We 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.
We 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.
We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form . If is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace with , which is defined as the smallest degenerate elliptic function above .
We show a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under volume and convexity constraints.We prove existence of minimizers, and we describe their shape as the volume tends to zero or to infinity.
We 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.
In the present paper, we motivate and describe a numerical approach in order to detect the creation of fractures in a facet of a crystal evolving by anisotropic mean curvature. The result appears to be in accordance with the known examples of facet-breaking. Graphical simulations are included.
We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution.
We study the thresholds for the emergence of various properties in random subgraphs of (ℕ, <). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two–dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which...
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