ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35 Kalita, Evgeni, Inst. Appl. Math. Mech., Donetsk, Ukraine New class of a priori estimates for nonlinear elliptic equations and systems of high order We consider nonlinear elliptic systems of the form div^{m}A(x, D^{m}u) = f(x), x ∈ R^{n}, with the natural energetic space W_{2}^{m} and standard structure conditions. For any a ∈ (0, a^{∗}) where a^{∗} ∈ (0, n) defined explicitely by the modulus of ellipticity of system we establish the estimate

∫ \frac{|D^{m}u(x)|^{2}}{ω(x)} ≤ c ∫ \frac{|I_{m}f(x)|^{2}}{ω(x)} , ω(x) = 1 + I_{n−a}σ(x),

with arbitrary nonnegative finite Borel measure σ on R^{n}, constant c does not depend on σ; I_{t} is the fractional integral of order t. Applications to solvability in spaces that are weaker than the natural energetic space of system will be presented. In particular, for any p > n/(n + m) the explicit restriction on the modulus of ellipticity provides the solvability of system for f ∈ L_{p}, p ≥ 1, and for f ∈ H_{p} (Hardy spaces), p ≤ 1, is given (known results provide solvability only for p = 2n/(n + 2m)).