Hasse principles for multinorm equations

Invariance Principles for Parabolic Equations with Random Coeecients Invariance Principles for Parabolic Equations with Random Coeecients

A general Hilbert-space-based stochastic averaging theory is brought forth in this note for arbitrary-order parabolic equations with (possibly long range dependent) random coeecients. We start with regularity conditions on the coeecients and Cauchy data of @ t u " (x; t) = X 0jkjj2p A k (x; t="; !)@ k x u " (x; t); u " (x; 0) = '(x) (1) which are only slightly stronger than those required to pr...

Variational Principles for Some Nonlinear Wave Equations

Generally speaking, there exist two basic ways to describe a physical problem [1]: (1) by differential equations (DE) with boundary or initial conditions; (2) by variational principles (VP). The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration...

Dual Variational Principles for Diffusion Equations

New Maximum Principles for Linear Elliptic Equations

We prove extensions of the estimates of Aleksandrov and Bakel′man for linear elliptic operators in Euclidean space R to inhomogeneous terms in L spaces for q < n. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and L estimates.

Maximum principles for elliptic dynamic equations

We consider second order partial dynamic operators of the elliptic type on time scales. We establish basic maximum principles and apply them to obtain the uniqueness of Dirichlet boundary value problems for dynamic elliptic equations, eg Poisson equation. Our special cases include the situation in which several variables are continuous and the other discrete. We conclude with open problems, we ...