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Un théorème de valeurs intermédiaires dans les espaces de Sobolev et application. (A mean value theorem in Sobolev spaces and applications). (French) Zbl 0557.35055
Let \(\Omega\) be a connected open subset of \({\mathbb{R}}^ n\) and \(u\in W^{1,1}(\Omega):\) It is shown that u(\(\Omega)\) does not have any open ”hole”. From this result follows the necessary and sufficient conditions for some elliptic semi-linear problems of monotone type with Neumann boundary conditions to have multiple solutions. Analogous results are obtained for time-periodic solutions of corresponding parabolic problems.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B10 Periodic solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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References:
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