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Jay
Jay
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Let $u:\Omega \to \mathbb R$$u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.

How can I compute the following distributional derivative?

$$ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx.$$

Let $u:\Omega \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.

How can I compute the following distributional derivative?

$$ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx.$$

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.

How can I compute the following distributional derivative?

$$ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx.$$

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Jay
Jay
  • 109
  • 7

Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.

How can I compute the following distributional derivative?

$$ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx.$$