If $u$ is a solution to the wave equation (Cauchy) then $|u(x,t)|le A/t$ for some $A$
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Let $u(x,t)$ be a solution for the Cauchy Problem
$$u_{tt}-Delta_xu = 0mbox{ in $mathbb{R}^3times mathbb{R}$}$$
$$u(x,0) = f(x)mbox{ in $mathbb{R}^3$}$$ $$u_t(x,0) = g(x) mbox{in
$mathbb{R}^3$}$$
where $f$ and $g$ are of class $C^3$ and $C^2$ respectively in
$mathbb{R}^3$ which are null in the complementar of a compact. Show
that there exists a constant $A$ such that
$$|u(x,t)|le A/t, xinmathbb{R}^3, tge 1$$
Find, also, an estimative for the constant $A$ in terms of $f$ and
$g$.
UPDATE:
I've found the solution 
but I need to understand why the intersection with the support is at most $4pi R^2$. As I understand, the support of the data is the support of $f$. The intersection of $S_t(x)$ with this support should be what?
I'm trying to imagine the complement of the ball $B(0,R)$ which contains the support of $f$ (is the support $g$ necessary?). I must take the intersection with $S_x(t)$ but I do not know what to do
real-analysis integration pde wave-equation
asked Nov 27 at 19:33
Lucas Zanella
92811330
add a comment |
up vote
1
down vote
favorite
Let $u(x,t)$ be a solution for the Cauchy Problem
$$u_{tt}-Delta_xu = 0mbox{ in $mathbb{R}^3times mathbb{R}$}$$
$$u(x,0) = f(x)mbox{ in $mathbb{R}^3$}$$ $$u_t(x,0) = g(x) mbox{in
$mathbb{R}^3$}$$
where $f$ and $g$ are of class $C^3$ and $C^2$ respectively in
$mathbb{R}^3$ which are null in the complementar of a compact. Show
that there exists a constant $A$ such that
$$|u(x,t)|le A/t, xinmathbb{R}^3, tge 1$$
Find, also, an estimative for the constant $A$ in terms of $f$ and
$g$.
UPDATE:
I've found the solution
but I need to understand why the intersection with the support is at most $4pi R^2$. As I understand, the support of the data is the support of $f$. The intersection of $S_t(x)$ with this support should be what?
I'm trying to imagine the complement of the ball $B(0,R)$ which contains the support of $f$ (is the support $g$ necessary?). I must take the intersection with $S_x(t)$ but I do not know what to do
real-analysis integration pde wave-equation
asked Nov 27 at 19:33
Lucas Zanella
92811330
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $u(x,t)$ be a solution for the Cauchy Problem
$$u_{tt}-Delta_xu = 0mbox{ in $mathbb{R}^3times mathbb{R}$}$$
$$u(x,0) = f(x)mbox{ in $mathbb{R}^3$}$$ $$u_t(x,0) = g(x) mbox{in
$mathbb{R}^3$}$$
where $f$ and $g$ are of class $C^3$ and $C^2$ respectively in
$mathbb{R}^3$ which are null in the complementar of a compact. Show
that there exists a constant $A$ such that
$$|u(x,t)|le A/t, xinmathbb{R}^3, tge 1$$
Find, also, an estimative for the constant $A$ in terms of $f$ and
$g$.
UPDATE:
I've found the solution
but I need to understand why the intersection with the support is at most $4pi R^2$. As I understand, the support of the data is the support of $f$. The intersection of $S_t(x)$ with this support should be what?
I'm trying to imagine the complement of the ball $B(0,R)$ which contains the support of $f$ (is the support $g$ necessary?). I must take the intersection with $S_x(t)$ but I do not know what to do
real-analysis integration pde wave-equation
asked Nov 27 at 19:33
Lucas Zanella
92811330
Let $u(x,t)$ be a solution for the Cauchy Problem
$$u_{tt}-Delta_xu = 0mbox{ in $mathbb{R}^3times mathbb{R}$}$$
$$u(x,0) = f(x)mbox{ in $mathbb{R}^3$}$$ $$u_t(x,0) = g(x) mbox{in
$mathbb{R}^3$}$$
where $f$ and $g$ are of class $C^3$ and $C^2$ respectively in
$mathbb{R}^3$ which are null in the complementar of a compact. Show
that there exists a constant $A$ such that
$$|u(x,t)|le A/t, xinmathbb{R}^3, tge 1$$
Find, also, an estimative for the constant $A$ in terms of $f$ and
$g$.
UPDATE:
I've found the solution
but I need to understand why the intersection with the support is at most $4pi R^2$. As I understand, the support of the data is the support of $f$. The intersection of $S_t(x)$ with this support should be what?
I'm trying to imagine the complement of the ball $B(0,R)$ which contains the support of $f$ (is the support $g$ necessary?). I must take the intersection with $S_x(t)$ but I do not know what to do
real-analysis integration pde wave-equation
real-analysis integration pde wave-equation
asked Nov 27 at 19:33
Lucas Zanella
92811330
asked Nov 27 at 19:33
Lucas Zanella
92811330
edited Dec 3 at 19:04
asked Nov 27 at 19:33
Lucas Zanella
92811330
asked Nov 27 at 19:33
Lucas Zanella
92811330
asked Nov 27 at 19:33
Lucas Zanella
92811330
92811330
add a comment |
add a comment |
1 Answer
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up vote
2
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We use the following fact.
- If $Csubsetmathbb R^3$ is a convex set contained in a ball $B_R$, then $mathcal H^2(partial C)leq mathcal H^2(partial B_R)=4pi R^2$.
