A theorem on the existence of weak solutions of the Cauchy problem for first order functional differential equations defined on the Haar pyramid is proved. The initial problem is transformed into a system of functional integral equations for the unknown function and for its partial derivatives with respect to spatial variables. The method of bicharacteristics and integral inequalities are applied. Differential equations with deviated variables and differential integral equations can be obtained...

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order $$D_{x}z(x,y)=f(x,y,z_{(x,y)},D_{y}z(x,y)),$$ where $z_{(x,y)}[-\tau ,0]\times [0,h]\to ℝ$ is a function defined by $z_{(x,y)}(t,s)=z(x+t,y+s)$, $(t,s)\in [-\tau ,0]\times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.

We investigate the solvability of the linear Neumann problem (1.1) with L¹ data. The results are applied to obtain existence theorems for a semilinear Neumann problem.

We present a simple proof of the fact that if $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 2$, which is convex and symmetric with respect to $k$ orthogonal directions, $1\le k\le N$, then the nodal sets of the eigenfunctions of the laplacian corresponding to the eigenvalues ${\lambda }_2,\mathrm{\cdots },{\lambda }_{k+1}$ must intersect the boundary. This result was proved by Payne in the case $N=2$ for the second eigenfunction, and by other authors in the case of convex domains in the plane, again for the second eigenfunction.

We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in ℝ$. We investigate about the persistence of the decorrelation between the...

We explain the relation between the weak asymptotics method introduced by the author and V. M. Shelkovich and the classical Maslov-Whitham method for constructing approximate solutions describing the propagation of nonlinear solitary waves.

Let Hₙ be the (2n+1)-dimensional Heisenberg group, let p,q ≥ 1 be integers satisfying p+q=n, and let $L={\sum }_{j=1}^p(X{²}_j+Y{²}_j)-{\sum }_{j=p+1}^n(X{²}_j+Y{²}_j)$, where X₁,Y₁,...,Xₙ,Yₙ,T denotes the standard basis of the Lie algebra of Hₙ. We compute explicitly a relative fundamental solution for L.