GRISVARD'S SHIFT THEOREM NEAR L-infinity AND YUDOVICH THEORY ON POLYGONAL DOMAINS

Let Omega subset of R-2 be a bounded, simply connected domain with boundary partial derivative Omega of class C-1,C-1 except at finitely many points S-j where partial derivative Omega is locally a corner of aperture alpha(j) <= pi/2. Improving on results of Grisvard [J. Monogr. Stud. Math. 24, Pitman, Boston, MA, 1985; J. Math. Pures Appl., 74 (1995), pp. 3-33], we show that the solution G(Omega)f to the Dirichlet problem on Omega with data f is an element of L-p(Omega) and homogeneous boundary conditions satisfies the estimates parallel to G(Omega)f parallel to(W2),(p(Omega)) <= Cp parallel to f parallel to L-p(Omega) for all 2 <= p < infinity, parallel to D(2)G(Omega)f parallel to(ExpL1(Omega)) <= C parallel to f parallel to L-infinity(Omega). The proof uses sharp L-p bounds for singular integrals on power weighted spaces inspired by the work of Buckley [Trans. Amer. Math. Soc., 340 (1993), pp. 253-272]. Our results lead to the extension of the Yudovich theory [V. I. Yudovich, Z. Vycisl. Mat. i Mat. Fiz., 3 (1963), pp. 1032-1066; Math. Res. Lett., 2 (1995), pp. 27-38] of existence, uniqueness, and regularity of weak solutions to the Euler equations on Omega x (0, T) to polygonal domains Omega as above