2017/05/26 by Daniela Giachetti, Giachetti, Daniela, Pedro J. Martínez−Aparicio +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1705.09527
We perform the homogenization of the semilinear elliptic problem\n \
begincases u^
varepsilon
geq 0 amp;
mboxin
;\n
Omega^
varepsilon,
displaystyle - div
,A(x) D u^
varepsilon =\nF(x,u^
varepsilon) amp;
mboxin
;
Omega^
varepsilon,
u^
varepsilon = 0 amp;\n
mboxon
;
partial
Omega^
varepsilon.
endcases In this\nproblem F(x,s) is a Carath 'eodory function such that 0 \≤ F(x,s) \≤\nh(x)/\Γ(s) a.e. x\∈\Ω for every s > 0, with h in some\nLr(\Ω) and \Γ a C1([0, +\∞[) function such that \Γ(0)\n= 0 and \Γ'(s) > 0 for every s > 0. On the other hand the open sets\n\Ω^\ε are obtained by removing many small holes from a fixed\nopen set \Ω in such a way that a "strange term" \μ u0 appears in the\nlimit equation in the case where the function F(x,s) depends only on x.\n We already treated this problem in the case of a "mild singularity", namely\nin the case where the function F(x,s) satisfies 0 \≤ F(x,s) \≤ h(x)\n( frac 1s + 1). In this case the solution u^\ε to the problem\nbelongs to H10 (\Ω^\ε) and its definition is a "natural" and\nrather usual one.\n In the general case where F(x,s) exhibits a "strong singularity" at u =\n0, which is the purpose of the present paper, the solution u^\ε to\nthe problem only belongs to H tiny loc1(\Ω^\ε) but in\ngeneral does not belongs to H10 (\Ω^\ε) any more, even if\nu^\ε vanishes on \∂\Ω^\ε in some sense.\nTherefore we introduced a new notion of solution (in the spirit of the\nsolutions defined by transposition) for problems with a strong singularity.\nThis definition allowed us to obtain existence, stability and uniqueness\nresults.\n