Quasilinear diffusion problems with singular coefficients with respect to the unknown

Abstract

We study a class of quasilinear elliptic problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown; the other coefficients being continuous with respect to the unknown (without any growth assumption). We introduce two equivalent notions of solutions for such problems and we prove an existence result in these frameworks. Under additional local assumptions on the coefficients, we also establish the uniqueness of the solution. In that case, and when the non-diagonal coefficients are bounded, this unique (generalized) solution is also the unique weak solution strictly less than the value where the diagonal coefficient blows up.

Nonlinear equations with unbounded heat conduction and integrable data

“…Let us point out that the energy condition that we obtain in the present paper is more precise that the one stated in [6].…”

“…In this subsection, we give the definition of a solution of (1.1). This definition is more precise than the one used in [6] in the the sense that it localizes the behavior of the energy near the zone where a solution may reach the value m (see [6]). …”

“…When dealing such problems, the main difficulty is indeed to give a sense to the flux A(t, x, u)Du on the set {(t, x) ; u(t, x) = m}, or more precisely to describe the behavior of the energy A ε (t, x, u ε )Du ε · Du ε for approximate solutions u ε . In the elliptic case where the matrix is independent of the space variable and with a diagonal blowing part the analysis was carried out in [6] for L 2 data where an L 2 estimate for A ε (u ε )Du ε was derived. Then the authors gave two formulations of problem of type (1.1), both of them using a sort of decoupling behavior of the solution on the subset {u < m} and on the subset {u = m}.…”

“…Here we just assume that f is in L 1 and then we have to handle the set {u = m}. We give a notion of renormalized solutions (which is more precise in the elliptic case than the one given in [6]) and we prove the existence of such solutions for L 1 data. The formulation of the problem consists (as in [6]) in considering the equation on the subset {u < m} on the one hand and in specifying the behavior of the energy near the subset {u = m} where the coefficients are singular on the other hand.…”

“…We give a notion of renormalized solutions (which is more precise in the elliptic case than the one given in [6]) and we prove the existence of such solutions for L 1 data. The formulation of the problem consists (as in [6]) in considering the equation on the subset {u < m} on the one hand and in specifying the behavior of the energy near the subset {u = m} where the coefficients are singular on the other hand. For this point of view, it is at least formally similar to the formulation for elliptic problems with measure data considered in [7], where one distinguishes the equation where the measure is smooth and the behavior of the energy where the measure is singular.…”