On a new class of fractional calculus of variations and related fractional differential equations. (Q2128417)

The authors analyze a class of fractional calculus of variations problems and their associated Euler-Lagrange equations. One of the main contribution in this work is that the fractional calculus of variations considered is based on a new developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange equations are well established. This is proved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which yield more general fractional differential equations.