arXiv subjects
Publications and source records attributed to Graziano Crasta.
We prove two rigidity results for a variational infinity ground state $u$ of an open bounded convex domain $Ω\subset \mathbb{R}^n$. They state that $u$ coincides with a multiple of the distance from the boundary of $Ω$ if either $|\nabla u|$ is constant on $\partial Ω$, or $u$ is of class $C ^ {1,1}$ outside the high ridge of $Ω$. Consequently, in both cases $Ω$ can be geometrically characterized as a "stadium-like domain".
In this paper we prove a nonautonomous chain rule formula for the distributional divergence of the composite function $\boldsymbol{v}(x)=\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a divergence--measure vector field and $u$ is a function of bounded variation. As an application, we prove a uniqueness result for scalar conservation laws with discontinuous flux.
We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.
In this paper we consider the chain rule formula for compositions $x\mapsto F(x, u(x))$ in the case when $u$ has a Sobolev or BV regularity and $F(x,z)$ is separately Sobolev, or BV, with respect to $x$ and $C^1$ with respect to $z$. Our results extend to this "nonautonomous" case the results known for compositions $x\mapsto F(u(x))$.
We give a complete characterization, as "stadium-like domains", of convex subsets $Ω$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $Ω$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet problem is of class $C^{1,1} (Ω)$.
We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.
Given an open bounded subset $Ω$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-Δ_{\infty} u = 1$ in $Ω$, subject to the homogeneous boundary condition $u = 0$ on $\partial Ω$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(Ω)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial Ω$; by using a suitable $P$-function we prove that, if $Ω$ satisfies the same assumptions as above and in addition contains a ball with touches $\partial Ω$ at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of $Ω$ coincide. In turn, in dimension $n=2$, this entails that $Ω$ must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class $C^2$.
We prove a uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain $Ω\subset \mathbb{R}^n$ in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in $Ω$ with constant source term admits a viscosity solution depending only on the distance from $\partial Ω$. This problem was previously addressed and studied by Buttazzo and Kawohl. In the light of some geometrical achievements reached in our recent paper "On the characterization of some classes of proximally smooth sets", we revisit the results obtained by Buttazzo and Kawohl and we prove strengthened versions of them, where any regularity assumption on the domain and on the solution is removed. Our results require a delicate analysis based on viscosity methods. In particular, we need to build suitable viscosity test functions, whose construction involves a new estimate of the distance function near singular points.
We provide a complete characterization of closed sets with empty interior and positive reach in $\mathbb{R}^2$. As a consequence, we characterize open bounded domains in $\mathbb{R}^2$ whose high ridge and cut locus agree, and hence $C^1$ planar domains whose normal distance to the cut locus is constant along the boundary. The latter results extends to convex domains in $\mathbb{R}^n$.
We consider a system of PDEs of Monge-Kantorovich type that, in the isotropic case, describes the stationary configurations of two-layers models in granular matter theory with a general source and a general boundary data. We propose a new weak formulation which is consistent with the physical model and permits us to prove existence and uniqueness results.
We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $$φ(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \in \partial Ω$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \to + \infty$ of Serrin's symmetry problem for the $p$-Laplacian), and equations in divergence form whose solutions depend only on the distance from the boundary in some subset of their domain.
We prove a uniqueness result of solutions for a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $Ω\subset\mathbf{R}^n$, and on the (possibly asymmetric) norm defined in $Ω$. In the special case when $Ω$ is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.
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.
We study the geodesics in a planar chessboard structure with two values 1 and $β>1$. The results for a fixed structure allow us to infer the properties of the Finsler metrics obtained, with an homogenization procedure, as limit of oscillating chessboard structures.
The aim of this paper is to give an existence result for a class of one-dimensional, non-convex, non-coercive problems in the Calculus of Variations. The main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity.
We consider a class of one--dimensional non--convex non--coercive problems in the Calculus of Variations. We prove an existence result for this class of problems using a Liapunov type theorem on the range of non--atomic measures.
Let $F:[0,T]\times\R^n\mapsto 2^{\R^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if $F$ satisfies the following Lipschitz Selection Property: \begin{itemize} \item[{(LSP)}] {\sl For every $t,x$, every $y\in \overline{co} F(t,x)$ and $\varepsilon>0$, there exists a Lipschitz selection $ϕ$ of $\overline{co}F$, defined on a neighborhood of $(t,x)$, with $|ϕ(t,x)-y|<\varepsilon$.} \end{itemize} then there exists a measurable selection $f$ of $ext F$\ such that, for every $x_0$, the Cauchy problem $$ \dot x(t)=f(t,x(t)),\qquad\qquad x(0)=x_0 $$ has a unique Caratheodory solution, depending continuously on $x_0$. We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class, for which (LSP) holds, consists of those continuous multifunctions $F$ whose values are compact and have convex closure with nonempty interior.