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This paper is concerned with analyzing a class of fractional calculus ofvariations problems and their associated Euler-Lagrange (fractionaldifferential) equations. Unlike the existing fractional calculus of variationswhich is based on the classical notion of fractional derivatives, thefractional calculus of variations considered in this paper is based on a newlydeveloped notion of weak fractional derivatives and their associated fractionalorder Sobolev spaces. Since fractional derivatives are direction-dependent,using one-sided fractional derivatives and their combinations leads to newtypes of calculus of variations and fractional differential equations as wellas nonstandard Neumann boundary operators. The primary objective of this paperis to establish the well-posedness and regularities for a class of fractionalcalculus of variations problems and their Euler-Lagrange (fractionaldifferential) equations. This is achieved first for one-sided Dirichlet energyfunctionals which lead to one-sided fractional Laplace equations, then for moregeneral energy functionals which give rise to more general fractionaldifferential equations.

On a New Class of Fractional Calculus of Variations and Related Fractional Differential Equations