Grisvard's Shift Theorem Near L∞ and Yudovich Theory on Polygonal Domains

Let Ω⊂R2 be a bounded, simply connected domain with boundary ∂Ω of class C 1,1 except at finitely many points S j where ∂Ω is locally a corner of aperture α j ≤ π2 . Improving on results of Grisvard [13, 14], we show that the solution GΩ f to the Dirichlet problem on Ω with data f ∈ Lp(Ω) and homogeneous boundary conditions satisfies the estimates ‖GΩ f ‖W2,p(Ω) ≤ Cp‖ f ‖Lp(Ω), ∀ 2≤ p <∞, ‖DGΩ f ‖ExpL1(Ω) ≤ C‖ f ‖L∞(Ω). The proof uses sharp Lp bounds for singular integrals on power weighted spaces inspired by the work of Buckley [5]. Our results allow for the extension of the Yudovich theory [31, 32] of existence, uniqueness and regularity of weak solutions to the Euler equations on Ω× (0,T) to polygonal domains Ω as above.