The Weak Lower Semicontinuity For A Type Of Nonlocal Integral Functionals

With abundant nonlinear problems emerging in natural science and engineering, theSobolev space appears its limitation in application. For instance, it does not work in theresearch on a type of nonlinear problems with variable exponential growth, which is afresh area for research. In the process of studying on this class of nonlinear problems, thevariable exponential Lebesgue and Sobolev spaces provide their theoretic support.In this paper, based on the variable exponent Sobolev spaceWk, p(x), we havestudied the weak lower semicontinuity for a type of nonlocal integral function. The weaklower semicontinuity of this kind of functions is meaningful and interesting. Besides, itcan be applied to obtain the existence of an equilibrium solution in nonlinear elasticity.As p (x)is a function, it is more complicated than the general Sobolev spaceW k, p(x).For example, p (x)is nonhomogeneous, which makes some methods and skills used inthe general spaces unable to be used in the p (x)case. In this paper, under standardgrowth conditions, we have discussed the weak lower semicontinuity and convexity ofthis type of functions. Furthermore, we give the proof of the equivalence of these twoproperties.The main contents are as follows:1. According to the results on Young Measures in W k,p(), we have discussedthe counterpart in W k,p(x)(). Thus, we transform the weak lower continuity of thefunctions to the issue of minimizing sequence for the function.2. According to the convex envelope and Young Measures in W k,p(x)(), we havediscussed the minimizing sequence for the function. Then we obtain some resultsconcerning convexity.3. We have discussed the equivalence of the weak lower semicontinuity andconvexity of the integral functions.