Fdtxjr

Multi tool use

Where is the License file location for Identity Server in Sitecore 9.1?

Do I need a return ticket to Canada if I'm a Japanese National?

ESPP--any reason not to go all in?

Create chunks from an array

How to recover against Snake as a heavyweight character?

Why do phishing e-mails use faked e-mail addresses instead of the real one?

How do you make a gun that shoots melee weapons and/or swords?

Why does a car's steering wheel get lighter with increasing speed

Draw this image in the TIKZ package

How can I portion out frozen cookie dough?

Should I file my taxes? No income, unemployed, but paid 2k in student loan interest

What would be the most expensive material to an intergalactic society?

How strong is the axiom of well-ordered choice?

Who has more? Ireland or Iceland?

Help! My Character is too much for her story!

Can I negotiate a patent idea for a raise, under French law?

Should I apply for my boss's promotion?

Precision notation for voltmeters

Was this cameo in Captain Marvel computer generated?

Tabular environment - text vertically positions itself by bottom of tikz picture in adjacent cell

Having the player face themselves after the mid-game

An Undercover Army

Short story about an infectious indestructible metal bar?

I am the light that shines in the dark

Inequality between infinity norm of Laplacian and Hessian

integral of Laplacian of a positive functionHow to compute Bochner laplacian $Delta=nabla^*nabla=sum nabla_e_i$?connection laplacian on general vector bundles$Delta_L(textim,delta^*_g)subsettextim,delta^*_g$ and $Delta_Lbig(textker,textBian(g)big)subsettextker,textBian(g)$?Smooth extension of a tangent vectorCommuting Laplacian and DivergenceSupremum of Operator norm of the differential of a co-ordinate mapping for a Riemannian manifoldSobolev spaces on Riemannian manifold and the LaplacianInequality of the laplacian involving the Ricci curvatureLaplacian is the trace of Hessian in local coordinates

0

$begingroup$

Let $M$ be a smooth compact riemannian manifold with Levi Civita connection and consider a smooth function $f: M to mathbbR$. Then the Laplacian of $f$
$$ Delta f = textdiv ( textgrad f) $$
end the Hessian of $f$
$$ nabla^2 f (X,Y) = langle nabla_X textgrad f , Y rangle $$
are well defined. I want to prove the inequality
$$ | Delta f|_infty le C| nabla^2 f |$$
for some positive constant $C$. I'm using
$$ | Delta f|_infty = sup_x in M |Delta f| quad quad |nabla^2 f|_infty = sup_x in M Biggl ( sup_u,v in T_xM setminus 0 frac Biggr ) $$

How can I prove it?

Looking at the problem in an abstract way, maybe I can consider a general $tau in T_2^0(M)$ and since the laplacian is the trace of the hessian I have

$$ |texttr(tau)|_infty = sup_x in M |texttr(tau(x))|= sup_x in M tau_ij(x)g^ij(x) le sup_x in M biggl ( sum_i,jtau_ij(x) max_ij g^ij(x) biggr ) =$$
$$ sup_x in M biggl ( max_ijg^ij(x) sum_i,j tau_ij(x) biggr ) = biggl ( sup_x in M max_i,j g^ij(x) biggr ) sup_x in M sum_i,jtau_ij(x) lesssim biggl ( sup_x in M max_i,j g^ij(x) biggr ) |tau|_infty$$
where the last inequality is due to the fact that all norms are equivalent in finite dimensional spaces. Moreover the quantity
$$ biggl ( sup_x in M max_i,j g^ij(x) biggr )$$
is finite being it the supremum of a continuous function on a compact set.

differential-geometry laplacian

share|cite|improve this question

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

asked 15 hours ago

Bremen000Bremen000

448210

$endgroup$

add a comment | 

0

$begingroup$

Let $M$ be a smooth compact riemannian manifold with Levi Civita connection and consider a smooth function $f: M to mathbbR$. Then the Laplacian of $f$
$$ Delta f = textdiv ( textgrad f) $$
end the Hessian of $f$
$$ nabla^2 f (X,Y) = langle nabla_X textgrad f , Y rangle $$
are well defined. I want to prove the inequality
$$ | Delta f|_infty le C| nabla^2 f |$$
for some positive constant $C$. I'm using
$$ | Delta f|_infty = sup_x in M |Delta f| quad quad |nabla^2 f|_infty = sup_x in M Biggl ( sup_u,v in T_xM setminus 0 frac Biggr ) $$

How can I prove it?

