Results 1 
3 of
3
The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use th ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
An L¹Theory Of Existence And Uniqueness Of Solutions Of Nonlinear Elliptic Equations
"... In this paper we study the questions of existence and uniqueness of solutions for equations of the form \GammaAu = F (x; u), posed in \Omega\Gamma an open subset of R N (bounded or unbounded) , with Dirichlet boundary conditions. A is a nonlinear elliptic operator modeled on the pLaplacian operat ..."
Abstract
 Add to MetaCart
In this paper we study the questions of existence and uniqueness of solutions for equations of the form \GammaAu = F (x; u), posed in \Omega\Gamma an open subset of R N (bounded or unbounded) , with Dirichlet boundary conditions. A is a nonlinear elliptic operator modeled on the pLaplacian operator \Delta p (u) = div (jDuj p\Gamma2 Du), with p ? 1, and F (x; u) is a Caratheodory function which is nonincreasing in u. Typical cases include F (x; u) = f(x) or F (x) = f(x) \Gamma fi(u), where fi is an increasing function with fi(0) = 0 (or even a maximal monotone graph with 0 2 fi(0)). We use an integrability assumption on F which in these cases means that f 2 L 1 (\Omega\Gamma9 The existence theory offers few difficulties when p ? N . Here we consider the case 1 ! p ! N and establish existence of a weak solution u. For p ? 2 \Gamma (1=N) and\Omega bounded the solution lies in the usual Sobolev space W 1;q 0 (\Omega\Gamma with 1 ! q ! p = N(p \Gamma 1)=(N \Gamma 1). However, when...