In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in}\quad \Omega\subset \mathbb{R}\times \mathbb{R}^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb{R}^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.

}, doi = {10.1007/s00205-010-0381-z}, url = {http://hdl.handle.net/1963/4911}, author = {Stefano Bianchini and Camillo De Lellis and Roger Robyr} } @article {2011, title = {Structure of level sets and Sard-type properties of Lipschitz maps}, number = {SISSA;51/2011/M}, year = {2011}, institution = {SISSA}, url = {http://hdl.handle.net/1963/4657}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {2011, title = {A uniqueness result for the continuity equation in two dimensions}, number = {SISSA;52/2011/M}, year = {2011}, institution = {SISSA}, url = {http://hdl.handle.net/1963/4663}, author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa} } @article {2010, title = {Estimates on path functionals over Wasserstein Spaces}, journal = {SIAM J. Math. Anal. 42 (2010) 1179-1217}, number = {SISSA;11/2009/M}, year = {2010}, abstract = {In this paper we consider the class a functionals (introduced in [Brancolini, Buttazzo, and Santambrogio, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415-434] $\\\\mathcal{G}_{r,p}$ defined on Lipschitz curves $\\\\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\\\\mu$, give conditions in order to assure the existence of a curve $\\\\gamma$ such that $\\\\gamma(0)=\\\\mu$, $\\\\gamma(1)=\\\\delta_{x_0}$, and $\\\\mathcal{G}_{r,p}(\\\\gamma)<+\\\\infty$. To this end, new estimates on $\\\\mathcal{G}_{r,p}(\\\\mu)$ are given, and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions.}, doi = {10.1137/100782693}, url = {http://hdl.handle.net/1963/3583}, author = {Stefano Bianchini and Alessio Brancolini} } @article {2010, title = {On the Euler-Lagrange equation for a variational problem : the general case II}, journal = {Math. Z. 265 (2010) 889-923}, number = {SISSA;75/2007/M}, year = {2010}, doi = {10.1007/s00209-009-0547-2}, url = {http://hdl.handle.net/1963/2551}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2010, title = {The Monge problem in geodesic spaces}, number = {SISSA;33/2010/M}, year = {2010}, url = {http://hdl.handle.net/1963/3873}, author = {Stefano Bianchini and Fabio Cavalletti} } @article {2010, title = {On optimality of c-cyclically monotone transference plans}, journal = {Comptes Rendus Mathematique 348 (2010) 613-618}, year = {2010}, publisher = {Elsevier}, abstract = {Abstract. This note deals with the equivalence between the optimality of a transport plan for the Monge-Kantorovich problem and the condition of c-cyclical monotonicity, as an outcome of the construction in [7]. We emphasize the measurability assumption on the hidden structure of linear preorder, applied also to extremality and uniqueness problems. Resume. Dans la presente note nous decrivons brievement la construction introduite dans [7] a propos de l\\\'equivalence entre l\\\'optimalite d\\\'un plan de transport pour le probleme de Monge-Kantorovich et la condition de monotonie c-cyclique ainsi que d\\\'autres sujets que cela nous amene a aborder. Nous souhaitons mettre en evidence l\\\'hypothese de mesurabilite sur la structure sous-jacente de pre-ordre lineaire.}, doi = {10.1016/j.crma.2010.03.022}, url = {http://hdl.handle.net/1963/4023}, author = {Stefano Bianchini and Laura Caravenna} } @article {2009, title = {The boundary Riemann solver coming from the real vanishing viscosity approximation}, journal = {Arch. Ration. Mech. Anal. 191 (2009) 1-96}, number = {SISSA;24/2006/M}, year = {2009}, abstract = {We study the limit of the hyperbolic-parabolic approximation $$ \\\\begin{array}{lll} v_t + \\\\tilde{A} ( v, \\\\, \\\\varepsilon v_x ) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in R^N\\\\\\\\ \\\\tilde \\\\beta (v (t, \\\\, 0)) = \\\\bar g \\\\\\\\ v (0, \\\\, x) = \\\\bar v_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nThe function $\\\\tilde \\\\beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\\\\tilde \\\\beta$ is not invertible.\\nThe data $\\\\bar g$ and $\\\\bar v_0$ are constant. When $\\\\tilde B$ is invertible, the previous problem takes the simpler form $$ \\\\left\\\\{ \\\\begin{array}{lll} v_t + \\\\tilde{A} \\\\big( v, \\\\, \\\\varepsilon v_x \\\\big) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in \\\\mathbb{R}^N\\\\\\\\ v (t, \\\\, 0) \\\\equiv \\\\bar v_b \\\\\\\\ v (0, \\\\, x) \\\\equiv \\\\bar{v}_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nAgain, the data $\\\\bar v_b$ and $\\\\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $\\\\tilde A$ can be 0.\\n Second, as pointed out before we take into account the possibility that $\\\\tilde B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.