1,721,178 research outputs found
Soluzioni di tipo barriera
No full textWe present the general theory of barrier solutions in the sense of De Giorgi, and we consider different applications to ordinary and partial differential equations. We discuss, in particular, the case of second order geometric evolutions, where the barrier solutions turn out to be equivalent to the well-known viscosity solutions
Convergence of an algorithm for the anisotropic and crystalline mean curvature flow
Full text linkWe give a simple proof of convergence of the anisotropic variant of a well-known algorithm for mean curvature motion, introduced in 1992 by Merriman, Bence, and Osher. The algorithm consists in alternating the resolution of the (anisotropic) heat equation, with initial datum the characteristic function of the evolving set, and a thresholding at level 1/2
Regularity of the obstacle problem for the parabolic biharmonic equation
No full textWe consider the obstacle problem for the parabolic biharmonic equation. We study the problem via an implicit time discretization, we prove the existence of a unique solution and discuss its regularity property
Regularity results for some 1-homogeneous functionals
No full textWe consider local minimizers for a class of 1-homogeneous integral functionals defined on BVloc(Omega), with Omega subset of R-2. Under general assumptions on the functional, we prove that the boundary of the subgraph of such minimizers is (locally) a lipschitz graph in a suitable direction. The proof of this statement relies on a regularity result holding for boundaries in R-2 which minimize an anisotropic perimeter. This result is applied to the boundary of sublevel sets of a minimizer u is an element of BV_loc(Omega)
Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations
No full textWe consider a mesoscopic model for phase transitions in a periodic medium and we construct multibump solutions. The rational perturbative case is dealt with by explicit asymptotics
Front propagation in infinite cylinders I. A variational approach
No full textIn their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on the real line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of a propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. Here we show that in problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the equation considered is a gradient flow in an exponentially weighted L(2)-space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. We show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a "pulled" front, or a "pushed" front. The choice of the scenario is completely characterized via a minimization problem
Volume constrained minimizers of the fractional perimeter with a potential energy
Full text linkWe consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate
Stability of crystalline evolutions
No full textIn this paper we analyze the stability properties of the Wulff-shape in the crystalline flow. It is well known that the Wulff-shape evolves self-similarly, and eventually shrinks to a point. We consider the flow restricted to the set of convex polyhedra, we show that the crystalline evolutions may be viewed, after a proper rescaling, as an integral curve in the space of polyhedra with fixed volume, and we compute the Jacobian matrix of this field. If the eigenvalues of such a matrix have real part different from zero, we can determine if the Wulff-shape is stable or unstable, i.e. if all the evolutions starting close enough to the Wulff-shape converge or not, after rescaling, to the Wulff-shape itself
The geometry of mesoscopic phase transition interfaces
No full textWe consider a mesoscopic model of phase transitions and investigate the geometric properties of the interfaces of the associated minimal solutions. We provide density estimates for level sets and, in the periodic setting, we construct minimal interfaces at a universal distance from any given hyperplane
Regularity results for boundaries in R^2 with prescribed anisotropic curvature
No full textIn this paper we consider the anisotropic perimeter
P-phi (E) = integral(partial derivative E) phi(nu(E)) dH(1)
defined on subsets E subset of R-2, where the anisotropy phi is a (possibly non-symmetric) norm on R-2 and nu(E) is the exterior unit normal vector to partial derivative E.
We consider quasi-minimal sets E (which include sets with prescribed curvature) and we prove that partial derivative E \ Sigma(E) is locally a bi-Lipschitz curve and the singular set Sigma(E) is closed and discrete.
We then classify the global P-phi-minimal sets. In particular we find that global minimal sets may have a singular point if and only if {phi <= 1} is a triangle or a quadrilateral and that sets with two singularities exist if and only if {phi <= 1} is a triangle.
We finally show that the boundary of a subset of R-2, which locally minimizes the anisotropic perimeter, plus a volume term (prescribed constant curvature) is contained, up to a translation and a rescaling, in the boundary of the Wulff shape determined by the anisotropy
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