Looking at the problem in an abstract way, maybe I can consider a general $tau in T_2^0(M)$ and since the laplacian is the trace of the hessian I have

$$ |texttr(tau)|_infty = sup_x in M |texttr(tau(x))|= sup_x in M tau_ij(x)g^ij(x) le sup_x in M biggl ( sum_i,jtau_ij(x) max_ij g^ij(x) biggr ) =$$
$$ sup_x in M biggl ( max_ijg^ij(x) sum_i,j tau_ij(x) biggr ) = biggl ( sup_x in M max_i,j g^ij(x) biggr ) sup_x in M sum_i,jtau_ij(x) lesssim biggl ( sup_x in M max_i,j g^ij(x) biggr ) |tau|_infty$$
where the last inequality is due to the fact that all norms are equivalent in finite dimensional spaces. Moreover the quantity
$$ biggl ( sup_x in M max_i,j g^ij(x) biggr )$$
is finite being it the supremum of a continuous function on a compact set.

differential-geometry laplacian

share|cite|improve this question

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

asked 15 hours ago

Bremen000Bremen000

448210

$endgroup$

add a comment | 

$begingroup$

Let $M$ be a smooth compact riemannian manifold with Levi Civita connection and consider a smooth function $f: M to mathbbR$. Then the Laplacian of $f$
$$ Delta f = textdiv ( textgrad f) $$
end the Hessian of $f$
$$ nabla^2 f (X,Y) = langle nabla_X textgrad f , Y rangle $$
are well defined. I want to prove the inequality
$$ | Delta f|_infty le C| nabla^2 f |$$
for some positive constant $C$. I'm using
$$ | Delta f|_infty = sup_x in M |Delta f| quad quad |nabla^2 f|_infty = sup_x in M Biggl ( sup_u,v in T_xM setminus 0 frac Biggr ) $$

How can I prove it?

Looking at the problem in an abstract way, maybe I can consider a general $tau in T_2^0(M)$ and since the laplacian is the trace of the hessian I have

$$ |texttr(tau)|_infty = sup_x in M |texttr(tau(x))|= sup_x in M tau_ij(x)g^ij(x) le sup_x in M biggl ( sum_i,jtau_ij(x) max_ij g^ij(x) biggr ) =$$
$$ sup_x in M biggl ( max_ijg^ij(x) sum_i,j tau_ij(x) biggr ) = biggl ( sup_x in M max_i,j g^ij(x) biggr ) sup_x in M sum_i,jtau_ij(x) lesssim biggl ( sup_x in M max_i,j g^ij(x) biggr ) |tau|_infty$$
where the last inequality is due to the fact that all norms are equivalent in finite dimensional spaces. Moreover the quantity
$$ biggl ( sup_x in M max_i,j g^ij(x) biggr )$$
is finite being it the supremum of a continuous function on a compact set.

differential-geometry laplacian

share|cite|improve this question

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

asked 15 hours ago

Bremen000Bremen000

448210

$endgroup$

share|cite|improve this question

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

asked 15 hours ago

Bremen000Bremen000

448210

share|cite|improve this question

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

asked 15 hours ago

Bremen000Bremen000

448210

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

edited 12 hours ago

Rodrigo de Azevedo

13.1k41960

asked 15 hours ago

Bremen000Bremen000

448210

asked 15 hours ago

Bremen000Bremen000

448210