]

To make sense of it I would recommend but I am not sure, Hodge (absolute) boundary conditions, on the unitball

The Hodge Laplacian commutes with the Leray projection

but in this case the Leray projection act the following way

[untold delta]

Functionanatolysis

[junior bloom]

\nu is normal yes?

[astral vine]

yes

[junior bloom]

Alright. Think if I added the zero condition to the tangential product anything would break?

Actually never mind, I'm not sure I want that.

This works for me.

[astral vine]

The Hodge Laplacian involves two BC

the other one is

[untold delta]

Functionanatolysis

[junior bloom]

Oh huh, now I think about it I have another problem because I'm an idiot. I don't know what $\mathbb P$ does to a nowhere differentiable function.

[untold delta]

teafortwo

[astral vine]

there is still a definition of it as one the whole space but each involved operator have to be treated more carefully

L² functions are sometimes nowhere differentiable too

[junior bloom]

If I can understand it in the sense of weak derivatives that would be ideal.

[astral vine]

You cannot makes sense of it for general distributions

You need at least it to be an Lp or a not too negative-index Sobolev fonctions

For many technical reasons

[junior bloom]

Yeah, I worked through some technica like that a few weeks ago, I didn't quite grasp what it was for. I think it's to keep this projection well-formed

Does it have a meaning in H^{-1}_0?

[astral vine]

No, sadly

it makes sense at most on

[untold delta]

Functionanatolysis

[junior bloom]

rip

[astral vine]

beyond that you require boundayr conditions

below that you require to be the exact dual of some Sobolev space with boundary conditions

which is somewhat very difficult to write explicitly except in the case of the upper Half space

[junior bloom]

If I have a weak solution of the form B(u,v) = (f, v) for some elliptic linear PDE operator (weak form) B satisfying Lax-Milgram, should I be able to understand Pu in terms of applications to v just by naively tossing it in the equation?

[astral vine]

I don't think so

if v is not in Hs like #advanced-pdes message

you won't be able to makes sense of Pv

[junior bloom]

oh this is for any test function v

[astral vine]

yes

[junior bloom]

so i can presumably pick the space to be restricted, except who knows wtf that means

[astral vine]

Restriction of weak elliptic problems have to be treated carefully

I don't know if it could work

[junior bloom]

this is unreal, i dont understand these methods lmao, back to the drawing board

i need to talk to a functional analyst at my uni, what a mess

[astral vine]

What do you have against Functional Analysts

[junior bloom]

ruining my day

the hardest part of spdes is again the pdes part

the s part is basically trivial

[astral vine]

Th real difficulty in this kind of problems is always about a fairly nice treatment of the linear non random part

[junior bloom]

Actually all of my woes come from the fact that my u(x,t)'s are always Holder, nowhere differentiable, and probably singular in the infinite oscillations way, which is how random fields tend to be.

When mixed with PDEs it's a recipe for tears.

[lilac barn]

Can somebody explain how does he obtain the last inequality (goes from second equality to the third inequality)? It seems like the "constant" C is doing all the work but I am not sure how to properly justify the last quantity.
V is a compact set in U and |h| ≤ dist(V, boundary of U)

[quaint herald]

the inner integral in the second line is bounded by the rhs integral in the third line for any particular t. integrating in a t interval of length 1 you get the same upper bound, then you are just summing up n such terms which you can modify the C for.

[lilac barn]

Oh right, I always forget that for Evans the constants can change in between lines. Thanks a lot!

[quaint herald]

yeah it's not just Evans, this is pretty standard practice when you don't need to be explicit about the C.

no worries

(Also you should understand why the first sentence I said is true, and this is precisely because V was chosen so that shifting it by h units in any direction keeps it inside U.)

[lilac barn]

Yes yes, my main worry in justifying the last line was why there isn't any n in the last line which as you say is because C is modified.

[quaint herald]

Right yep that is my take.

[junior bloom]

I feel this is why the various version of \leq were introduced, like the squiggly or bowed line ones.

[obsidian wraith]

can someone give me some help with bootstraping arguments for regularity theory on parabolic pde (specifically in mild solutions)?

[astral vine]

I can.

Let me know

[tired hollow]

Hello.

Are there any serious, fruitful, contemporary approaches to PDE research that make usage of techniques from algebra, category theory, or modal theory?

