Boundary Behavior of Solutions of a Class of Genuinely Nonlinear Hyperbolic Systems

We study the set of boundary singularities of arbitrary classical solutions of a certain class of genuinely nonlinear hyperbolic systems written in Riemann # ‚ # invariant form, , where denotes the directional derivative in the direction H V œ ! H 5 5 5 / 5 œ " # 3 ÐV ßV Ñ 5 )5 " # , , . The specific conditions that we place on the defining functions ) are ( ) and ( ) i ii ) ) # " # œ  E  1 l ` ÐVÑ `V # )5 5 l  F V − E F , for some positive constants , . ‘ We show for any system of this kind there is a such that for any locally Lipschitz 7  " solution in a smoothly bounded domain the set of points of at which fails to V K `K V have a nontangential limit has Hausdorff dimension at most , and, on the other hand, for 7 any such system for which the we construct a solution on a half) ‘ 5 _ # _ − G Ð Ñ G V plane for which the set of points of at which fails to have a nontangential limit ‡ ‡ ` V has positive Hausdorff dimension. This result is immediately applicable to constant principal strain mappings, which are defined in terms of a system of this kind for which )" is a linear function of and . V V " # 1. ntroduction I For hyperbolic systems in two independent variables and , most often B > associated with space and time, one usually studies the Cauchy problem in which one seeks a solution , for which coincides with a given , the ?ÐBß >Ñ > € ! ?ÐBß !Ñ ? ÐBÑ ! questions considered including well-posedness, global existence, blow-up and behavior of solutions as . In the nonlinear case discussion is often limited to initial data with > p _ a small range and even for such data, generalized solutions must be considered. In this paper we concern ourselves with the following inverse question for a certain family of genuinely nonlinear hyperbolic systems: What can be said about # ‚ # the boundary values of an classical solution in a domain ? Here "classical" arbitrary K can be taken to mean , although the treatment we give will be valid for locally G Lipschitz solutions. In the first place, we are interested in systems for which there is no a priori limit on the range of characteristic directions, that is, systems such that for a characteristic given parametrically by , can potentially cover any interval DÐ=Ñ ÖD Ð=Ñ× arg w of , in contrast to what is implicitly the case in the standard . ‘ space-time context Secondly, we are interested in statements valid for all solutions rather than ones known to arise from some form of initial value problem. Because of this generality even in geometrically simple domains such as disks or half-planes characteristics can be quite contorted curves. the set of Although the specific focus of this paper is the size of boundary points at which arbitrary solutions can fail to have nontangential limits, it would be reasonable to investigate other aspects of their behavior and that of the associated characteristics. In any event, given the nonstandard nature of the boundary value question and of several of the issues that arise in dealing with it, we shall begin with a somewhat detailed discussion of a system for which it is physically meaningful, namely the system which describes smooth planar mappings with constant principal stretches cps-mappings , about which we have previously written ([ChG],[G1]-[G5]). It Ð Ñ is in fact the study of the boundary behavior of such mappings that is the main goal of this paper, and we have only chosen to work in a wider context because it is possible to do so