Boundary Behavior of Solutions of a Class of Genuinely Nonlinear

We study the set of boundary singularities of arbitrary classical solutions of genuinely nonlinear planar hyperbolic systems of the form , where # ‚ # H V œ ! H 5 5 5 denotes differentiation in the direction , , , and where the defining / 5 œ " # 3 ÐV ßV Ñ )5 " # functions satisfy )5 ( ) , and i ) ) # " # œ  1 ( ) ii E  l ` ÐVÑ `V # )5 5 l  F V − 5 œ " # , for all , , ‘ for some positive constants , . We show that for any system of this kind there is a E F 7  " V K such that for any locally Lipschitz solution in a smoothly bounded domain , the set of points of at which fails to have a nontangential limit has Hausdorff `K V dimension at most , and, on the other hand, for any such system for which the 7 ) ‘ ‡ 5 _ # _ − G Ð Ñ G V , we construct a solution on a half-plane for which the set of points of at which fails to have a nontangential limit has positive Hausdorff dimension. ` V ‡ These results are 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 plane domain ? Here arbitrary K "classical" can be taken to mean , although the treatment we give will be valid for G locally Lipschitz solutions. In the first place, we are interested in systems whose formulation imposes no a priori limit on the range of characteristic directions, that is, systems such that for a characteristic given parametrically by , can DÐ=Ñ ÖD Ð=Ñ× arg w potentially cover all of , in contrast to what is implicitly the case in the standard ‘ spacetime context solutions rather than . Secondly, we are interested in statements valid for all 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