}, doi = {10.1007/s00205-008-0177-6}, url = {http://hdl.handle.net/1963/1831}, author = {Stefano Bianchini and Laura Spinolo} } @article {2009, title = {A connection between viscous profiles and singular ODEs}, journal = {Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41}, number = {SISSA;05/2008/M}, year = {2009}, url = {http://hdl.handle.net/1963/2555}, author = {Stefano Bianchini and Laura Spinolo} } @article {2009, title = {On the extremality, uniqueness and optimality of transference plans}, journal = {Bull. Inst. Math. Acad. Sin. (N.S.) 4 (2009) 353-458}, number = {SISSA;46/2009/M}, year = {2009}, abstract = {We consider the following standard problems appearing in optimal mass transportation theory: when a transference plan is extremal; when a transference plan is the unique transference plan concentrated on a set A,; when a transference plan is optimal.}, url = {http://hdl.handle.net/1963/3692}, author = {Stefano Bianchini and Laura Caravenna} } @article {2009, title = {The Monge problem for distance cost in geodesic spaces}, number = {SISSA;50/2009/M}, year = {2009}, url = {http://hdl.handle.net/1963/3703}, author = {Stefano Bianchini and Fabio Cavalletti} } @article {2008, title = {Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems}, number = {SISSA;83/2008/M}, year = {2008}, url = {http://hdl.handle.net/1963/3400}, author = {Stefano Bianchini and Laura Spinolo} } @inbook {2008, title = {Transport Rays and Applications to Hamilton{\textendash}Jacobi Equations}, booktitle = {Nonlinear PDE{\textquoteright}s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, C{\'e}dric Villani. Eds. Luigi Ambrosio, Giuseppe Savar{\'e}. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20}, year = {2008}, note = {This volume collects the notes of the CIME course Nonlinear PDE{\textquoteright}s and\\r\\napplications held in Cetraro (Italy) on June 23{\textendash}28, 2008. The school consisted\\r\\nin 5 series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), Felix Otto (Bonn University), Cedric Villani (Ecole Normale Superieure de Lyon).}, publisher = {Springer}, organization = {Springer}, abstract = {The aim of these notes is to introduce the readers to the use of the Disintegration Theorem for measures as an effective tool for reducing problems in transport equations to simpler ones. The basic idea is to partition Rd into one dimensional sets, on which the problem under consideration becomes one space dimensional (and thus much easier, hopefully).}, isbn = {978-3-642-21718-0}, doi = {10.1007/978-3-642-21861-3_1}, url = {http://hdl.handle.net/1963/5463}, author = {Stefano Bianchini and Matteo Gloyer} } @article {2007, title = {Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy}, journal = {Comm. Pure Appl. Math. 60 (2007) 1559-1622}, number = {SISSA;83/2005/M}, year = {2007}, abstract = {We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition.}, doi = {10.1002/cpa.20195}, url = {http://hdl.handle.net/1963/1780}, author = {Stefano Bianchini and Bernard Hanouzet and Roberto Natalini} } @article {2007, title = {On the Euler-Lagrange equation for a variational problem}, journal = {Discrete Contin. Dynam. Systems A 17 (2007) 449-480}, number = {SISSA;95/2005/M}, year = {2007}, url = {http://hdl.handle.net/1963/1792}, author = {Stefano Bianchini} } @article {2007, title = {Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem}, year = {2007}, institution = {SISSA}, url = {http://preprints.sissa.it/handle/1963/35315}, author = {Stefano Bianchini} } @article {2006, title = {On Bressan\\\'s conjecture on mixing properties of vector fields}, journal = {Self-Similar Solutions of Nonlinear PDE / Ed. Piotr Biler and Grzegorz Karch. - Banach Center Publ. 74 (2006) 13-31}, number = {SISSA;70/2005/M}, year = {2006}, url = {http://hdl.handle.net/1963/1806}, author = {Stefano Bianchini} } @article {2006, title = {Glimm interaction functional for BGK schemes}, number = {SISSA;69/2005/M}, year = {2006}, url = {http://hdl.handle.net/1963/1770}, author = {Stefano Bianchini} } @article {2005, title = {Vanishing viscosity solutions of nonlinear hyperbolic systems}, journal = {Ann. of Math. 161 (2005) 223-342}, number = {SISSA;86/2001/M}, year = {2005}, publisher = {Annals of Mathematics}, abstract = {We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$.}, url = {http://hdl.handle.net/1963/3074}, author = {Stefano Bianchini and Alberto Bressan} } @article {2003, title = {A note on singular limits to hyperbolic systems of conservation laws}, journal = {Commun. Pure Appl. Ana., 2003, 2, 51-64}, number = {SISSA;85/00/M}, year = {2003}, publisher = {SISSA Library}, abstract = {In this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. \\nUnder the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation.