Wellposedness and regularity of a variable-order space-time fractional diffusion equation

We prove wellposedness of a variable-order linear space-time fractional diffusion equation in multiple space dimensions. In addition we prove that the regularity of its solutions depends on the behavior of the variable order (and its derivatives) at time [Formula: see text], in addition to the usual smoothness assumptions. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at [Formula: see text] or have certain singularity at [Formula: see text] like its constant-order fractional analogue if the variable order has a non-integer value at time [Formula: see text].

A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems

We prove partial regularity for minimizers to elasticity type energies with [Formula: see text]-growth, [Formula: see text], in a geometrically linear framework in dimension [Formula: see text]. Therefore, the energies we consider depend on the symmetrized gradient of the displacement field. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than [Formula: see text], and actually [Formula: see text] in the autonomous case (full regularity is well-known in dimension [Formula: see text]). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions [Formula: see text] and [Formula: see text].

A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis

The time-fractional diffusion partial differential equations (tFPDEs) (of order 0 < α < 1) properly model the anomalous diffusive transport or memory effects. Recent work [23] showed that the first-order time derivatives of their solutions have a singularity of O(tα−1) near the initial time t = 0, which makes the error estimates of their numerical approximations in the literature that were proved under full regularity assumptions of the true solutions inappropriate. A sharp error estimate was proved for a finite difference method (FDM) with a graded partition for a one-dimensional tFPDE without artificial regularity assumptions on true solutions, [23]. Motivated by the derivation of the tFPDE from stochastic continuous time random walk (CTRW), we present a modified tFPDE and prove that it has full regularity on the entire time interval (including t = 0) and that its FDM on a uniform time partition has an optimal-order convergence rate only under the assumptions of the regularity of the initial condition and right-hand source term. Numerical experiments show that with the same initial data, the solutions of the modified tFPDE and the classical tFPDE converge to each other as time increases, but the solution of the former does not have the singularity as that to the classical tFPDE near time t = 0.

Erratum: "Full regularity for a C*-algebra of the canonical commutation relations"

The proof of the main theorem of the paper [1] contains an error. We are grateful to Professor Ralf Meyer (Mathematisches Institut, Georg-August Universität Göttingen) for pointing out this mistake.

Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)

Let {\boldsymbol{L}} be a second order uniformly elliptic operator, and consider the equation {\boldsymbol{L}u=f} under the boundary condition {u=0}. We assume data f in generical subspaces of continuous functions {D_{\overline{\omega}}} characterized by a given modulus of continuity{\overline{\omega}(r)}, and show that the second order derivatives of the solution u belong to functional spaces {D_{\widehat{\omega}}}, characterized by a modulus of continuity{\widehat{\omega}(r)} expressed in terms of {\overline{\omega}(r)}. Results are optimal. In some cases, as for Hölder spaces, {D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity occurs. In particular, full regularity occurs for the new class of functional spaces {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular case, the classical Hölder spaces {C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}. Few words, concerning the possibility of generalizations and applications to non-linear problems, are expended at the end of the introduction and also in the last section.