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Green’s function estimates for a singularly perturbed convection-diffusion problem. (English) Zbl 1235.35087

It is considered the following problem \[ -\varepsilon(u_{xx}+u_{yy})-(a(x,y)u)_x+b(x,y)u=f(x,y), \,\, (x,y)\in \Omega, \]
\[ u(x,y)=0,\,\, (x,y)\in \partial \Omega,\quad (\Omega=(0,1)^2). \] The authors obtain upper and lower \(L_1\)-estimates of Green’s function and its first- and second-order derivatives. These estimates are sharp with respect to the parameter \(\varepsilon\). Green’s function is represented in explicit form and contains the modified Bessel function \(K_0\).

MSC:

35J08 Green’s functions for elliptic equations
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
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