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An integral formula of hyperbolic type for solutions of the Dirac equation on Minkowski space with initial conditions on a hyperboloid

Martin Sikora (2010)

Archivum Mathematicum

The Dirac equation for spinor-valued fields f on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet H + of the hyperboloid. In particular, we derive an integral formula expressing the value of f in a chosen point p as an integral over a compact cycle given by the intersection of the null cone with H + in the Minkowski space 𝕄 .

Angular limits of the integrals of the Cauchy type

Josef Král, Dagmar Medková (1997)

Czechoslovak Mathematical Journal

Integrals of the Cauchy type extended over the boundary ∂ A of a general compact set A in the complex plane are investigated. Necessary and sufficient conditions on ∂ A are established guaranteeing the existence of angular limits of these integrals at a fixed z ∈ ∂ A for all densities satisfying a Hölder-type condition at z .

Approximations by the Cauchy-type integrals with piecewise linear densities

Jaroslav Drobek (2012)

Applications of Mathematics

The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity ω ( · ) satisfies lim sup s ↓ 0 ω ( s ) ln 1 s = 0 are considered on these boundaries. Functions satisfying the Hölder condition of order α , 0 < α ≤ 1 , belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise...

Calderón's problem for Lipschitz classes and the dimension of quasicircles.

Kari Astala (1988)

Revista Matemática Iberoamericana

In the last years the mapping properties of the Cauchy integralCΓf(z) = 1/(2πi) ∫Γ [f(ξ) / ξ - z] dξhave been widely studied. The most important question in this area was Calderón's problem, to determine those rectifiable Jordan curves Γ for which CΓ defines a bounded operator on L2(Γ). The question was solved by Guy David [Da] who proved that CΓ is bounded on L2(Γ) (or on Lp(Γ), 1 &lt; p &lt; ∞) if and only if Γ is regular, i.e.,H1(Γ ∩ B(z0,R) ≤ CRfor every z0 ∈ C, R &gt; 0 and for...

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