Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. (English) Zbl 0803.35046

Boccardo, L.; Giachetti, D.; Diaz, J. I.; Murat, F.

J. Differ. Equations 106, No. 2, 215-237 (1993).

This paper is devoted to the study of renormalized solutions of \[ -\text{div} \bigl( a(x,u,Du) \bigr) - \text{div} \bigl( \Phi (u) \bigr) + g(x,u) = f, \tag{*} \] on a bounded open subset of \(\mathbb{R}^ N\) with zero boundary conditions. The operator \(A\) defined on \(W^{1,p} (\Omega)\) (with \(p>1)\) by \((a\) is a vector valued Caratheodory function) \(Au = \text{div} (a(x,u,Du))\) is a nonlinear Leray-Lions operator such that there exist \(c_ i>0\) \((i=1,2,3)\) such that \[ \bigl | a(x,t, \xi) \bigr | \leq c_ 1 | t |^{p-1} + c_ 1 | \xi |^{p-1} + d(x), \quad d \in L^{p/(p-1)} (\Omega) \] \[ \bigl[ a(x,t,\xi) - a(x,t, \overline \xi) \bigr] [\xi - \overline \xi] > 0\;(\xi \neq \overline \xi),\quad a(x,t, \xi) \xi \geq c_ 3 | \xi |^ p \] \(\forall (t, \xi, \overline \xi) \in \mathbb{R}^{2N + 1}\), a.e. \(x \in \Omega\), \(f\) and \(\Phi\) belong respectively to \(W^{-1,p/(p-1)} (\Omega)\) and \((C^ 0(\mathbb{R}))^ N\) and \(g\) satisfies some more technical conditions.
It is proven that there exists a renormalized solution to \((*)\), i.e. a function \(u\) of \(W_ 0^{1,p} (\Omega)\) such that \[ - \biggl[\text{div} \bigl( a(x,u,Du) \bigr) \biggr] h(u) - \text{div} \bigl( \Phi (u)h(u)\bigr) + \Phi (u)h'(u)Du+ g(x,u)h(u) = fh(u) \] in \({\mathcal D}' (\Omega)\), \(\forall h \in C^ 1_ c (\Omega)\). Properties of these solutions are then studied: a sufficient condition for a renormalized solution to be a usual weak solution of \((*)\); energy identities; the largest class of possible test functions. \(L^ s\)-regularity of renormalized solutions is at least studied in the general case, and in a case where a further coerciveness condition (a growth condition) on \(g\) provides more regularity. Application of these two regularity results to the existence of weak solutions is finally given.

Reviewer: J.Dolbeault (Paris)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
35D10	Regularity of generalized solutions of PDE (MSC2000)

Keywords:

renormalized solutions; nonlinear Leray-Lions operator; largest class of possible test functions