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Michael Hardy
Michael Hardy
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Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = dist(x, \partial \Omega)$$ \delta(x) = \operatorname{dist}(x, \partial \Omega)$. For $ x \in \Omega$ (but i am interested in the case where $ x$ is close to the boundary) consider $$ I(u)(x) =\int_{ y \in \Omega, \delta(x) \le |y-x| \le 1} \frac{ \nabla u(x) \cdot (y-x)}{|x-y|^{N+1}} dy$$$$ I(u)(x) =\int_{ y \in \Omega, \delta(x) \le |y-x| \le 1} \frac{ \nabla u(x) \cdot (y-x)}{|x-y|^{N+1}} \, dy$$. I am interested in trying to estimate $I(u)(x)$ for $x$ near the boundary.

Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = dist(x, \partial \Omega)$. For $ x \in \Omega$ (but i am interested in the case where $ x$ is close to the boundary) consider $$ I(u)(x) =\int_{ y \in \Omega, \delta(x) \le |y-x| \le 1} \frac{ \nabla u(x) \cdot (y-x)}{|x-y|^{N+1}} dy$$. I am interested in trying to estimate $I(u)(x)$ for $x$ near the boundary.

Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = \operatorname{dist}(x, \partial \Omega)$. For $ x \in \Omega$ (but i am interested in the case where $ x$ is close to the boundary) consider $$ I(u)(x) =\int_{ y \in \Omega, \delta(x) \le |y-x| \le 1} \frac{ \nabla u(x) \cdot (y-x)}{|x-y|^{N+1}} \, dy$$. I am interested in trying to estimate $I(u)(x)$ for $x$ near the boundary.

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Math604
Math604
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Estimating a potential

Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = dist(x, \partial \Omega)$. For $ x \in \Omega$ (but i am interested in the case where $ x$ is close to the boundary) consider $$ I(u)(x) =\int_{ y \in \Omega, \delta(x) \le |y-x| \le 1} \frac{ \nabla u(x) \cdot (y-x)}{|x-y|^{N+1}} dy$$. I am interested in trying to estimate $I(u)(x)$ for $x$ near the boundary.