Nie, Dongdong; Xie, Feng Singularly perturbed semilinear elliptic boundary value problems with discontinuous source term. (English) Zbl 1353.35034 Bound. Value Probl. 2016, Paper No. 164, 17 p. (2016). Summary: A class of singularly perturbed semilinear elliptic boundary value problems with discontinuous source term on two different domains is considered in this article. The formal asymptotic solution is constructed by using the method of boundary layer functions. Furthermore, the uniform validity of the solutions is proved by using the maximum principle. Finally, as an illustration, an example is presented. Cited in 3 Documents MSC: 35B25 Singular perturbations in context of PDEs 35C20 Asymptotic expansions of solutions to PDEs 35R05 PDEs with low regular coefficients and/or low regular data 35J20 Variational methods for second-order elliptic equations 35J61 Semilinear elliptic equations Keywords:boundary layer function; maximum principle PDFBibTeX XMLCite \textit{D. Nie} and \textit{F. Xie}, Bound. Value Probl. 2016, Paper No. 164, 17 p. (2016; Zbl 1353.35034) Full Text: DOI References: [1] Ayadi, MA, Bchatnia, A, Hamouda, M, Messaoudi, S: General decay in a Timoshenko-type system with thermoelasticity with second sound. Adv. Nonlinear Anal. 4, 263-284 (2015) · Zbl 1329.35301 [2] Colli, P, Gilardi, G, Sprekels, J: A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions. Adv. Nonlinear Anal. 4, 311-325 (2015) · Zbl 1327.35162 [3] Radulescu, V, Repovš, D: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015) · Zbl 1343.35003 · doi:10.1201/b18601 [4] De Jager, EM, Jiang, F: The Theory of Singular Perturbations. Elsevier, Amsterdam (1996) [5] Babuška, I: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207-213 (1970) · Zbl 0199.50603 · doi:10.1007/BF02248021 [6] Babuška, I: Solution of problems with interface and singularities. Tech. Note BN-789, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland (1974) · Zbl 1187.65128 [7] Brayanov, IA: Numerical solution of a mixed singularly perturbed parabolic-elliptic problem. J. Math. Anal. Appl. 320, 361-380 (2006) · Zbl 1098.65086 · doi:10.1016/j.jmaa.2005.06.098 [8] Brayanov, IA: Numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem with discontinuous coefficients. Appl. Math. Comput. 182, 631-643 (2006) · Zbl 1113.65106 [9] O’Riordan, E: Opposing flows in a one dimensional convection-diffusion problem. Cent. Eur. J. Math. 10, 85-100 (2012) · Zbl 1259.65122 · doi:10.2478/s11533-011-0121-0 [10] Huang, Z: Tailored finite point method for the interface problem. Netw. Heterog. Media 4, 91-106 (2009) · Zbl 1187.65128 · doi:10.3934/nhm.2009.4.91 [11] Roos, HG, Zarin, H: A second-order scheme for singularly perturbed differential equations with discontinuous source term. J. Numer. Math. 10, 275-289 (2002) · Zbl 1023.65077 · doi:10.1515/JNMA.2002.275 [12] Lin, H, Xie, F: Singularly perturbed second order semilinear boundary value problems with interface conditions. Bound. Value Probl. 2015, 47 (2015) · Zbl 1315.34060 · doi:10.1186/s13661-015-0309-5 [13] Vasil’eva, AB, Butuzov, VF, Kalachev, LV: The Boundary Function Method for Singular Perturbation Problems. SIAM, Philadelphia (1995) · Zbl 0823.34059 · doi:10.1137/1.9781611970784 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.