Global regularity of the\overline \partial-neumann problem on an annulus between two pseudoconvex manifolds which satisfy property (P)problem on an annulus between two pseudoconvex manifolds which satisfy property (P)

LetX be a complex manifold of dimensionn≥3. Let Ω₁, Ω₂ be two open pseudoconvex submanifolds with smooth boundary such that Ω₁ ? Ω₂ ?X . Let Ω = Ω₂ \ $\overline \Omega_1 $ . Assume thatbΩ₁ andbΩ₁ satisfy Catlin's condition (P). Then the compactness estimate for (p, q)-forms with 0<q<n?1 holds for the $\overline \partial$ -Neumann problem on Ω. This result implies that given a $\overline \partial$ -closed (p, q)-form α with 0<q<n?1, which isC ^∞ on $\overline \Omega$ and which is cohomologous to zero on Ω, the canonical solutionu of the equation $\overline \partial$ u=α is smooth on $\overline \Omega$ .     

The \bar \partial -problem on q-pseudoconvex domains with applications

For a q-pseudoconvex domain Ω in ? ⁿ , 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $ -problem with exact support in Ω. Moreover, we solve the $\bar \partial $ -problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.     

L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

Global analytic hypoellipticity of the\bar \partial-Neumann problem on circular domains

In this paper we show that if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that \(\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0\) for allz near the boundary, then the solutionu to the \(\bar \partial\) -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ? is analytic hypoelliptic.     

On a Spectral Sequence for Twisted Cohomologies

Let(Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex(Ω*(M), d + H∧) and its associated twisted de Rham cohomology H*(M, H). The authors show that there exists a spectral sequence {Ep,qr, dr} derived from the filtration Fp(Ω*(M)) = i≥pΩi(M) of Ω*(M), which converges to the twisted de Rham cohomology H*(M, H). It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well,which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.     

Decay of solutions of the wave equation with a local degenerate dissipation

We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation:u _tt ? Δu+a(x)u _t =0 in Ω × [0, ∞) with the boundary conditionu/?Ω, wherea(x) is a nonnegative function on $\bar \Omega $ satisfying $$a(x) > a.e. x \in \omega and\smallint _\omega \frac{1}{{a(x)^P }}dx< \infty for some 0< p< 1$$ for an open set $\omega \subset \bar \Omega $ including a part of ?Ω with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.     

Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results

We consider for smooth pseudoconvex bounded domains Ω ? ?ⁿ of finite type as local analytic invariants on the boundary the growth orders of the Bergman kernel and the Bergman metric and the best possible order of subellipticity ε₁ > 0 for the \(\bar \partial - Neumann\) problem. Furthermore, we consider as local geometric invariants on ?Ω the order of extendability, the exponent of extendability, the 1-type, and the multitype. Various new inequalities between these invariants are proved, giving in particular analytic information from geometric input. On the other hand, a careful consideration of several series of examples of such domains Ω shows that starting fromn ≥ 3 (essentially) each of these invariants is independent of the remaining ones.     

Some pinching theorems for minimal submanifolds in \mathbb{S}^m (1) \times \mathbb{R}

Let Mn be an n-dimensional compact minimal submanifolds in Sm(1)×R.We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively.In fact,we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.     

Biharmonic Submanifolds with Parallel Mean Curvature in \mathbb{S}^{n}\times\mathbb{R}

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces M ⁿ (c)×?, where M ⁿ (c) is a space form with constant sectional curvature c, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^{n}(c)\times\mathbb{R}$ .     

Local cohomology modules of a smooth \mathbb{Z}-algebra have finitely many associated primes

Let R be a commutative Noetherian ring that is a smooth \(\mathbb {Z}\) -algebra. For each ideal \(\mathfrak {a}\) of R and integer k, we prove that the local cohomology module \(H^{k}_{\mathfrak {a}}(R)\) has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.     

