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 $v(x)=B(x,u(x))$, where $u:]a,b[\to\R^d$ has bounded variation, $B(x,\cdot)$ is continuously differentiable and $B(\cdot,u)$ 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 $(\boldsymbol{A}, Du)$ when $\boldsymbol{A}$ is a bounded divergence measure vector field and $u$ 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 $u:R^N oR$ is a scalar function of bounded variation, $B(cdot,t)$ has bounded variation and $B(x,cdot)$ 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 $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline u\geq 0& \text{in}\ \Omega, \newline u=0 & \text{on}\ \partial \Omega, \end{cases}$$ where, $\Delta_{1}$ is the $1$-laplace operator, $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty}(\Omega)$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-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 BVBV 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