9 problems

Bouleau–Hirsch energy image density conjecture. Every Dirichlet structure satisfies the energy image density property.

Energy measure singularity dichotomy conjecture. The energy measures associated with the Dirichlet form are singular with respect to the reference measure $m$.

Equivalence conjecture. For all finitely ramified fractals, the condition

Let $F$ be a finitely ramified fractal considered in the paper, let $\nu$ be its energy measure, and let $Z(x)$ be the matrix defining the weak gradient metric. The generalized Rie…

Let $K$ be the Sierpiński gasket. For a nonconstant harmonic function $h$ on $K$, let $uh=muh/muh(K)$ be its normalized energy measure. For nonconstant harmonic functions $h1,h2$…

Let a metric measure space carry a $p$-energy with $p$-walk dimension $\beta_p$, and let its associated $p$-energy measure be defined relative to the underlying measure. Singularit…

Let $(K,d,m)$ be the Sierpiński carpet, let $d_{\mathrm{ARC}}(K,d)$ be its Ahlfors regular conformal dimension, and suppose this dimension is attained. Let $\mathcal{V}_0$ be the b…

Let $(K,d,m)$ be the Sierpiński carpet, let $p=d_{\mathrm{ARC}}(K,d)$ be an attained Ahlfors regular conformal dimension, and let $\Gamma_p\langle f\rangle$ denote the $p$-energy m…

Let $u$ be a solution of the Euler equation developing a locally self-similar singularity at $(x_0,T)$, and let $\mathcal{E}_T$ be the limiting energy measure at time $T$. Write…