The Laplacian on Cylindrical Domains

The aim of this paper is to prove elliptic regularity and parabolic maximal regularity of the Laplacian with mixed boundary conditions on domains Ω carrying a cylindrical structure. More precisely, we consider Ω to be given as the Cartesian product of whole or half spaces, a cube ${\mathcal{Q}}$ , and a standard domain V having compact boundary. Taking advantage of this structure we apply operator-valued Fourier multiplier results to transfer ${\mathcal{H}^{\infty}}$ -calculus results known for the Laplacian in L ^p (V) to the Laplacian in L ^p (Ω). This approach turns out to inherit elliptic regularity, i.e. the domain of the Dirichlet Laplacian equals ${W^{2,p}(\Omega) \cap W_0^{1,p}(\Omega)}$ , for instance. This is surprising since Ω may be unbounded and non-convex with boundary neither compact nor of class C ^1,1 at the same time. More generally, we consider the following mixture of boundary conditions: on every smooth part of the boundary Dirichlet or Neumann boundary conditions are imposed and on parts related to ${\mathcal{Q}}$ generalized periodic boundary conditions are included. Via ${\mathcal{R}}$ -sectoriality we deduce maximal regularity in the parabolic sense which seems to be new for this general class of boundary conditions. Parabolic equations with such a mixture of boundary conditions on such type of domains appear for example in models describing growth of biological cells.     

Singularly perturbed nonlinear Dirichlet problems involving critical growth

We consider the following singularly perturbed nonlinear elliptic problem: $$\begin{array}{ll}-\varepsilon^{2}\Delta u + u=f(u),\; u > 0\, {\rm on}\, \Omega,\; u = 0\, {\rm on}\, \partial \Omega,\end{array}$$ where Ω is a bounded domain in ${\mathbb{R}^N (N \ge 3)}$ with a boundary ${\partial \Omega \in C^2}$ and the nonlinearity f is of critical growth. In this paper, we construct a solution ${u_\varepsilon}$ of the above problem which exhibits one spike near a maximum point of the distance function from the boundary ?Ω under a critical growth condition on f. Our result complements the study made in [9] in the sense that, in that paper, only the subcritical growth was considered.     

Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization $\tilde{J}^d(Y)$ of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in $\tilde{J}^d(Y)$ . We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space M confirming the Vafa–Intriligator formulae in the nodal case.     

Extension of functions from Sobolev spaces

By definition, the domain Ω ??ⁿ belongs to the class EW _p ^l if there exists a continuous linear extension operator . An example is given of a domain Ω ??² with compact closure and Jordan boundary, having the following properties: (1) The curve ?Ω is not a quasicircle, has finite length and is Lipschitz in a neighborhood of any of its points except one. (2) Ω ε EW _p ¹ for p<2. and Ω ? EW _p ¹ for p?2. (3) for p>2 and for p?2.     

Biharmonic properly immersed submanifolds in Euclidean spaces

We consider a complete biharmonic immersed submanifold M in a Euclidean space ${\mathbb{E}^N}$ . Assume that the immersion is proper, that is, the preimage of every compact set in ${\mathbb{E}^N}$ is also compact in M. Then, we prove that M is minimal. It is considered as an affirmative answer to the global version of Chen’s conjecture for biharmonic submanifolds.     

Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain Ω in ${{\mathbb R}^N}$ and a nonnegative potential V in Ω such that V(x) d(x, ?Ω)² is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator L _V := Δ ? V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of ${z \in \partial \Omega}$ . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for ${z \in \partial \Omega}$ to be finely regular. An intermediate result consists in a majorization of ${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$ for u positive harmonic in Ω and ${A \subset \Omega}$ . Conditions for almost everywhere regularity in a subset A of ?Ω are also given as well as an extension of the main results to a notion of fine ${\mathcal{ L}_1 \vert \mathcal{L}_0}$ -regularity, if ${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$ being two potentials, with V ₀ ≤ V ₁ and ${\mathcal{L}}$ a second order elliptic operator.     

Uniqueness of H-surfaces in {\mathbb{H}}^2 \times \mathbb{R},{{\vert H\vert \leq 1/2}} , with boundary one or two parallel horizontal circles

We prove that a H-surface M in ${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$ , inherits the symmetries of its boundary $\partial M,$ when $\partial M$ is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.     

Metodi variazionali nello studio di problemi al contorno con parte non lineare discontinua

The main purpose of this paper is to use variational methods in the study of problems of following type: $$\left\{ {\begin{array}{*{20}c} {Lu = \lambda f(x,u) in \Omega } \\ {u = 0 on \partial \Omega } \\ \end{array} } \right.$$ Here Ω is supposed to be a bounded domain with smooth boundary ? Ω,L an elliptic operator, λ∈IR andf(x, t) a real function defined on \(\tilde \Omega \times IR\) having one or several simple discontinuities ont. Mainly we are interested in solutions which satisfy (.) a. e., which are most meaningful in physical problems, and we prove various existence theorems for several choices ofL, f and λ. The main difficulty consists in the fact that the functionals related to (.) are not Fréchet differentiable in every point, sincef is discontinuous.