Generalized coarea formula and fractal sets

Abstract

For any domain ω of R^N (N≥1), the classGC (ω) of functionals Λ:L ¹ (ω) → [0, +∞] fulfilling the followinggeneralized coarea formula is introduced:

• (1) Λ(u) = ∫_R Λ(H _s (u))ds (≤ + ∞) ∀u ∈L ¹(Ω) whereH _s (ξ)=0 if ξ <s andH _s (ξ)=1 if ξ ≥s, for any ξ,s ∈ R. Examples are

• (2) $V\left( u \right): = \int_\Omega {\left| {\nabla u} \right|: = \mathop {sup}\limits_{\eta \in C_c^1 \left( \Omega \right)^N ,\left| \eta \right| \leqslant 1} \int_\Omega {u div \eta dx} }$,

• (3) $\Lambda _r \left( u \right): = \int {\int_{\Omega ^2 } {\left| {u\left( x \right) - u\left( y \right)} \right| \cdot \left| {x - y} \right|^{ - \left( {N + r} \right)} dxdy \forall r \in \left] {0,1} \right[} }$,

• (4) $\tilde \Lambda _r \left( u \right): = \int {\int_{\Omega \times R^ + } {\left( {\mathop {ess sup u}\limits_{B_h \left( x \right) \cap \Omega } - \mathop { ess inf u}\limits_{B_h \left( x \right) \cap \Omega } } \right)h^{ - \left( {1 + r} \right)} dxdh \forall r \in \left] {0,1} \right[} }$.

The main properties of this class of functionals are here investigated.\gL _r and$\tilde \Lambda _r$ also allow us to construct two new definitions offractional dimension for set boundaries. Applications to models ofsurface tension effects in two-phase systems are then outlined. In particular,\gL _r and$\tilde \Lambda _r$ allow us to represent very irregular phase interfaces.