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H\(^ 1\) versus C\(^ 1\) local minimizers. (English. Abridged French version) Zbl 0803.35029

Summary: We consider functionals of the form \(\Phi (u) = (1/2) \int_ \Omega | \nabla u |^ 2 - \int_ \Omega F(x,u)\). Under suitable assumptions we prove that a local minimizer of \(\Phi\) in the \(C^ 1\) topology must be a local minimizer in the \(H^ 1\) topology. This result is especially useful when the corresponding equation admits a sub and super solution.

MSC:

35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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