The Euler Equation for Functionals with Linear Growth

The Cauchy problem for linear growth functionals

Asymptotic Formulas for Solutions of Half-linear Euler-weber Equation

p−1 p p−1 . This equation is a special case of a general half-linear second order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, (2) where Φ(x) := |x| sgn x, p > 1, and r, c are continuous functions, r(t) > 0 (in the studied equation (1) we have r(t) ≡ 1). Let us recall that similarly as in the linear case, which is a special case of (2) for p = 2 and equation (2) then reduces to the linear...

Almost multiplicative linear functionals and approximate spectrum

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

A Parabolic Quasilinear Problem for Linear Growth Functionals

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f (x;) = p 1 + kk 2 , which corresponds with the time-dependent minimal surface equation. We also study the asimpto...

Lyapunov Functionals for the Enskog Equation

Two Lyapunov functionals are presented for the Enskog equation. One is to describe interactions between particles with various velocities and another is to measure the L distance between two classical solutions. The former yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the L stability of glob...