scholarly journals On the canonical solution of $\protect \hspace{0.0pt}\protect \hspace{0.0pt}\protect \overline{\protect \hspace{0.0pt}\partial }$ on polydisks

2020 ◽  
Vol 358 (5) ◽  
pp. 523-528
Author(s):  
Muzhi Jin ◽  
Yuan Yuan
Keyword(s):  
Canonical Solution

The use of Riemann problems in solving a class of transcendental equations

1973 ◽  
Vol 73 (1) ◽  
pp. 111-118 ◽  
Author(s):  
E. E. Burniston ◽  
C. E. Siewert
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Explicit Solutions ◽  
Riemann Boundary Value Problem ◽  
Riemann Problems ◽  
Riemann Boundary ◽  
Canonical Solution ◽  
Transcendental Equations ◽  
Explicit Expressions

AbstractA method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

Acoustic Emission From a Brief Crack Propagation Event

1979 ◽  
Vol 46 (1) ◽  
pp. 107-112 ◽  
Author(s):  
J. D. Achenbach ◽  
J. G. Harris
Keyword(s):  
Acoustic Emission ◽  
Brittle Fracture ◽  
Crack Propagation ◽  
Acoustic Emissions ◽  
Elastic Solid ◽  
Propagation Speed ◽  
Crack Edge ◽  
Arbitrary Distribution ◽  
Canonical Solution ◽  
Phase Functions

Acoustic emissions produced by elementary processes of deformation and fracture at a crack edge are investigated on the basis of elastodynamic ray theory. To obtain a two-dimensional canonical solution we analyze wavefront motions generated by an arbitrary distribution of climbing edge dislocations emanating from the tip of a semi-infinite crack in an unbounded linearly elastic solid. These wavefront results are expressed in terms of emission coefficients which govern the variation with angle, and phase functions which govern the intensity of the wavefront signals. Explicit expressions for the emission coefficients are presented. The coefficients have been plotted versus the angle of observation, for various values of the crack propagation speed. The phase functions are in the form of integrals over the emanating dislocation distributions. Specific dislocation distributions correspond to brittle fracture and plastic yielding at the crack tip, respectively. Acoustic emission is most intense for brittle fracture, when the particle velocities experience wavefront jumps which are proportional to the stress-intensity factors prior to fracture. An appropriate adjustment of the canonical solution accounts for curvature of a crack edge. Such effects as focussing, finite duration of the propagation event, and finite dimensions of the crack are briefly discussed. As a specific example, the first signals generated by brittle Mode I propagation of an elliptical crack are calculated.

Global solvability and regularity for ∂¯ on an annulus between two weakly convex domains which satisfy property (P)

2019 ◽  
Vol 12 (03) ◽  
pp. 1950041
Author(s):  
Sayed Saber
Keyword(s):  
Neumann Problem ◽  
Complex Manifold ◽  
Global Solvability ◽  
Closed Formula ◽  
Convex Domains ◽  
Weakly Convex ◽  
Canonical Solution ◽  
Dimension Formula ◽  
Satisfy Property

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be a weakly [Formula: see text]-convex and [Formula: see text] be a weakly [Formula: see text]-convex in [Formula: see text] with smooth boundaries such that [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] satisfy property [Formula: see text]. Then the compactness estimate for [Formula: see text]-forms [Formula: see text] holds for the [Formula: see text]-Neumann problem on the annulus domain [Formula: see text]. Furthermore, if [Formula: see text] is [Formula: see text]-closed [Formula: see text]-form, which is [Formula: see text] on [Formula: see text] and which is cohomologous to zero on [Formula: see text], the canonical solution [Formula: see text] of the equation [Formula: see text] is smooth on [Formula: see text].

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