We consider viscosity solutions for second order differential-functional equations of parabolic type. Initial value and mixed problems are studied. Comparison theorems for subsolutions, supersolutions and solutions are considered.
Let be a differential operator with constant coefficients depending analytically on a parameter . Assume that the family P(,D) is of constant strength. We investigate the equation where is a given analytic function of with values in some space of distributions and the solution is required to depend analytically on , too. As a special case we obtain a regular fundamental solution of P(,D) which depends analytically on . This result answers a question of L. Hörmander.
In this article, we prove the partial exact controllability of a nonlinear system. We use semigroup formulation together with fixed point approach to study the nonlinear system.
We apply Robin penalization to Dirichlet optimal control problems governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.
We apply Robin penalization to Dirichlet optimal control problems governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.
The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...