Existence of evolutionary variational solutions via the calculus of variations

In this paper we introduce a purely variational approach to time dependent problems, yielding the existence of global parabolic minimizers, that is∫0T∫Ω[u⋅∂tφ+f(x,Du)]dxdt⩽∫0T∫Ωf(x,Du+Dφ)dxdt, whenever T>0T>0 and φ∈C0∞(Ω×(0,T),RN). For the integrand f:Ω×RNn→[0,∞]f:Ω×RNn→[0,∞] we merely assume convexity with respect to the gradient variable and coercivity. These evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals. In the simplest situation, with some growth conditions on f, the method provides the existence of global weak solutions to Cauchy–Dirichlet problems of parabolic systems of the type∂tu−divDξf(x,Du)=0in Ω×(0,∞).