We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^−1/4−ε; in the smoothness direction, a sufficient condition in Hölder classes is q∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^−1/4 and belong to C^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.

It is shown that if a function u satisfies a backward parabolic inequality in an open set Ω∉Rⁿ⁺¹ and vanishes to infinite order at a point (x₀·t₀) in Ω, then u(x, t₀)=0 for all x in the connected component of x₀ in Ω⌢(Rⁿ×{t₀}).

This paper gives a partial answer to a problem raised by Griffiths in [4], which is a kind of converse of Abel’s theorem.

We present a homogeneous real analytic hypersurface in C³, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube Γ_C over the two-dimensional real cone in R³ is also homogeneous and has a seven-dimensional CR automorphism group. However, our example is not biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of Γ_C are, in contrast, noncommutative.

The continuity of weak solutions of elliptic partial differential equations $div \mathcal{A}(x,\nabla u) = 0$ is considered under minimal structure assumptions. The main result guarantees the continuity at the point x₀ for weakly monotone weak solutions if the structure of A is controlled in a sequence of annuli $B(x_0 ,R_j )\backslash \bar B(x_0 ,r_j )$ with uniformly bounded ratio R_j/r_j such that lim_j→∞R_j=0. As a consequence, we obtain a sufficient condition for the continuity of mappings of finite distortion.

We study a semilinear elliptic equation of the form $ - \Delta u + u = f(x,u), u \in H_0^1 (\Omega ),$ where f is continuous, odd in u and satisfies some (subcritical) growth conditions. The domain Ω⊂R^N is supposed to be an unbounded domain (N≥3). We introduce a class of domains, called strongly asymptotically contractive, and show that for such domains Ω, the equation has infinitely many solutions.

We prove that the linearized Riesz transforms and the imaginary powers of the Laplacian are H¹-bounded on complete Riemannian manifolds satisfying the doubling property and the Poincaré inequality, where H¹ denotes the Hardy space on M.

LetD be a relatively compact domain in C² with smooth connected boundary ∂D. A compact set K⊂∂D is called removable if any continuous CR function defined on ∂D/K admits a holomorphic extension to D. If D is strictly pseudoconvex, a theorem of B. Jöricke states that any compact K contained in a smooth totally real disc S⊂∂D is removable. In the present article we show that this theorem is true without any assumption on pseudoconvexity.

We prove the following Hartogs-Bochner type theorem: Let M be a connected C² hypersurface of P_n(C) (n≥2) which divides P_n(C) in two connected open sets Ω₁ and Ω₂. Suppose that M has at most one open CR orbit. Then there exists i∈{1,2} such that C¹ CR functions defined on M extends holomorphically to Ω_i.