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We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if . Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate
for every u ∈ EXP(). Similarly, we consider the distance from in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then
for every u ∈ EXP(). We also prove that...
For conductivity problems in dimension N = 2, we prove a variant of a classical result: if a sequence of matrices H-converges to (or in other terms if converges to in the sense of homogenization) and if tends to a.e., then one has .
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