67 research outputs found
A chain rule formula in BV and applications to conservation laws
In this paper we prove a new chain rule formula for the distributional
derivative of the composite function , where
has bounded variation, is continuously differentiable and
has bounded variation. We propose an application of this formula
in order to deal in an intrinsic way with the discontinuous flux appearing in
conservation laws in one space variable.Comment: 26 page
Anzellotti's pairing theory and the Gauss--Green theorem
In this paper we obtain a very general Gauss-Green formula for weakly
differentiable functions and sets of finite perimeter. This result is obtained
by revisiting Anzellotti's pairing theory and by characterizing the measure
pairing when is a bounded divergence
measure vector field and is a bounded function of bounded variation.Comment: 27 page
Lower semicontinuity for non autonomous surface integrals
Some lower semicontinuity results are established for nonautonomous surface integrals depending in a discontinuous way on the spatial variable. The proof of the semicontinuity results is based on some suitable approximations from below with appropriate functionals
Nonautonomous chain rules in BV with Lipschitz dependence
The aim of this paper is to state
a nonautonomous chain rule in BV with Lipschitz dependence, i.e. a formula for the distributional derivative of the composite function v(x)=B(x,u(x)), where is a scalar function of bounded variation, has bounded variation and is only a Lipschitz continuous function.
We present a survey of recent developments on the nonautonomous chain rules in BV.
Formulas of this type are an useful tool especially in view to applications to lower semicontinuity for integral
functional (see cite{DC,dcfv,DCFV2,dcl}) and to the conservation laws with discontinuous flux (see cite{CD,CDD,CDDG})
An extension of the pairing theory between divergence-measure fields and BV functions
In this paper we introduce a nonlinear version of the notion of Anzellotti's
pairing between divergence--measure vector fields and functions of bounded
variation, motivated by possible applications to evolutionary quasilinear
problems. As a consequence of our analysis, we prove a generalized Gauss--Green
formula.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1708.0079
The Dirichlet problem for singular elliptic equations with general nonlinearities
In this paper, under very general assumptions, we prove existence and
regularity of distributional solutions to homogeneous Dirichlet problems of the
form where, is the -laplace operator,
is a bounded open subset of with Lipschitz boundary,
is a continuous function which may become singular at , and
is a nonnegative datum in with suitable small norm.
Uniqueness of solutions is also shown provided is decreasing and . As
a by-product of our method a general theory for the same problem involving the
-laplacian as principal part, which is missed in the literature, is
established. The main assumptions we use are also further discussed in order to
show their optimality
Representation formulas for pairings between divergence-measure fields and BV functions
The purpose of this paper is to find pointwise representation formulas for the density of the pairing between divergence-measure fields and BV functions, in this way continuing the research started in [17, 20]. In particular, we extend a representation formula from an unpublished paper of Anzellotti [7] involving the limit of cylindrical
averages for normal traces, and we exploit a result of [35] in order to derive another representation in terms of limits of averages in half balls
Representation formulas for pairings between divergence-measure fields and functions
The purpose of this paper is to find pointwise representation formulas for
the density of the pairing between divergence-measure fields and BV functions,
in this way continuing the research started in [17,20]. In particular, we
extend a representation formula from an unpublished paper of Anzellotti [7]
involving the limit of cylindrical averages for normal traces, and we exploit a
result of [35] in order to derive another representation in terms of limits of
averages in half balls.Comment: 24 page
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