Limit of the average of the laplacian of a test function and a continuous function
$begingroup$
Let $f$ be a continuous function, and $B(x,r)$ the ball of center $x$ and radius $r>0$ in $ mathbb{R}^{m} $ with $mgeq1$. We know that there exists a test function $phi_{r}(t)$ that is between 0 and 1, its support is in $B(x,r)$ and is equal to 1 on the closure of $B(x,r/2)$. Let $theta$ be a smooth function whose Laplacian is identically 1 everywhere. What is
$$lim_{rto 0}frac{1}{lambda r^{m}}int_{B(x,r)}f(t)Delta(phi_{r}(t)theta(t))dt?$$ ( $lambda$ is the volume of the unit ball in $ mathbb{R}^{m} $ and $Delta$ is the laplacian)
real-analysis distribution-theory
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
$endgroup$
add a comment |
$begingroup$
Let $f$ be a continuous function, and $B(x,r)$ the ball of center $x$ and radius $r>0$ in $ mathbb{R}^{m} $ with $mgeq1$. We know that there exists a test function $phi_{r}(t)$ that is between 0 and 1, its support is in $B(x,r)$ and is equal to 1 on the closure of $B(x,r/2)$. Let $theta$ be a smooth function whose Laplacian is identically 1 everywhere. What is
$$lim_{rto 0}frac{1}{lambda r^{m}}int_{B(x,r)}f(t)Delta(phi_{r}(t)theta(t))dt?$$ ( $lambda$ is the volume of the unit ball in $ mathbb{R}^{m} $ and $Delta$ is the laplacian)
real-analysis distribution-theory
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
$endgroup$
add a comment |
$begingroup$
Let $f$ be a continuous function, and $B(x,r)$ the ball of center $x$ and radius $r>0$ in $ mathbb{R}^{m} $ with $mgeq1$. We know that there exists a test function $phi_{r}(t)$ that is between 0 and 1, its support is in $B(x,r)$ and is equal to 1 on the closure of $B(x,r/2)$. Let $theta$ be a smooth function whose Laplacian is identically 1 everywhere. What is
$$lim_{rto 0}frac{1}{lambda r^{m}}int_{B(x,r)}f(t)Delta(phi_{r}(t)theta(t))dt?$$ ( $lambda$ is the volume of the unit ball in $ mathbb{R}^{m} $ and $Delta$ is the laplacian)
real-analysis distribution-theory
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
$endgroup$
Let $f$ be a continuous function, and $B(x,r)$ the ball of center $x$ and radius $r>0$ in $ mathbb{R}^{m} $ with $mgeq1$. We know that there exists a test function $phi_{r}(t)$ that is between 0 and 1, its support is in $B(x,r)$ and is equal to 1 on the closure of $B(x,r/2)$. Let $theta$ be a smooth function whose Laplacian is identically 1 everywhere. What is
$$lim_{rto 0}frac{1}{lambda r^{m}}int_{B(x,r)}f(t)Delta(phi_{r}(t)theta(t))dt?$$ ( $lambda$ is the volume of the unit ball in $ mathbb{R}^{m} $ and $Delta$ is the laplacian)
real-analysis distribution-theory
real-analysis distribution-theory
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
edited Dec 19 '18 at 6:28
M. Rahmat
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
asked Dec 19 '18 at 6:12
M. RahmatM. Rahmat
291212
291212
add a comment |
add a comment |
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