}, url = {http://hdl.handle.net/1963/1542}, author = {Stefano Bianchini} } @article {2002, title = {A center manifold technique for tracing viscous waves}, journal = {Commun. Pure Appl. Anal. 1 (2002) 161-190}, number = {SISSA;85/2001/M}, year = {2002}, publisher = {American Institute of Mathematical Sciences}, abstract = {In this paper we introduce a new technique for tracing viscous travelling profiles. To illustrate the method, we consider a special 2 x 2 hyperbolic system of conservation laws with viscosity, and show that any solution can be locally decomposed as the sum of 2 viscous travelling profiles. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data.}, url = {http://hdl.handle.net/1963/3075}, author = {Stefano Bianchini and Alberto Bressan} } @article {2002, title = {On a Lyapunov functional relating shortening curves and viscous conservation laws}, journal = {Nonlinear Anal. 51 (2002) 649-662}, number = {SISSA;123/99/M}, year = {2002}, publisher = {Elsevier}, abstract = {We study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law.}, doi = {10.1016/S0362-546X(01)00848-3}, url = {http://hdl.handle.net/1963/1337}, author = {Stefano Bianchini and Alberto Bressan} } @article {2002, title = {On the Stability of the Standard Riemann Semigroup}, journal = {P. Am. Math. Soc., 2002, 130, 1961}, number = {SISSA;71/00/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1528}, author = {Stefano Bianchini and Rinaldo M. Colombo} } @article {2001, title = {A case study in vanishing viscosity}, journal = {Discrete Cont. Dyn. Syst. 7 (2001) 449-476}, year = {2001}, publisher = {American Institute of Mathematical Sciences}, url = {http://hdl.handle.net/1963/3091}, author = {Stefano Bianchini and Alberto Bressan} } @article {2001, title = {A Glimm type functional for a special Jin-Xin relaxation model}, journal = {Ann. Inst. H. Poincare\\\' Anal. Non Lineaire 18 (2001), no. 1, 19-42}, number = {SISSA;140/99/M}, year = {2001}, publisher = {Elsevier}, doi = {10.1016/S0294-1449(00)00124-4}, url = {http://hdl.handle.net/1963/1355}, author = {Stefano Bianchini} } @article {2001, title = {Stability of L-infinity solutions for hyperbolic systems with coinciding shocks and rarefactions}, journal = {Siam J. Math. Anal., 2001, 33, 959}, number = {SISSA;65/00/M}, year = {2001}, publisher = {SISSA Library}, abstract = {We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\\\\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \\\\mathcal{S}_t u_0$, defined on initial data $u_0 \\\\in L^\\\\infty$. The semigroup $\\\\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\\\\text{loc}}$ topology. Moreover $\\\\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.}, doi = {10.1137/S0036141000377900}, url = {http://hdl.handle.net/1963/1523}, author = {Stefano Bianchini} } @article {2000, title = {BV solutions for a class of viscous hyperbolic systems}, journal = {Indiana Univ. Math. J. 49 (2000) 1673-1714}, year = {2000}, publisher = {Indiana University Mathematics Journal}, doi = {10.1512/iumj.2000.49.1776}, url = {http://hdl.handle.net/1963/3194}, author = {Stefano Bianchini and Alberto Bressan} } @article {2000, title = {The semigroup generated by a Temple class system with non-convex flux function}, journal = {Differential Integral Equations 13 (2000) 1529-1550}, number = {SISSA;107/98/M}, year = {2000}, publisher = {Khayyam Publishing}, abstract = {We consider the Cauchy problem for a nonlinear n {\texttimes} n system of conservation laws of Temple class, i.e. with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.}, url = {http://hdl.handle.net/1963/3221}, author = {Stefano Bianchini} } @article {2000, title = {On the shift differentiability of the flow generated by a hyperbolic system of conservation laws}, journal = {Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329-350}, number = {SISSA;60/99/M}, year = {2000}, publisher = {American Institute of Mathematical Sciences}, doi = {10.3934/dcds.2000.6.329}, url = {http://hdl.handle.net/1963/1274}, author = {Stefano Bianchini} } @article {1999, title = {Extremal faces of the range of a vector measure and a theorem of Lyapunov}, journal = {J. Math. Anal. Appl. 231 (1999) 301-318}, number = {SISSA;4/98/M}, year = {1999}, publisher = {Elsevier}, doi = {10.1006/jmaa.1998.6260}, url = {http://hdl.handle.net/1963/3370}, author = {Stefano Bianchini} } @article {1999, title = {Vanishing viscosity solutions of hyperbolic systems on manifolds}, number = {SISSA;24/99/M}, year = {1999}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1238}, author = {Stefano Bianchini and Alberto Bressan} } @article {1999, title = {The vector measures whose range is strictly convex}, journal = {J. Math. Anal. Appl. 232 (1999) 1-19}, number = {SISSA;108/96/M}, year = {1999}, publisher = {Elsevier}, doi = {10.1006/jmaa.1998.6215}, url = {http://hdl.handle.net/1963/3546}, author = {Stefano Bianchini and Carlo Mariconda} }