An extremum theory of the residual functional in Sobolev spaces \(W^{m,p}({\Omega{}})\)
Consider a properly posed problem of the partial differential equation \(L(u)=f\) and of the boundary value \(B(u)=g\) in the Sobolev space, where \(L\) and \(B\) are quasilinear and linear differential operators, respectively. In the paper under review a minimum existence theorem in the Sobolev space for the residual functional \[ J(u):=\| L(u)- f\|_ p^ p+\| B(u)-g\|_ p^ p \] is treated. Also, the weak compactness, force condition, lower semi-continuity and convexity of the residual functional are proved.
scientific article; zbMATH DE number 96725