LetD be a strictly pseudoconvex domain in Cⁿ. We prove that $\bar \partial u = \phi , \phi $ , ϕ a $\bar \partial $ (0,1)-form, admits solutions in L^p(∂D), 1≤p<∞ and in BMO, under certain Wolff type conditions of ϕ. Some such results (for 1< p<∞) have previously been obtained by Amar in the ball, but under slightly stronger hypotheses. As a corollary we obtain a H^p-corona result for two generators.

A function G in a Bergman space A^p, 0< p<∞, in the unit disk D is called A^p-inner if |G|^p−1 annihilates all bounded harmonic functions in D. Extending a recent result by Hedenmalm for p=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $\Delta \Phi = \chi _D (|G|^p - 1),$ where χ_D denotes the characteristic function of D and G is an arbitrary A^p-inner function, is continuous in C, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space A² to a general A^p-setting.

In the semi-classical regime we study the resonances of the operator P_t=h²Δ+V+t·δV in some small neighborhood of the first spectral band of P₀. Here V is a periodic potential, δV a compactly supported potential and t a small coupling constant. We construct a meromorphic multivalued continuation of the resolvent of P_t, and define the resonances to be the poles of this continuation. We compute these resonances and study the way they turn into eigenvalues when t crosses a certain threshold.

The theory of Weierstrass points and gap sequences for linear series on smooth curves is generalized to smooth families of curves with geometrically irreducible fibers, and over an arbitrary base scheme.

This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂Rⁿ be a bounded domain with a C² boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution. $\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .$

Applying simple techniques, we derive basic inequalities and show uniform boundedness for the set of solutions. Distance estimates as well as uniqueness results are obtained in special cases, e.g. we show that if ∂Ω and ∂D are two quadrature surfaces for a fixed measure μ and Ω is convex, then D⊂Ω. The main observation, however, is that if ∂Ω is a quadrature surface for μ≥0 and xε∂Ω, then the inward normal ray to ∂Ω at x intersects the convex hull of supp μ. We also study relations between quadrature surfaces and quadrature domains. D is said to be a quadrature domain with respect to a mesure μ if there is a solution to the following overdetermined Cauchy problem: $\Delta u = 1 - \mu in D, andu = |\nabla u| = 0 on \partial D.$

Finally, we apply our results to a problem of electrochemical machining.