On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data

This work focuses on the nonhomogeneous nonlocal double phase problem L_au(x)=f(x,u,D_s^pu,D_a,t^qu)in Ω, where Ω⊂R^N is a bounded domain with Lipschitz boundary, 0<s,t<1<p≤q<∞ with tq≤sp and the operator L_a is defined as L_au(x)=2P.V.∫R^N|u(x)−u(y)|^p−2(u(x)−u(y))K_s,p(x,y)+2P.V.∫R^Na(x,y)|u(x)−u(y)|^q−2(u(x)−u(y))K_t,q(x,y)dy. We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on f is also obtained using the notion of De Giorgi classes.