We are concerned with an infinite dimensional variational inequality which is connected with the dynamic oligopolistic market equilibrium problem. We will provide existence theorems and show, under minimal assumptions on the data, the Lipschitz continuity of the solution. Moreover a general duality theory is provided overcoming the difficulty of the voidness of the interior of the ordering cone which defines the cone constraints
We consider the weighted traffic equiUbrium problem introduced in [4] and we apply the infinite-dimensional duality theorem developed in [2], obtaining the existence of Lagrange variables, which allow to describe the behavior of the weighted traffic. Moreover, making use of a regularity result we present a descritization method to compute the weighted traffic equilibrium solution.
Strong solvability in Sobolev spaces is proved for a unilateral boundary value problem for nonlinear parabolic operators. The operator is assumed to be of Carathéodory type and to satisfy a suitable ellipticity condition; only measurability with respect to the independent variable X is required.The main tools of the proof are an estimate for the second derivatives of functions which satisfy the unilateral boundary conditions and the monotonicity of the operator −ut with respect to ∆u for the same functions.
Mathematics Subject Classification (2000). Primary 35K60; Secondary 35K85.
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