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Register a new userExistence and uniqueness of propagating terraces
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Thomas Giletti
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This work focuses on dynamics arising from reaction-diffusion equations , where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment, under very wide and generic conditions. We are especially concerned with their existence, uniqueness, and their spatial structure. Our goal is to provide insight into the intricate dynamics arising from multistable non-linearities.
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There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the Riesz characteristic, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Amp`ere equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Holder continuity of subsolutions when the Riesz characteristic p satisfies $1 leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the geometric cases. A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Amp`ere case are completely classified.
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