Complex Variables and Elliptic Equations

Publisher: Taylor & Francis


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    Complex variables and elliptic equations (Online), Complex variables
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Publications in this journal

  • [show abstract] [hide abstract]
    ABSTRACT: We study the geometric properties of the local homeomorphisms satisfying some generalized modular inequalities. We establish in Theorem 1 that a local homeomorphism f:D included in Rn →Rn satisfying condition (N) and having local ACLq inverses, with q>1, satisfy important modular inequalities. We generalize in this class of mappings known theorems from the theory of quasiregular mappings like Zoric’s theorem and the estimate of the radius of injectivity.
    Complex Variables and Elliptic Equations 12/2013;
  • [show abstract] [hide abstract]
    ABSTRACT: We extend to differential forms the system of partial differential equations ∂l Hu = f defined for functions as follows ⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ ∂u ∂z j + i 4 k=�n−m k=1 � ∂ Bk ∂z j ∂u ∂ζk + ∂ Bk ∂z j ∂u ∂ζ k � = f j 1 ≤ j ≤ m, ∂u ∂ζ k = fm+k 1 ≤ k ≤ n − m. This leads to the ∂l H −cohomology, and then we will prove the corresponding Dolbeault–Grothendieck lemma
    Complex Variables and Elliptic Equations 09/2013; 58(9).
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    ABSTRACT: We improve our previous generalizations to Arsove's and Ko\lodziej's and Thorbi\"ornson's results concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second.
    Complex Variables and Elliptic Equations 09/2013; 59 (2014)(2):149-161.
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    ABSTRACT: The paper, mainly deals with the question of existence of the solutions of the generalized Cauchy–Riemann system in function spaces, in particular, in the space of bounded analytic functions. Without summability assumption on the coefficients of the equation, existence of the solutions for the whole complex plane is proved
    Complex Variables and Elliptic Equations 09/2013;
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    ABSTRACT: We study the Dirichlet problem for general degenerate Beltrami equation ${\overline {\partial}}f\, =\, \mu {\partial f}+\nu {\overline {\partial f}}$ in the unit disk D in C. Given an arbitrary analytic function A with isolated singularities in D, we find criteria for the existence of a solution f of the form f = A◦ω where ω stands for a regular W1,1loc (D) himeomorphism of D onto itself. Some criteria for the existence of regular solutions are given.
    Complex Variables and Elliptic Equations 01/2013;
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    ABSTRACT: It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh–Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity.
    Complex Variables and Elliptic Equations 09/2012; 59(2):162-171.
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    ABSTRACT: We characterize the compactness of the linear fractionally induced commutator in terms of the function theoretic properties of and ψ. We show that in the automorphic case the commutator is compact if and only if and ψ are simple rotations of the unit disc. On the other hand, when one of the inducing maps is not an automorphism of the disc, we show that the commutator is non-trivially compact if and only if the inducing maps are both parabolic with the same boundary fixed point or they are both hyperbolic with the same boundary fixed point and their other fixed points are conjugate reciprocals.
    Complex Variables and Elliptic Equations 06/2012; 57(6):677-686.
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    ABSTRACT: Let K   be compact and \K connected, P(K) the uniform closure of all polynomial functions restricted to K and A(K) the uniform algebra of all functions continuous on K and holomorphic on int(K). It is known (Mergelyan) that P(K) = A(K) if and only if \K is connected. Let ∂K be the topological boundary of K and (P(K)) = {z  K: there exists f  P(K) such that f (z) = 1, |f(w)| < 1 for w  K\{z}}, i.e., the set of peak points relative to P(K). A long standing problem is to determine the family of K's such that (P(K)) = ∂K. We give new results for the solution to this problem by using two different methods. The first one uses no notions of capacity and only employs results from classic complex variable theory and topology. We prove that for every K, there is a dense subset of ∂K (‘escape’ points) contained in (P(K)) such that each escape point has a peaking function, that is a homeomorphism on K and is conformal on int(K). The second uses the Curtis Peak Point Criterion which depends on properties of capacity. From this, we prove not only that (P(K))  ∂K (long known), but finally that (P(K)) = ∂K for all K.
