Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach

摘要

Let Ω i and Ω o be two bounded open subsets of mathbbRn{{mathbb{R}}^{n}} containing 0. Let G i be a (nonlinear) map from ¶Wi×mathbbRn{partialOmega^{i}times {mathbb{R}}^{n}} to mathbbRn{{mathbb{R}}^{n}} . Let a o be a map from ∂Ω o to the set M_n(mathbbR){M_{n}({mathbb{R}})} of n × n matrices with real entries. Let g be a function from ∂Ω o to mathbbRn{{mathbb{R}}^{n}} . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from ]1-(2),+¥[×M_n(mathbbR){]1-(2),+infty[times M_{n}({mathbb{R}})} to M_n(mathbbR){M_{n}({mathbb{R}})} . Then we consider the problem $left{ {ll} {{rm div}}, (T(omega,Du))=0 &quad {{rm in}} ;Omega^{o} setminusepsilon{{rm cl}} Omega^{i}, -T(omega,Du(x))nu_{epsilonOmega^{i}}(x)=frac{1}{gamma(epsilon)}G^{i}({x}/{epsilon}, gamma(epsilon)epsilon^{-1} ({rm log} , epsilon)^{-delta_{2,n}} u(x)) & quad forall x in epsilonpartialOmega^{i}, T(omega, Du(x)) nu^{o}(x)=a^{o}(x)u(x)+g(x) & quad forall x in partial Omega^{o}, right.$left{ begin{array}{ll} {{rm div}}, (T(omega,Du))=0 &quad {{rm in}} ;Omega^{o} setminusepsilon{{rm cl}} Omega^{i}, -T(omega,Du(x))nu_{epsilonOmega^{i}}(x)=frac{1}{gamma(epsilon)}G^{i}({x}/{epsilon}, gamma(epsilon)epsilon^{-1} ({rm log} , epsilon)^{-delta_{2,n}} u(x)) & quad forall x in epsilonpartialOmega^{i}, T(omega, Du(x)) nu^{o}(x)=a^{o}(x)u(x)+g(x) & quad forall x in partial Omega^{o}, end{array} right.