Local approximation of superharmonic and superparabolic functions in nonlinear potential theory (Q360555)

Two equations are studied. The supersolutions of the equations \[ \text{div}(|\nabla u|^{p-2}\nabla u)= 0 \] and \[ {\partial v\over\partial t}= \text{div}(|\nabla v|^{p-2}\nabla v) \] are defined via comparison principles and they are, by definition, lower semicontinuous. Here \(p\geq 2\) and \(u= u(x)\), \(v= v(x,t)\). Using auxiliary obstacle problems, the authors show that such supersolutions are limits of much more regular supersolutions, which, actually, are Hölder continuous. In the parabolic case the assumption \(v\in L^{p-2}_{\text{loc}}\) is needed to avoid supersolutions that are not generating Riesz measures. A useful application is given for uniformly bounded sequences \(v_1,v_2,v_3,\dots\)\ .