[lilac barn]

Taken from: https://math.stackexchange.com/questions/2658368/how-is-category-theory-used-to-study-differential-equations

[astral vine]

Cohomology/Homology of Sobolev Spaces on domains are very important to study PDEs which have strong relations geometry. Sometimes even Homology Cobordism, and Algebraic Topology are also very important (especially for PDEs on manifolds).

There are also some links between Algebraic Geometry and some PDEs like Korteweg-de Vries

[lilac barn]

Can somebody tell me what prerequisites would one have to study (from Evans preferably) to be able to prepare for these topics?

[astral vine]

The description is too vague in my opinion to give some appropriate recommendations

[tired hollow]

Cute. 🙄

[tired hollow]

Thank you.

[lilac barn]

idm vague recommendations, I just want to prepare for these topics, where prepare means that I would be able to understand the basic terminology being used in these topics. My guess is 5,6,7,8,9 from Evans should get be equipped?

[astral vine]

Probably, you should read carefully Chapter 7, Section 7.1 and 7.4 for Navier-Stokes and it would be sufficient if you know nice properties of the convolution with the Gaussian on the whole space. Evans is quite far from Schauder-type estimates, and other mentioned topics share words at most like "Calculus of Variations" which may include very different kind of maths, and a lot is not covered by Evans.

Some stuff relies more on Numerical Analysis for instance

which is not treated by Evans

[lilac barn]

Any other textbook recommendations you have in mind? Perhaps Taylor?

[astral vine]

Books I have in mind have too much prerequirements to be very well adapted

Like deep harmonic Analysis stuff and so on

[lilac barn]

Aah, I guess then the class itself will be first covering the prerequisites because the class that it lists as a prereq only did Evans 5,6,8,9

[astral vine]

Probably yeah

As I said, the description is very vague and can concern like hundreds of topics and their different ways to cover them

For instance, just the Navier-Stokes strong and weak solutions

There are like at least 15 ways in my mind to cover it, depending on the class of Solution you are looking at, how you construct those, on domains, or on the whole space, if on domains what kind of BCs, etc...

[junior bloom]

The core component of what you want to look at is probably Hodge theory, but there are also approaches to PDEs using symmetry groups, and also geometric problems attacked via GAGA-like correspondences.

[fickle cedar]

hi is this linear or semi linear PDE ?

[twilit rover]

It is linear in u

[twilit rover]

Is (5.1.15-5.1.16) in Jost PDE wrong

on page 91

It would be correct if the integration was written dS(x)dt

Here is the reference https://antivirus.uclv.edu.cu/update/libros/Mathematics and Statistics/Partial Differential Equations - Jrgen Jost%2C 3rd ed. 2013 - 978-1-4614-4809-9.pdf

[junior bloom]

@astral vine Yet more silly questions about the space ${u:\nabla\cdot u=0}$. Is there a natural characterization for PDE operators whose evolutions that start witha state in this space remain the space for all time?

[untold delta]

teafortwo

[junior bloom]

Er, where I have $\partial_t u= \mathcal{L}u$.

[untold delta]

teafortwo

[astral vine]

What kind of caracterization ?

[junior bloom]

I'm curious if there is a nicer understanding of the class of $\mathcal{L}$s that preserve belongingness to ${u:\nabla\cdot u=0}$

[astral vine]

Domain of L being left invariant by the Helmoltz-Leray projection

[untold delta]

teafortwo

[junior bloom]

Yeah.

[astral vine]

That's the only thing you can say

[junior bloom]

Hm, alright.

[astral vine]

Other wise you can build plenty of shitty different operators

you can even Build operators that keep the the divergence free condition through the evolution, with coeffcients in the W^{-1,\infty}

you read perfectly -1

you lost a full derivative

https://arxiv.org/pdf/2011.13771.pdf

[junior bloom]

Oof. Alright, that's a good "counterexample" to my hope that these operators are nice.

Appreciated.

[astral vine]

Notice that this is already as worst as possible in the whole space

then imagine adding boundary conditions

then on poorly regular domain

everything falls completely in the range of pure madness

[junior bloom]

I give up working with boundary anythings

It's impossible

[astral vine]

It is totally doable

but it requires very different tools

[untold delta]

Amorous aka Lucifer

[tired hollow]

@gloomy thicket this doesn't look like PDE's in the slightest

[twilit rover]

I get that $H^{s}'$ can be identified with $H^{-s}$ by taking the $L^2$ inner product of Fourier transforms, but how does that prove the boundedness of the map.

