NONREGULAR BOUNDARY VALUE PROBLEMS WITH A PARAMETER
Keywords:
nonlinear PDE, Cauchy problem, elliptic operators.DOI:
https://doi.org/10.17654/0975045225002Abstract
In this paper, we discuss boundary value problems for a class of nonlinear elliptic equations with data on a boundary surface. We denote by y the unknown function which is supposed to take its values in We assume that only some components of y are given on the whole boundary surface implying the designation of nonregular boundary value problems. To derive an interesting necessary condition for the solvability of our original problem, we construct an appropriate Cauchy problem for nonlinear elliptic equations which we solve by making use of a parameter which is small enough.
Received: October 15, 2024
Revised: October 28, 2024
Accepted: November 18, 2024
References
A. Alsaedy and N. Tarkhanov, Normally solvable nonlinear boundary value problems, Nonlinear Analysis 95 (2014), 468-482.
https://doi.org/10.1016/j.na.2013.09.024.
F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc. 69(4) (1963), 862-874. https://doi.org/10.2307/1994223.
A. P. Calderón, Boundary value problems for elliptic equations, Outlines of the Joint Soviet-American Symp. on Part. Diff. Eq., Novosibirsk, 1963, pp. 303-304.
M. Denche and A. Kourta, Boundary value problem for second-order differential operators with nonregular integral boundary conditions, Rocky Mountain J. Math. 36 (2006), 893-913. http://www.jstor.org/stable/44239147.
A. A. Dezin, General Questions of the Theory of Boundary Value Problems, Nauka, Moscow, 1980.
C. Ebmeyer, Mixed boundary value problems for nonlinear elliptic systems in n-dimensional Lipschitzian domains, Z. Anal. Anwend. 18(3) (1999), 539-555. DOI: 10.4171/ZAA/897.
S. Glebov, O. Kiselev and N. Tarkhanov, Nonlinear equations with small parameter, Waves and Boundary Problems, de Gruyter, Berlin, Boston, Volume 2, 2018. https://doi.org/10.1515/9783110534979.
I. Ly, A Cauchy problem for the Cauchy-Riemann operator, Afr. Mat. 32 (2021), 69-76.
I. Ly and N. Tarkhanov, The Cauchy problem for nonlinear elliptic equations, Nonlinear Analysis 70(7) (2009), 2494-2505.
https://doi.org/10.1016/j.na.2008.03.034.
I. Ly and N. Tarkhanov, Local boundary problem for Dirac type operators, Sib. Math. J. 51(5) (2010), 847-859.
I. Ly and S. Tao, Nonregular boundary value problem for the Cauchy-Riemann operator, Advances in Pure Mathematics 10 (2020), 623-630.
doi: 10.4236/apm.2020.1011038.
Ya. A. Roitberg, Elliptic Boundary Value Problems in Generalized Functions, Kluwer Academic Publishers, Dordrecht NL, 1996.
https://doi.org/10.1007/978-94-011-5410-9.
M. Schechter, Negative norms and boundary problems, Ann. Math. 72(3) (1960), 581-593. https://doi.org/10.2307/1970230.
A. Shlapunov and N. Tarkhanov, Duality by reproducing kernels, IJMMS 6 (2003), 327-395. http://eudml.org/doc/50277.
N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995.
url=https://api.semanticscholar.org/CorpusID:116973503.
L. R. Volevich and A. R. Shirikyan, Stable and unstable manifolds for nonlinear elliptic equations with parameter, Trans. Moscow Math. Soc. 61 (2000), 97-138.
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