The Diederich–Fornaess index and the global regularity of the \bar{\partial }-Neumann problem

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), the constant ${\mathcal {E}}_s$ which is needed to control the commutator of a totally real vector field $T_{{\mathcal {E}}}$ with $\bar{\partial }^*$ in order to have $H^s$ a-priori estimates for the Bergman projection $B_k, k\ge q-1$ , on a smooth $q$ -convex domain $D\subset \subset {\mathbb {C}}^{n}$ . This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), yields regularity of $B_k$ in specific Sobolev degree $s$ . Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function $r$ of $D$ , the operators $(T^+)^{-\frac{\delta }{2}}$ and $(-r)^{\frac{\delta }{2}}$ . We are thus able to extend to general degree $k\ge 0$ of $B_k$ , the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999) which only holds for $q=1$ and $k=0$ : if for the Diederich–Fornaess index $\delta $ of $D$ , we have $(1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s$ , then $B_k$ is $H^s$ -regular.     

Optimal Lp estimates for the\bar \partial -equation on complex ellipsoids in ℂn-equation on complex ellipsoids in ℂn

In this paper we prove the best possible L^p estimates for the-equation on complex ellipsoids in ℂⁿ, and provide examples to show why they cannot be improved.     

On the\bar \partial equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

For a large class of subharmonicφ, the equation is studied in . Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL ^p norms are derived as a corollary. A class of counterexamples, related to the failure of to be analytic-hypoelliptic on certain CR manifolds, is discussed. Communicated by Steven Krantz     

Limits on an extension of Carleson’s\bar \partial - Theorem

We study a theorem essentially due to Carleson about solving the $\bar \partial - equation$ on the unit disk. We show that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, our main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. We exhibit a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to ∞. Since Carleson’s $\bar \partial - theorem$ depends on the notion of a Carleson measure, we also discuss Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.     

On Supnorm Estimates for $\bar{\partial}$ on Infinite Type Convex Domains in ℂ2

In this paper, we study the \(\bar{\partial}\) equation on some convex domains of infinite type in ?². In detail, we prove that supnorm estimates hold for infinite exponential type domains, provided the exponent is less than 1.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

Hölder andL p -estimates for the\bar \partial -equation on non-smooth strictlyq-convex domainson non-smooth strictlyq-convex domains

A solution operator for the \(\bar \partial \) -equation on strictlyq-convex domains with nonsmooth boundary is constructed. It is proved that the solution satisfies optimal 1/2-Hölder andL ^p estimates.     

On the spectrum of the Cesàro operator C 1 on b\bar v_0 \cap \ell _\infty

In this paper we have investigated the spectrum of the Cesàro operator C ₁ which is regarded as an operator on the sequence space $b\bar v_0 \cap \ell _\infty $ the space of statistically null bounded variation sequences.     

Hölder estimates for the\bar \partial - equation in some domains of finite typein some domains of finite type

We obtain Hölder estimates for the $\bar \partial - equation$ on some domains of finite type in ?ⁿ using proper mapping techniques. The domains considered are domains of finite type in the sense of D’Angelo and are defined by local coordinate expressions satisfying certain algebraic geometric conditions which prevent the existence of complex analytic varieties in the boundary of the domain. Using a proper mapping which is given by the finite type condition and which carries all the information about the intrinsic geometry of the boundary, we transform the finite type points into strongly pseudoconvex ones. At these strongly pseudoconvex points we compute an explicit solution using the Henkin integral formula and we obtain estimates that we are able to pull back to the original domain. We achieve this by exploiting the branching behavior of the proper mapping. We also construct some biholomorphic numerical invariants associated with some of the domains under consideration.     

Spectrum of Normal Weighted Composition Operators on the Fock Space Over ℂN

By using commutative C^*-algebra techniques, spectrum and essential spectrum of normal weighted composition operators on the Fock space over C^N are completely characterized. As an application, spectrum of self-adjoint weighted composition operators on the Fock space are obtained also.     

On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains

Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R ³. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg in R ³ and Sohr-von Wahl in exterior domains to general domains.     

Solvability in the large on (0, ∞) of the fundamental initial-boundary value problem for the two-dimensional equations of the motion of Oldroyd fluids

The unqiue global solvability for t (0, ) is proved for the system describing the two-dimensional motion of an Oldroyd fluid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 69–72, 1986.     

A Note on the π—regularity of Rings

LetRbeassociativewithanidentity.RingRiscalledaπ-regularringifandonlyifforeveryelementainR,thereexistsomen1,c∈Rsuchthatan=ancan.RingRiscalledaunitπ-regularringifandonlyifforanyelementainR,thereissomem1suchthatam=amuamforaunitu∈R.Manyringsareunitπ-regularrings,suchassemicommutativeπ-regularrings,finitesemicommutativerings,stronglyπ-regularringsandunit-regularrings.Risstronglyπ-regularifforeverya∈Rthereexistn1andc∈Rsuchthatan=an 1c.Ifthein-dexnaboveisoneforalla∈R,thenRiscalledabel…     

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ^−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.     

Homogeneous initial–boundary value problem of the Rosenau equation posed on a finite interval

This paper considers the IBVP of the Rosenau equation $\{\begin{array}{c}{\partial }_{t}u+{\partial }_t{\partial }_x^{4}u+{\partial }_{x}u+u{\partial }_{x}u=0\text{,}x\in (0,1),t>0\text{,}\\ u(0,x)=u_0\left(x\right)\\ u(0,t)={\partial }_x^{2}u(0,t)=0\text{,}u(1,t)={\partial }_x^{2}u(1,t)=0\text{.}\end{array}$ It is proved that this IBVP has a unique global distributional solution $u\in C([0,T];H^s(0,1))$ as initial data $u_0\in H^s(0,1)$ with $s\in [0,4]$. This is a new global well-posedness result on IBVP of the Rosenau equation with Dirichlet boundary conditions.