Research Of The Fractional MEMS Equation With Fringing Field

In this thesis,firstly,we redefine the fractional Laplacian by extension method,and then we give the fractional subsolutions and supersolutions method and its proof,finally we consider the existence of solutions of the fractional MEMS equation with fringing field.In Chapter three,we mainly discuss the fractional Laplacian with 1/2 order,first of all,we extend ? in RN to a cylinder D in one more dimension,in addition,we define a new space ?(?)in the sense of traces and the harmonic extension of functions in this space.A series of equivalence proofs will confirm that,we can change a nonlocal problem with Dirichlet boundary condition to a local problem with Neumann boundary condition,and then we establish the definition of weak solutions of equations with 1/2 order.After giving the proof of fractional subsolutions and supersolutions method,we obtain an approach to discussing the existence of solutions of the fractional MEMS equation with fringing field,but when ? is small enough,we turn to a similar equation.