Semielliptic singularities
Semigroup commutators under differences (II).
This is the second instalment of my previous paper with the same title, [1]. This paper consists of two different parts. The first part is devoted to improvements of the results developed in [1]. These improvements are described in section 0.1 below and developed in sections 1 to 5, and 9 to 10; they are in fact technically distinct from [1] and rely on a systematic use of microlocalisation in the context of Hörmander-Weyl calculus. These paragraphs can therefore be read quite independently from...
Semilinear diffraction of conormal waves (joint work with Melrose and Sa Barreto)
Semilinear Elliptic Equations on Unbounded Domains.
Semilinear hyperbolic systems in one space dimension with strongly singular initial data.
Semilinear waves with cusp singularities
Separable solutions of quasilinear Lane–Emden equations
For and either or , we prove the existence of solutions of in a cone , with vertex 0 and opening , vanishing on , of the form . The problem reduces to a quasilinear elliptic equation on and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.
Series solutions of time-fractional PDEs by homotopy analysis method.
Sharp condition for global existence and blow-up on Klein-Gordon equation.
Sheaf theory and regularity. Application to local and microlocal analysis
A review of some methods in sheaf theory is presented to make precise a general concept of regularity in algebras or spaces of generalized functions. This leads to the local analysis of the sections of sheaves or presheaves under consideration and then to microlocal analysis and microlocal asymptotic analysis.
Sign-changing and extremal constant-sign solutions of nonlinear elliptic Neumann boundary value problems.
Similarity and rescaling.
Simmetria delle soluzioni di equazioni ellittiche semilineari in
Nella prima parte di questa Nota si dimostrano dei risultati di simmetria unidimensionale e radiale per le soluzioni di in . Questi risultati sono legati a due congetture (De Giorgi, 1978 e Gibbons, 1994) riguardanti la classificazione delle soluzioni dell’equazione in . Si dimostra, in particolare, la seguente generalizzazione della congettura di Gibbons: se e se l’insieme degli zeri di è limitato nella direzione , allora , ovvero, è unidimensionale. Nella seconda parte si considerano...
Simplifying numerical solution of constrained PDE systems through involutive completion
When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations...
Simplifying numerical solution of constrained PDE systems through involutive completion
When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations...
Singular dynamics for semiconcave functions
Singular -harmonic functions and related quasilinear equations on manifolds.
Singular perturbation for the Dirichlet boundary control of elliptic problems
A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...
Singular perturbation for the Dirichlet boundary control of elliptic problems
A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...
Singular solutions of the Briot-Bouquet type partial differential equations