ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35, 43, 47 Muscalu, Camil, Brown University, USA A Harnack Type Theorem for Elliptic Systems with L^{∞} Coefficients The following theorem will be presented: “ Let Ω ⊆ ℝ^{n} be a domain, m ∈ ℕ and λ, q > 0. Then, there exists r(= r(λ, q)) > 1 such that for every 0 < p < q, whenever u⃗_{1}, u⃗_{2}, ..., u⃗_{m} are weak solutions of a strongly elliptic system with m equations of ellipticity λ satisfying (u⃗_{1}u⃗_{2}...u⃗_{m}) ∈ P_{r} a. e. and Ω′ ⊆ Ω subdomain, the following inequalities hold:

sup_{1≤j≤m} ∥|u⃗_{j}|∥_{L^{q}(Ω′)} ≤ C inf_{1≤j≤m} ∥|u⃗_{j}|∥_{L^{p}(Ω′)}

sup_{1≤j≤m} ∥|u⃗_{j}|^{−1}∥_{L^{q}(Ω′)} ≤ C inf_{1≤j≤m} ∥|u⃗_{j}|^{−1}∥_{L^{p}(Ω′)}

where C(= C(n, m, λ, q, p, Ω, Ω′)) is a positive constant. ” Here “(u⃗_{1}u⃗_{2}...u⃗_{m}) ∈ P_{r}” means that the matrix whose j-th column is u⃗_{j} satisfies a certain “uniform positiveness” assumption. Examples concerning the sharpness of the result will also be discussed. Reference: C. Muscalu: On Harnack Principle for Strongly Elliptic Systems with Non-smooth Coefficients, preprint March 1998.