Linear and Nonlinear Perturbations of the Operator Div
This book presents results onboundary-value problems for L and the theory of nonlinear perturbations of L. Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundary-value problems for the operator L. an analog of the Weyl decomposition is proved.
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Linear Perturbations of the Operator div
Nonlinear Perturbations of the Operator div
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assume assumptions of Theorem ball Banach spaces belongs boundary condition bounded domain Cauchy sequence change of variables Cl(u coincides compact support consider continuous derivatives continuously differentiable mapping critical regular point defined denote depends linearly diffeomorphism domain with boundary equation equivalent exists extension operator formula Function spaces functional 10.1 functional with constraint Hence holds identity mapping implicit function theorem implies integral Jacobi matrix Lagrange multipliers last equality Lemma Lq(u mapping F matrix multiplicative inequality natural number nonnegative nonsingular nonzero solution normed spaces notation obtain open set operator div orthogonal point z(x problem 4.1 prove quadratic form r-times continuously differentiable relation right-hand side satisfies the estimates satisfy the conditions scalar set of solutions Sobolev space solenoidal vector-valued function solvability sufficient conditions sufficiently small neighborhood surface structure symmetric matrix Theorem 5.1 vanishes vector-valued functions VT(X zero solution