The {\bar{\partial}} -Neumann problem on product domains in {\mathbb{C}^{n}}

Let ${\Omega=\Omega_{1}\times\cdots\times\Omega_{n}\subset\mathbb{C}^{n}}$ , where ${\Omega_{j}\subset\mathbb{C}}$ is a bounded domain with smooth boundary. We study the solution operator to the ${\overline\partial}$ -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the ${\overline\partial}$ -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the ${\overline\partial}$ -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.     

Atomic intersection of σ-fields and some of its consequences

Let ${(\Omega, \mathcal{F}, P)}$ be a probability space. For each ${\mathcal{G}\subset\mathcal{F}}$ , define ${\overline{\mathcal{G}}}$ as the σ-field generated by ${\mathcal{G}}$ and those sets ${F\in \mathcal{F}}$ satisfying ${P(F)\in\{0,1\}}$ . Conditions for P to be atomic on ${\cap_{i=1}^k\overline{\mathcal{A}_i}}$ , with ${\mathcal{A }_1,\ldots,\mathcal{A}_k\subset\mathcal{F}}$ sub-σ-fields, are given. Conditions for P to be 0-1-valued on ${\cap_{i=1}^k \overline{\mathcal{A}_i}}$ are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.     

Local {W^{2,m(\cdot)}_{loc}}regularity for p(.)-Laplace equations

We study ${W^{2,m(\cdot)}_{loc}}$ regularity for local weak solutions of p(·)-Laplace equations where ${p\in C^1(\Omega) \cap C(\overline{\Omega})}$ and ${\min_{x\in \overline{\Omega}} p(x) > 1}$ .     

Cohomology of Some Complex Laminations

In this paper we solve the ${\overline{\partial }}$ -problem along the leaves for two types of laminations: (i) Some open sets Ω of ${{\mathbb C}\times B}$ (where B is any differentiable manifold) endowed with the canonical foliation that is, the foliation whose leaves are the sections ${\Omega ^t=\{ z\in {\mathbb C}:(z,t)\in \Omega \}}$ . We construct a solution to the equation ${\overline{\partial }h=fd\overline z}$ for any function ${f:\Omega\longrightarrow {\mathbb C}}$ of class ${C^{s}\,(s\in \mathbb{N}\cup\{ \infty \}),\,C^\infty}$ along the leaves and satisfies some growth conditions near the singularities. (ii) A complex lamination by Riemann surfaces obtained by suspending a homeomorphism of a closed set of the Euclidean space ${\mathbb{C}\times \mathbb{R}}$ .     

Rellich inequalities with weights

Let Ω be a cone in ${\mathbb {R}^{n}}$ with n ≥? 2. For every fixed ${\alpha \in \mathbb {R}}$ we find the best constant in the Rellich inequality ${\int\nolimits_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx \ge C\int\nolimits_{\Omega}|x|^{\alpha-4}|u|^{2}dx}$ for ${u \in C^{2}_{c}(\overline\Omega\setminus\{0\})}$ . We also estimate the best constant for the same inequality on ${C^{2}_{c}(\Omega)}$ . Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

The geometric Neumann problem for the Liouville equation

Let Ω denote the upper half-plane ${\mathbb{R}_+^2}$ or the upper half-disk ${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$ of center 0 and radius ${\varepsilon}$ . In this paper we classify the solutions ${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$ to the Neumann problem $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ where ${K, c_1, c_2 \in \mathbb{R}}$ , with the finite energy condition ${\int_{\Omega} e^v < \infty}$ As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Uniqueness for Inverse Boundary Value Problems by Dirichlet-to-Neumann Map on Subboundaries

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on subboundary ${\partial \Omega \setminus \Gamma_{-}}$ to Neumann data on subboundary ${\partial \Omega \setminus \Gamma_{+}}$ . First we prove uniqueness results in three dimensions under some conditions such as ${\overline{\Gamma_{+}\cup\Gamma_{-}}= \partial\Omega}$ Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given ${\Gamma_{-} = \Gamma_{+}}$ Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

Sufficient conditions for regularity of area-preserving deformations

We discuss some basic regularity properties of the area-preserving deformations ${u:\Omega \mapsto \mathbb{R}^2}$ that have minimal elastic energy ${\int \limits_\Omega|\nabla u|^2}$ among a suitable class of admissible vectorfields defined on a smooth, bounded domain ${\Omega\subset \mathbb{R}^2}$ . Although we restrict ourselves to the quadratic stored energy function and 2-space, most of our results extend to three dimensional setting with convex stored energy function.     

Generalized order and type of entire functions and best approximation in L p -norm

Mohammed Harfaoui characterized the generalized growth of entire functions of several complex variables by means of best polynomial approximation and interpolation on a compact set ${K \subset C^{n}}$ with respect to the set ${\overline \Omega _r = \{z \in C^{n}:{\rm exp} (V_k ) \leq r\}}$ where ${V_k = {\rm sup} \{(1/d) \ln |P_d |, P_d}$ a polynomial of degree ${\leq d, ||P_d ||_K \leq 1\}}$ . Harfaoui considered only the generalized order. In this paper, we characterize the generalized type of entire functions.     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category can Always be Replaced by Forcing Over a Complete Heyting Algebra

It is assumed that a Kripke–Joyal semantics ${\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}$ A = C , Cov , F , ? has been defined for a first-order language ${\mathcal{L}}$ L . To transform ${\mathbb{C}}$ C into a Heyting algebra ${\overline{\mathbb{C}}}$ C ¯ on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of ${\mathbb{C}}$ C . A pretopology ${\overline{{\rm Cov}}}$ Cov ¯ is defined on ${\overline{\mathbb{C}}}$ C ¯ using the pretopology on ${\mathbb{C}}$ C . A sheaf ${\overline{{\it F}}}$ F ¯ is made up of sections of F that obey functoriality. A forcing relation ${\overline{\Vdash}}$ ? ¯ is defined and it is shown that ${\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }$ A ¯ = C ¯ , Cov ¯ , F ¯ , ? ¯ is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of ${\mathcal{A}}$ A . That is to say, an object a of ${\mathbb{C}Ob}$ C O b forces a sentence with respect to ${\mathcal{A}}$ A if and only if the maximal a-crible forces it with respect to ${\overline{\mathcal{A}}}$ A ¯ . This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra.     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces