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I just started reading about fractional Laplacian. I am curious on the following questions

  1. Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value property if not whether inequality in one side holds always?

  2. Also if one has $f(x)\in\mathbb{R}^n$ and $g(y)\in\mathbb{R}$ then whether $(-\Delta)^s(fg)=(-\Delta)^sf\cdot g+(-\Delta)^sg\cdot f$ holds? I know this holds for usual Laplacian whereas doesn't hold for higher-order Laplacian.

I am trying to build some intuition on fractional Laplacian so any comment or reference will be very helpful.

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