On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system.
On annealed elliptic Green's function estimates
We consider a random, uniformly elliptic coefficient field on the lattice . The distribution of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function satisfy optimal annealed estimates which are and , respectively, in probability, i.e., they obtained bounds on and . In particular, the elliptic Green’s function satisfies optimal annealed bounds. In their recent work, the authors...
On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in the space variables
On approximation of solutions to some -dimensional integrable systems by Bäcklund transformations.
On bilinear restriction type estimates and applications to nonlinear wave equations
I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “bilinear estimates”. In addition to the theory, which is now quite well developed, I plan to discuss a more general point of view concerning the theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss...
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations
Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations∗∗∗
Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...
On contact equivalence of holomorphic Monge-Ampére equations.
On continuation of regular solutions of linear partial differential equations
On decomposable Monge-Ampère equations.
On evolution Galerkin methods for the Maxwell and the linearized Euler equations
The subject of the paper is the derivation and analysis of evolution Galerkin schemes for the two dimensional Maxwell and linearized Euler equations. The aim is to construct a method which takes into account better the infinitely many directions of propagation of waves. To do this the initial function is evolved using the characteristic cone and then projected onto a finite element space. We derive the divergence-free property and estimate the dispersion relation as well. We present some numerical...
On existence of a unique generalized solution to systems of elliptic PDEs at resonance
The Dirichlet boundary value problem for systems of elliptic partial differential equations at resonance is studied. The existence of a unique generalized solution is proved using a new min-max principle and a global inversion theorem.
On existence of the weak solution for nonlinear diffusion equation
The paper concerns the existence of bounded weak solutions of a anonlinear diffusion equation with nonhomogeneous mixed boundary conditions.
On existence of the weak solution for non-linear partial differential equations of elliptic type. II.
On extrema of functionals
On formal theory of differential equations. I.
On formal theory of differential equations. II.
On formal theory of differential equations. III.
Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy...
On functional linear partial differential equations in Gevrey spaces of holomorphic functions.
We investigate existence and unicity of global sectorial holomorphic solutions of functional linear partial differential equations in some Gevrey spaces. A version of the Cauchy-Kowalevskaya theorem for some linear partial -difference-differential equations is also presented.
On functions, whose lines of steepest descent bend proportionally to level lines