[untold delta]

IlIIllIIIlllIIIIllll
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[twilit rover]

well I guess it means $M_{\phi}^* : H^{-s} \to H^{-s}$ is bounded

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

But formally M* = M

Assume we know that

[untold delta]

Functionanatolysis

[twilit rover]

It's true that $M_{\phi}^* = M_{\phi}$ on $S(\mathbb{R}^n) \subset H^{-k}$.

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

well I can see $\langle \phi u, v \rangle = \langle u, \phi v \rangle$ for $u \in H^k$, $v \in S$.

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

I guess what is left is to show that $M_{\phi}^*v = \phi v$ for $v \in H^{-k}$

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

for this I take $v_j \in S$ with $v_j \to v$ in $H^{-k}$

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

hence $v_j \to v$ in $S'$

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

and $\phi v_j \to \phi v$ in $S'$

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

but from boundedness of $M_{\phi}^*$ it follows that $M_{\phi}^v_j = \phi v_j \to M_{\phi}^ v$ in $H^{-k}$

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

hence in $S'$ also

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

so $M_{\phi}^*v = \phi v$ by uniqueness of weak limits

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

Yes

but usally people don't check that

because since Schwartz function are dense in H^{-k}

there is unique bounded extension on whole H^{-k}

so that that's only possible (reasonable) meaning of the multiplicaiton by varphi

[twilit rover]

but we need to know that the bounded extension is actually $M_{\phi}$

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

That's only possible reasonable meaining

But your proof is correct

[twilit rover]

so far all we could say is that $M_{\phi}^*$ is some bounded map on $H^{-k}$ that acts as multiplication on $S(R^n)$

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

Yes but since there is only one unique bounded extension

people don't care about it

it is a common thing in Functional Analysis/PDE

you are right trying to check it

[twilit rover]

but I don't yet know that $v \mapsto \phi v$ is bounded on $H^{-k}$.

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

that's what I am trying to prove

[astral vine]

it is for Schwartz function

and bounded extension is UNIQUE

always

[twilit rover]

right but then I need to verify that the extension is still multiplication by $\phi$

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

It cannot be somthing else

[twilit rover]

Why not

all we get is a unique extension

[astral vine]

because of boundedness

If you are worried about that, you would be worried about all PDE/F.A. papers

no one ever verify this kind of things

because non continuous linear functional are useless

(there is only few cases where it is very important)

[twilit rover]

I mean I don't even know that $M_{\phi}$ maps H^{-k}$ to itself

[untold delta]

IlIIllIIIlllIIIIllll
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[astral vine]

Mvarphi is selfajoint on L²

[twilit rover]

I only know that it has some bounded extension that does

[astral vine]

L² is dense in H^{-k}

Mvarphi * must be Mvarphi

On L²

on H^{-k}

etc.

If you can give a unique/canonical sense to some limiting procedure via density argument then you give the limit as the definition

always

that's like writing stuff like

$$\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot\xi} \dd \xi = \delta_{0}(x-y)$$

[untold delta]

Functionanatolysis

[astral vine]

It would make no sense

but that's the only possible reasonable meaning of it

[twilit rover]

it's equality as distributions

[astral vine]

even on L² functions

by limiting procedure

[astral vine]

yes

Then we don't make the disctinction on L² functions or whatever

It is always like that

if you feel bad about the fact that it is not checked, then feel free to check it

[twilit rover]

Well, usually it has been proving an estimate about a linear functional, and then silently extending it uniquely

but the thing above makes a further claim

that the extension has a certain form

[astral vine]

but A LOT of things like that are not checked because we only we want to deal with bounded operators

[twilit rover]

what exactly is not checked

that $M_{\phi}^*v = \phi v$ even for $v \in H^{-k}$?

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

Yes, because of the uniqueness of the extension

[twilit rover]

but the uniqueness doesn't imply the above statement

[astral vine]

I get your troubles

and I agree, but my point is , there is no other possible sense to it$

[twilit rover]

I think that I must go through $S'$ to obtain the above

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

Yes you compeleted somewhat the proof

but you shouldn't be worried about that

[twilit rover]

@astral vine What should I be worried about

Now I think I need Bochner integral

[quaint vapor]

This might just be a simple algebra thing, but the main question is over calculus of variations

How do they get here from the variable change?

[quaint vapor]

<@&286206848099549185>

[desert nexus]

pull the w' into the square root and distribute after substituting dw = w'dz