With your notation, let $E_x(t)$ be the ball bounded by $S_x(t)$ and let $B_R$ the ball that contains $mathrm{supp}(f)$ and $mathrm{supp}(g)$. Then $E_x(t)cap B_R$ is convex and contained in $B_R$. Its surface is composed of two pieces, one of which is $partial E_x(t)cap B_R = S_x(t)cap B_R$. Therefore
$$
mathcal H^2bigl(S_x(t)cap B_Rbigr)
leq mathcal H^2bigl(partial(E_x(t)cap B_R)bigr)
leq mathcal H^2(partial B_R)=4pi R^2 .
$$
Edit
Obviously, I applied the initial fact with $C=E_x(t)cap B_R$.
answered Dec 3 at 19:18
Federico
4,192512
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
|
show 1 more comment
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
We use the following fact.
- If $Csubsetmathbb R^3$ is a convex set contained in a ball $B_R$, then $mathcal H^2(partial C)leq mathcal H^2(partial B_R)=4pi R^2$.
With your notation, let $E_x(t)$ be the ball bounded by $S_x(t)$ and let $B_R$ the ball that contains $mathrm{supp}(f)$ and $mathrm{supp}(g)$. Then $E_x(t)cap B_R$ is convex and contained in $B_R$. Its surface is composed of two pieces, one of which is $partial E_x(t)cap B_R = S_x(t)cap B_R$. Therefore
$$
mathcal H^2bigl(S_x(t)cap B_Rbigr)
leq mathcal H^2bigl(partial(E_x(t)cap B_R)bigr)
leq mathcal H^2(partial B_R)=4pi R^2 .
$$
Edit
Obviously, I applied the initial fact with $C=E_x(t)cap B_R$.
answered Dec 3 at 19:18
Federico
4,192512
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
|
show 1 more comment
up vote
2
down vote
accepted
We use the following fact.
- If $Csubsetmathbb R^3$ is a convex set contained in a ball $B_R$, then $mathcal H^2(partial C)leq mathcal H^2(partial B_R)=4pi R^2$.
With your notation, let $E_x(t)$ be the ball bounded by $S_x(t)$ and let $B_R$ the ball that contains $mathrm{supp}(f)$ and $mathrm{supp}(g)$. Then $E_x(t)cap B_R$ is convex and contained in $B_R$. Its surface is composed of two pieces, one of which is $partial E_x(t)cap B_R = S_x(t)cap B_R$. Therefore
$$
mathcal H^2bigl(S_x(t)cap B_Rbigr)
leq mathcal H^2bigl(partial(E_x(t)cap B_R)bigr)
leq mathcal H^2(partial B_R)=4pi R^2 .
$$
Edit
Obviously, I applied the initial fact with $C=E_x(t)cap B_R$.
answered Dec 3 at 19:18
Federico
4,192512
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
|
show 1 more comment
up vote
2
down vote
accepted
up vote
2
down vote
accepted
We use the following fact.
- If $Csubsetmathbb R^3$ is a convex set contained in a ball $B_R$, then $mathcal H^2(partial C)leq mathcal H^2(partial B_R)=4pi R^2$.
With your notation, let $E_x(t)$ be the ball bounded by $S_x(t)$ and let $B_R$ the ball that contains $mathrm{supp}(f)$ and $mathrm{supp}(g)$. Then $E_x(t)cap B_R$ is convex and contained in $B_R$. Its surface is composed of two pieces, one of which is $partial E_x(t)cap B_R = S_x(t)cap B_R$. Therefore
$$
mathcal H^2bigl(S_x(t)cap B_Rbigr)
leq mathcal H^2bigl(partial(E_x(t)cap B_R)bigr)
leq mathcal H^2(partial B_R)=4pi R^2 .
$$
Edit
Obviously, I applied the initial fact with $C=E_x(t)cap B_R$.
answered Dec 3 at 19:18
Federico
4,192512
We use the following fact.
- If $Csubsetmathbb R^3$ is a convex set contained in a ball $B_R$, then $mathcal H^2(partial C)leq mathcal H^2(partial B_R)=4pi R^2$.
With your notation, let $E_x(t)$ be the ball bounded by $S_x(t)$ and let $B_R$ the ball that contains $mathrm{supp}(f)$ and $mathrm{supp}(g)$. Then $E_x(t)cap B_R$ is convex and contained in $B_R$. Its surface is composed of two pieces, one of which is $partial E_x(t)cap B_R = S_x(t)cap B_R$. Therefore
$$
mathcal H^2bigl(S_x(t)cap B_Rbigr)
leq mathcal H^2bigl(partial(E_x(t)cap B_R)bigr)
leq mathcal H^2(partial B_R)=4pi R^2 .
$$
Edit
Obviously, I applied the initial fact with $C=E_x(t)cap B_R$.
answered Dec 3 at 19:18
Federico
4,192512
edited Dec 3 at 19:25
answered Dec 3 at 19:18
Federico
4,192512
answered Dec 3 at 19:18
Federico
4,192512
answered Dec 3 at 19:18
Federico
4,192512
4,192512
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
|
show 1 more comment
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
What is $mathcal H^2$?
– Lucas Zanella
Dec 3 at 19:19
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Two-dimensional Hausdorff measure. The surface area
– Federico
Dec 3 at 19:20
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Why is $C$, that is, $S_t(x)$ in my example, contained in $B_R$? I thought it was not necessairy contained
– Lucas Zanella
Dec 3 at 19:22
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
Read carefully. Where did I say that $S_x(t)subset B_R$?
– Federico
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
I thought $C$ was $S_x(t)$
– Lucas Zanella
Dec 3 at 19:23
|
show 1 more comment
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