Boundedness of the L2 norm of the boundary derivative of the Greens function.
$begingroup$
I wish to prove that an integral is bounded. For $delta in (0,1)$ and $textbf{c} in (delta - 1, 1-delta)^2$ we define our domain to be $Omega := (-1,1)^2 setminus B_delta (textbf{c})$. Let $textbf{n}in mathbb{S}^1$ be the outward pointing normal to $partial Omega$.
Let $G$ be the Greens function for the Laplace operator, which means that
$int_{Omega} Delta G(textbf{x},textbf{y})f(textbf{y}) dtextbf{y} = f(textbf{x})$.
I wish to prove that the following integral is bounded,
$intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.
calculus multivariable-calculus greens-function
asked Jan 4 at 22:46
A.R.PA.R.P
11
$endgroup$
add a comment |
$begingroup$
I wish to prove that an integral is bounded. For $delta in (0,1)$ and $textbf{c} in (delta - 1, 1-delta)^2$ we define our domain to be $Omega := (-1,1)^2 setminus B_delta (textbf{c})$. Let $textbf{n}in mathbb{S}^1$ be the outward pointing normal to $partial Omega$.
Let $G$ be the Greens function for the Laplace operator, which means that
$int_{Omega} Delta G(textbf{x},textbf{y})f(textbf{y}) dtextbf{y} = f(textbf{x})$.
I wish to prove that the following integral is bounded,
$intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.
calculus multivariable-calculus greens-function
asked Jan 4 at 22:46
A.R.PA.R.P
11
$endgroup$
add a comment |
$begingroup$
I wish to prove that an integral is bounded. For $delta in (0,1)$ and $textbf{c} in (delta - 1, 1-delta)^2$ we define our domain to be $Omega := (-1,1)^2 setminus B_delta (textbf{c})$. Let $textbf{n}in mathbb{S}^1$ be the outward pointing normal to $partial Omega$.
Let $G$ be the Greens function for the Laplace operator, which means that
$int_{Omega} Delta G(textbf{x},textbf{y})f(textbf{y}) dtextbf{y} = f(textbf{x})$.
I wish to prove that the following integral is bounded,
$intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.
calculus multivariable-calculus greens-function
asked Jan 4 at 22:46
A.R.PA.R.P
11
$endgroup$
I wish to prove that an integral is bounded. For $delta in (0,1)$ and $textbf{c} in (delta - 1, 1-delta)^2$ we define our domain to be $Omega := (-1,1)^2 setminus B_delta (textbf{c})$. Let $textbf{n}in mathbb{S}^1$ be the outward pointing normal to $partial Omega$.
Let $G$ be the Greens function for the Laplace operator, which means that
$int_{Omega} Delta G(textbf{x},textbf{y})f(textbf{y}) dtextbf{y} = f(textbf{x})$.
I wish to prove that the following integral is bounded,
$intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.
calculus multivariable-calculus greens-function
calculus multivariable-calculus greens-function
asked Jan 4 at 22:46
A.R.PA.R.P
11
asked Jan 4 at 22:46
A.R.PA.R.P
11
asked Jan 4 at 22:46
A.R.PA.R.P
11
asked Jan 4 at 22:46
A.R.PA.R.P
11
asked Jan 4 at 22:46
A.R.PA.R.P
11
11
add a comment |
add a comment |
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