Hyperbolic operators with non-Lipschitz coefficients

A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients

“…The improvement with respect to paper [5], however, was obtained defining a new energy, which involved (by derivation in time) second derivatives of the approximated coefficients. In paper [10], Colombini and Lerner considered instead the case when the coefficients a jk depend both on time and space variables. In particular, they assumed an isotropic punctual log-Lipschitz condition, i.e.…”

“…They assumed the same isotropic log-Lipschitz condition of [10] on the coefficients of the second order part of L, while b j and c j were supposed to be α-Hölder continuous (for some α ∈ ]1/2, 1[ ) and d to be only bounded. The authors headed towards questions such as local-in-space existence and uniqueness, and finite propagation speed.…”

“…The authors headed towards questions such as local-in-space existence and uniqueness, and finite propagation speed. Note that the restriction on α was due to the local-inspace study the authors performed (in particular, in order to have traces on the boundary well-defined): see for instance papers [10] and [9], were no bounds on the Hölder regularity were required.…”

“…In particular, according with the results of [10], [11] and [9], we will assume the first order coefficients to be Hölder continuous in the space variable, and the zero order coefficient to be just bounded. From an energy estimate analogous to (1.6), we will infer, for smooth in space coefficients, the H ∞ well-posedness of the related Cauchy problem with a finite loss of derivatives.…”

“…This means that we will focus on log-Lipschitz continuous a jk 's and Hölder continuous b j 's, while the 0-th order coefficient c can be taken just bounded (see also [10], [11] and [9] about these hypothesis).…”