A variational approach to parabolic equations under general and p,q-growth conditions

By Paolo Marcellini. 2020

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Abstract We consider variational solutions to the Cauchy-Dirichlet problem ∂ t u = div D ξ f ( x , u , D u ) − D u f ( x , u , D u ) in Ω T u = u 0 on ∂ par Ω T where the function f = f x , u , ξ , f : R n × R N × R N × n → [ 0 , ∞ ) , is convex with respect to u , ξ and coercive in ξ ∈ R N × n , but it not necessarily satisfies a growth condition from above. A motivation to consider a class of such energy functions f can be also easily found in the stationary case, where a large literature in the calculus of variations is devoted to the minimization of p , q -growth problems [45] and to double phase problems [23] , [24] , [4] , [5] , [6] . In the parabolic context the notion of variational solution (see the references from [8] to [15] ) is compatible with the lack of the same polynomial growth from below and from above.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7623208be819c0ca816567f05890c56fb7c83a93

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Lipschitz Bounds and Nonautonomous Integrals (2020, methodology): “for instance the paper [44] in the case of uniformly elliptic integrals. As for the nonuniformly elliptic setting, this approach has been used by Marcellini and coauthors, see for instance the survey [59] and references therein for a general overview. We also mention that, over the last years, Sobol”