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- Karakhanyan, Aram L., et al.
(författare)
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Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization
- 2016
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Ingår i: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 55:6
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Tidskriftsartikel (refereegranskat)abstract
- We consider the intersection of a convex surface Gamma with a periodic perforation of R-d, which looks like a sieve, given by T epsilon = boolean OR(d)(k is an element of Z) {epsilon k + a epsilon T} where T is a given compact set and a epsilon << epsilon is the size of the perforation in the epsilon-cell (0, epsilon)(d) subset of R-d. When epsilon tends to zero we establish uniform estimates for p- capacity, 1 < p < d, of the set Gamma n T-epsilon. Additionally, we prove that the intersections Gamma boolean AND {epsilon k + a(epsilon)T}(k) are uniformly distributed over Gamma and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of Gamma and the perforations, in a given bounded domain, is homogenizable when p < 1+ d/4. This result is new even for the classical Laplace operator.
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- Karakhanyan, Aram L., et al.
(författare)
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On a conjecture of De Giorgi related to homogenization
- 2017
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Ingår i: Annali di Matematica Pura ed Applicata. - : Springer. - 0373-3114 .- 1618-1891. ; 196:6, s. 2167-2183
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Tidskriftsartikel (refereegranskat)abstract
- For a periodic vector field F, let solve the dynamical system dX(epsilon)/dt = F (X-epsilon/epsilon). In (Set Valued Anal 2(1-2):175-182, 1994) Ennio De Giorgi enquiers whether from the existence of the limit one can conclude that . Our main result settles this conjecture under fairly general assumptions on F, which in some cases may also depend on t-variable. Once the above problem is solved, one can apply the result to the corresponding transport equation, in a standard way. This is also touched upon in the text to follow.
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