    Complex Variables and Elliptic Equations 06/2012; 57(6):611-624.
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    ABSTRACT: In this work, we investigate the approximation problems in weighted Smirnov–Orlicz classes. We prove a direct theorem for polynomial approximation of functions in certain subclasses of weighted Smirnov–Orlicz classes. The direct theorem is proved in terms of the modulus of smoothness. Also, from the main theorem some results are obtained. In the proof of the main theorem, a Jackson-Dzjadyk polynomial is used.
    Complex Variables and Elliptic Equations 05/2012; 57(5):567-577.
  • [show abstract] [hide abstract]
    ABSTRACT: We study an initial value problem for the one-phase Hele–Shaw problem with zero surface tension. We establish local well-posedness for the initial value problem in Sobolev space. Furthermore, we obtain that, on average in time, the solution gains 1/2 derivative of smoothness in spatial variable compared to the initial data.
    Complex Variables and Elliptic Equations 02/2012; 57(Nos. 2–4):351-368.
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    ABSTRACT: Formulas for heat kernels are found for degenerate elliptic operators by finding the probability density of the associated Ito diffusion. The formulas involve an integral of a product between a volume function and an exponential term.
    Complex Variables and Elliptic Equations 02/2012; 57(Nos. 2–4):155-168.
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    ABSTRACT: This article deals with a system of quasilinear elliptic variational inequalities whose leading differential operator is a diagonal (p 1, p 2)-Laplace operator with 1 < p 1, p 2 < ∞, and whose lower order vector field f = (f 1, f 2) is a gradient field, which is not subject to any growth restriction, in particular, it may have supercritical growth, and thus coercivity is violated. The novelty of this article is to establish a variational approach of the, in general, noncoercive elliptic system of variational inequalities by introducing the concept of trapping region for such type of problems. The trapping region allows us to transform the given noncoercive system of variational inequalities into an associated ‘truncated’ system to which variational methods can be applied. We are going to prove that the ‘truncated’ system possesses solutions, which can be characterized as the critical points of a suitably constructed (nonsmooth) energy functional, and any critical point is shown to be a solution of the original problem within the trapping region. Moreover, applications to quasilinear elliptic systems under Dirichlet boundary conditions as well as to elliptic systems under obstacle constraints are treated by the theory developed in this article.
    Complex Variables and Elliptic Equations 02/2012; 57(Nos. 2–4):169-183.
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    ABSTRACT: The power matrix is the matrix of the coefficients of the power series at 0 of powers of an analytic function. Composition corresponds to matrix multiplication. We generalize the power matrix by replacing power series with Faber polynomial expansions. We show that composition corresponds to multiplication of the generalized power matrices, for both simply and doubly connected domains. We apply this to give some matrix product formulas for the coefficients of conformal welding maps of an analytic homeomorphism of an analytic curve. In particular, in some sense one can solve for the coefficients of the conformal welding maps in terms of the generalized power matrix of the analytic homeomorphism.
    Complex Variables and Elliptic Equations 01/2012;
  • [show abstract] [hide abstract]
    ABSTRACT: In Lampert [On the boundary regularity of biholomorphic mappings, Contributions to several complex variables, Aspects Math. E9 (1986), pp. 193–215] and D'Angelo [Several Complex Variables and the Geometry of real Hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993, ISBN: 0-8493-8272-6] Lempert and D'Angelo showed that germs of real analytic sets in ℂ of infinite type contain a complex curve. In this article we discuss a very special case of their result, germs of real analytic pseudoconvex domains in ℂ. We reprove their theorem using a geometric construction which sheds light on the intricate structure of such boundaries in the presence of complex curves of high order tangency. The proof of Lempert and D'Angelo is somewhat more of an ideal theoretic nature.
    Complex Variables and Elliptic Equations 01/2012; 57(6):705-717.
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    ABSTRACT: We study the Carathéodory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly, incisive results are obtained for convex domains.
    Complex Variables and Elliptic Equations 01/2012;

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