Boundedness of the L2 norm of the boundary derivative of the Greens function.












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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,



$intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.










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    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,



    $intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.










    share|cite|improve this question







    New contributor




    A.R.P is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      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,



      $intlimits_{Omega} ointlimits_{partial Omega}(nabla_{textbf{y}}G(textbf{x},textbf{y})cdot textbf{n})^2 dS_{textbf{y}} dtextbf{x}$.










      share|cite|improve this question







      New contributor




      A.R.P is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      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






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      share|cite|improve this question







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      asked Jan 4 at 22:46









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