Boccardo, Lucio; Buttazzo, Giuseppe Quasilinear elliptic equations with discontinuous coefficients. (English) Zbl 0679.35035 Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 82, No. 1, 21-28 (1988). Summary: We prove an existence result for equations of the form \[ -D_ i(a_{ij}(x,u)D_ ju)=f\quad in\quad \Omega,\quad u\in H^ 1_ 0(\Omega), \] where the coefficients \(a_{ij}(x,s)\) satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable s. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients \(a_{ij}(x,s)\) are supposed only Borel functions. Cited in 1 ReviewCited in 1 Document MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 49J20 Existence theories for optimal control problems involving partial differential equations 35D05 Existence of generalized solutions of PDE (MSC2000) Keywords:quasilinear elliptic equations; Dirichlet problems; semicontinuity; calculus of variations × Cite Format Result Cite Review PDF