This is the expanded text of a lecture about viscosity solutions of degenerate elliptic equations delivered at the XVI Congresso UMI. The aim of the paper is to review some fundamental results of the theory as developed in the last twenty years and to point out some of its recent developments and applications.
The aim of this paper is to prove the existence of solutions to the Poisson equation in weighted Sobolev spaces, where the weight is the distance to some distinguished axis, raised to a negative power. Therefore we are looking for solutions which vanish sufficiently fast near the axis. Such a result is useful in the proof of the existence of global regular solutions to the Navier-Stokes equations which are close to axially symmetric solutions.
In this paper we want to show how well-known results from the theory of (regular) elliptic boundary value problems, function spaces and interpolation, subordination in the sense of Bochner and Dirichlet forms can be combined and how one can thus get some new aspects in each of these fields.
The author proves the existence of the multi-parameter asymptotic error expansion to the usual five-point difference scheme for Dirichlet problems for the linear and semilinear elliptic PDE on the so-called uniform and nearly uniform domains. This expansion leads, by Richardson extrapolation, to a simple process for accelerating the convergence of the method. A numerical example is given.
The author proves the existence of the multi-parameter asymptotic error expansion to the five-point difference scheme for Dirichlet problems for the linear and semilinear elliptic PDE on general domains. By Richardson extrapolation, this expansion leads to a simple process for accelerating the convergence of the method.
We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the metric Laplacian, and we consider in particular Riemannian or Finsler manifolds, Carnot-Carathéodory spaces, Gaussian spaces. The second one deals with the optimal shape of a graph when the minimization cost is of spectral type. The third one is the optimization problem for a Schrödinger potential in suitable classes.
The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives ideas for generalization...
The paper concerns the existence of weak solutions to nonlinear elliptic equations of the form A(u) + g(x,u,∇u) = f, where A is an operator from an appropriate anisotropic function space to its dual and the right hand side term is in with 0 < m < 1. We assume a sign condition on the nonlinear term g, but no growth restrictions on u.