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We aim to study very classical problems in the Calculus of Variations: existence and regularity of minima of scalar multidimensional functionals. The existence problem is usually solved by means of the Direct Method of the Calculus of Variations whenever suitable convexity and growth assumptions are satisfied. The regularity problem has been deeply studied under either growth assumption and/or the uniform convexity of the lagrangian.
We aim to study these problems starting from the regularity of the boundary datum using the so called Hilbert-Haar approach that has been recently renewed. New results have been proved dropping the classical growth and/or uniform convexity assumptions for functional depending just on the gradient. Some open problems still remain in this framework and some recent results and ongoing research allows us to believe that it is possible to generalize these approach to langrangians depending also on lower order terms.
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Mathematics - Mathematical analysis
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