TY  - JOUR
A1  - Chelkh, W.
A1  - Ly, Ibrahim
A1  - Tarkhanov, Nikolai
T1  - A remark on the Laplace transform
JF  - Siberian Mathematical Journal
N2  - The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces.
KW  - Fourier-Laplace transform
KW  - distributions with one-sided support
KW  - holomorphic function
KW  - analytic functional
Y1  - 2020
U6  - https://doi.org/10.1134/S0037446620040151
SN  - 0037-4466
SN  - 1573-9260
VL  - 61
IS  - 4
SP  - 755
EP  - 762
PB  - Consultants Bureau, Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Ly, Ibrahim
T1  - A Cauchy problem for the Cauchy-Riemann operator
JF  - Afrika Matematika
N2  - We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.
KW  - nonlinear PDI
KW  - Cauchy problem
KW  - Zaremba problem
Y1  - 2020
U6  - https://doi.org/10.1007/s13370-020-00810-4
SN  - 1012-9405
SN  - 2190-7668
VL  - 32
IS  - 1-2
SP  - 69
EP  - 76
PB  - Springer
CY  - Heidelberg
ER  - 
TY  - JOUR
A1  - Ly, Ibrahim
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Rado theorem for p-harmonic functions
JF  - Boletin de la Sociedad Matemática Mexicana
N2  - Let A be a nonlinear differential operator on an open set X subset of R-n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A(u) = 0 in XS of class F satisfies this equation weakly in all of X. For the most extensively studied classes F, we show conditions on S which guarantee that S is removable for F relative to A.
KW  - Quasilinear equations
KW  - Removable sets
KW  - p-Laplace equation
Y1  - 2016
U6  - https://doi.org/10.1007/s40590-016-0109-7
SN  - 1405-213X
SN  - 2296-4495
VL  - 22
SP  - 461
EP  - 472
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Ly, Ibrahim
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Generalized Beltrami equations
JF  - Complex variables and elliptic equations
N2  - We enlarge the class of Beltrami equations by developing a stability theory for the sheaf of solutions of an overdetermined elliptic system of first-order homogeneous partial differential equations with constant coefficients in Rn.
KW  - quasiconformal mapping
KW  - Beltrami equation
Y1  - 2015
U6  - https://doi.org/10.1080/17476933.2013.876759
SN  - 1747-6933
SN  - 1747-6941
VL  - 60
IS  - 1
SP  - 24
EP  - 37
PB  - Routledge, Taylor & Francis Group
CY  - Abingdon
ER  - 
TY  - JOUR
A1  - Ly, Ibrahim
A1  - Tarkhanov, Nikolaj Nikolaevič
T1  - Asymptotic expansions at nonsymmetric cuspidal points
JF  - Mathematical notes
N2  - We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.
KW  - Dirichlet problem
KW  - singular points
KW  - asymptotic expansions
Y1  - 2020
U6  - https://doi.org/10.1134/S0001434620070238
SN  - 0001-4346
SN  - 1573-8876
VL  - 108
IS  - 1-2
SP  - 219
EP  - 228
PB  - Springer Science
CY  - New York
ER  -