Fdtxjr

Should I take out a loan for a friend to invest on my behalf?

Conservation of Mass and Energy

What's wrong with this bogus proof?

Are tamper resistant receptacles really safer?

Why the color red for the Republican Party

Is it possible to avoid unpacking when merging Association?

PTIJ: Should I kill my computer after installing software?

In the late 1940’s to early 1950’s what technology was available that could melt a LOT of ice?

How to write ı (i without dot) character in pgf-pie

How does one describe somebody who is bi-racial?

Latex does not go to next line

Single word request: Harming the benefactor

Can I pump my MTB tire to max (55 psi / 380 kPa) without the tube inside bursting?

Examples of a statistic that is not independent of sample's distribution?

How can The Temple of Elementary Evil reliably protect itself against kinetic bombardment?

Why is computing ridge regression with a Cholesky decomposition much quicker than using SVD?

Is "conspicuously missing" or "conspicuously" the subject of this sentence?

Database Backup for data and log files

Bash script should only kill those instances of another script's that it has launched

Recommendation letter by significant other if you worked with them professionally?

Plausibility of Mushroom Buildings

How can I get players to stop ignoring or overlooking the plot hooks I'm giving them?

How are instrumentation amplifiers constructed on the semiconductor level?

Should I tell my boss the work he did was worthless

Can this operator taking boundary conditions for the wave equation be extended to $L^p$?

Weak holomorphicity implies smooth and holomorphic.Sequence of solutions to heat equationProving that the smooth, compactly supported functions are dense in $L^2$.What is the motivation for “continuity in the sense of distributions”?notations in navier stokes equationA question on convergence in Sobolev norm Vs convergence at a point of isolated discontinuityTsirelson norm constructionWeak solutions to parabolic equations - Continuous dependence of weak solution in timeContinuity of fourier transform with complex argumentFourier transform $mathcal F colon (mathcal S(mathbb R^d), lVert cdot rVert_1) to L^1(mathbb R^d)$ unbounded?

5

4

$begingroup$

We denote by $mathcalD(mathbbR^3)$ the collection smooth function in $mathbbR^3$ which are compactly supported.

Fix $t>0$ and consider the linear operator $L:mathcalD(mathbbR^3) to mathcalD(mathbbR^3)$ which maps $fmapsto partial_tu(cdot, t)$ where $u$ is a solution to
$$
begincases
u_tt - Delta u = 0\
u(x, 0) = 0\
u_t(x,0) = f.
endcases
$$
For $p>1$, $pneq 2$ (for $p=2$ the result is well known), can $L$ be extended to a map from $L^p(mathbbR^3)to L^p(mathbbR^3)$?

I am not quite sure how to approach the problem, but I figured I should look for a sequence of funtions $(f_n)subseteq mathcalD(mathbbR^3)$ such that $lVert f_n rVert_L^p(mathbbR^3)$ is uniformly bounded but
$$lVert partial_t u_n(cdot, t)rVert_L^p(mathbbR^3)toinfty.$$
I'm not quite sure how to find such a sequence or if this is even the correct approach. Any input is appreciated!

real-analysis functional-analysis pde

share|cite|improve this question

asked 2 days ago

QuokaQuoka

1,331313

$endgroup$

add a comment | 

5

4

$begingroup$

We denote by $mathcalD(mathbbR^3)$ the collection smooth function in $mathbbR^3$ which are compactly supported.

Fix $t>0$ and consider the linear operator $L:mathcalD(mathbbR^3) to mathcalD(mathbbR^3)$ which maps $fmapsto partial_tu(cdot, t)$ where $u$ is a solution to
$$
begincases
u_tt - Delta u = 0\
u(x, 0) = 0\
u_t(x,0) = f.
endcases
$$
For $p>1$, $pneq 2$ (for $p=2$ the result is well known), can $L$ be extended to a map from $L^p(mathbbR^3)to L^p(mathbbR^3)$?

I am not quite sure how to approach the problem, but I figured I should look for a sequence of funtions $(f_n)subseteq mathcalD(mathbbR^3)$ such that $lVert f_n rVert_L^p(mathbbR^3)$ is uniformly bounded but
$$lVert partial_t u_n(cdot, t)rVert_L^p(mathbbR^3)toinfty.$$
I'm not quite sure how to find such a sequence or if this is even the correct approach. Any input is appreciated!

real-analysis functional-analysis pde

share|cite|improve this question

asked 2 days ago

QuokaQuoka

1,331313

$endgroup$

add a comment | 

$begingroup$

We denote by $mathcalD(mathbbR^3)$ the collection smooth function in $mathbbR^3$ which are compactly supported.

Fix $t>0$ and consider the linear operator $L:mathcalD(mathbbR^3) to mathcalD(mathbbR^3)$ which maps $fmapsto partial_tu(cdot, t)$ where $u$ is a solution to
$$
begincases
u_tt - Delta u = 0\
u(x, 0) = 0\
u_t(x,0) = f.
endcases
$$
For $p>1$, $pneq 2$ (for $p=2$ the result is well known), can $L$ be extended to a map from $L^p(mathbbR^3)to L^p(mathbbR^3)$?

I am not quite sure how to approach the problem, but I figured I should look for a sequence of funtions $(f_n)subseteq mathcalD(mathbbR^3)$ such that $lVert f_n rVert_L^p(mathbbR^3)$ is uniformly bounded but
$$lVert partial_t u_n(cdot, t)rVert_L^p(mathbbR^3)toinfty.$$
I'm not quite sure how to find such a sequence or if this is even the correct approach. Any input is appreciated!

real-analysis functional-analysis pde

share|cite|improve this question

asked 2 days ago

QuokaQuoka

1,331313

$endgroup$

share|cite|improve this question

asked 2 days ago

QuokaQuoka

1,331313

share|cite|improve this question

asked 2 days ago

QuokaQuoka

1,331313

asked 2 days ago

QuokaQuoka

1,331313

asked 2 days ago

QuokaQuoka

1,331313