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Riku
Riku
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Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (as it would be in the case $Du = D^{jump} u$) but of "fractal type" (for example Koch curve)? Or maybe many jump discontinuities across Koch-type curves oriented in the same direction?

If this is the case, how can one make this intuition rigorous?

 

Related questions are asked in Heuristic and graphic representation of BV functions and their singularities and Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative.

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (as it would be in the case $Du = D^{jump} u$) but of "fractal type" (for example Koch curve)? Or maybe many jump discontinuities across Koch-type curves oriented in the same direction?

If this is the case, how can one make this intuition rigorous?

Related questions are asked in Heuristic and graphic representation of BV functions and their singularities and Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative.

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (as it would be in the case $Du = D^{jump} u$) but of "fractal type" (for example Koch curve)? Or maybe many jump discontinuities across Koch-type curves oriented in the same direction?

If this is the case, how can one make this intuition rigorous?

 

Related questions are asked in Heuristic and graphic representation of BV functions and their singularities and Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative.

Source Link
Riku
Riku
  • 839
  • 6
  • 17

Alberti rank-one theorem and irregular jump discontinuities

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (as it would be in the case $Du = D^{jump} u$) but of "fractal type" (for example Koch curve)? Or maybe many jump discontinuities across Koch-type curves oriented in the same direction?

If this is the case, how can one make this intuition rigorous?

Related questions are asked in Heuristic and graphic representation of BV functions and their singularities and Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative.