Strong Solvability of Nonlinear Elliptic Problems with Discontinuous Coefficients

Sofia Giuffrè D.I.M.E.T. Faculty of Engineering, University of Reggio Calabria, Italy

                      e-mail giuffre@ing.unirc.it

Our aim is to present some results about strong solvability of boundary value problems for nonlinear elliptic equations with discontinuous coefficients.

In particular we prove existence results for planar oblique derivative problems, studying different behaviours of the field l which describes boundary condition, and with a n-dimensional boundary value problem, n ≥ 2, with unilateral boundary conditions, which derives from classical Signorini problem.

The merely Carathéodory condition on the operators is not enough to ensure solvability of boundary value problems. There are examples of not solvability even in the linear case for Dirichlet problem.

In the above problems we assume the Carathéodory operators to satisfy an ellipticity condition that does not imply continuity of the operator with respect to the independent variable x, but, in view of the Carathéodory character, only measurability with respect to x is required.

First we study an oblique derivative problem for the operator A(x, u, Du, D^{2}u), with boundary condition \frac{∂u}{∂l} + σu = 0, σ > 0. We assume the operator to growth quadratically with respect to the gradient, l to be C^{1}-piecewice on the boundary ∂Ω and l · χ ≥ 0, where χ is the unit outward normal to ∂Ω, that is the field l is of neutral type. Using Leray-Shauder fixed point and a semilinear variant of Aleksandrov-Pucci maximum principle proved by Trudinger, after a step by step process, we obtain W^{2,q}(Ω) solvability, q ∈]2, q̃[.

Later we study a problem for the operator A (x, D^{2}u), with boundary condition \frac{∂u}{∂l} = 0, where the field l is of emergent type, that is l · χ changes its sign on the trajectories of l through the points of tangency. Prescribing the values of u at the points of tangency, by the aid of a technique based on a fixed point theorem, we obtain an existence and uniqueness theorem in W^{2,2}(Ω).

Finally we study a n-dimensional problem, n ≥ 2, for the operator A (x, D^{2}u) − λu, λ > 0, with unilateral boundary conditions, that arises from elastic-plastic problems. We prove an uniqueness and existence result in W^{2,2}(Ω), using an estimate for the second derivatives of the functions which satisfy the unilateral boundary conditions, in which it has been possible to prove that the constant is equal to 1.