## 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)

### Keywords:

local minimizers of nonlinear functionals