Sentence examples for results of regularity from inspiring English sources

end{aligned} (1.3) We assume (tilde{Q}, tilde{phi}in C^{infty}(partialOmega)), thus we have (tilde{Q}_{0}, tilde{phi}_{0}in C^{infty}(bar{ Omega})) by standard results of regularity.

In the first part of the paper we prove various results on regularity of Feynman Kac functionals of Hunt processes associated with time-dependent semi-Dirichlet forms.

Then, in order to have a weaker hypothesis on f, we prove a result of regularity for the composition of the Nemytskii operator with a Stepanov-like weighted pseudo almost periodic function, assuming a compactness condition.

As Kato remarks, in the atomic case, there were no previous positive results on regularity of eigenfunctions if (N ge 2) although it was known that certain series expansions did not work.

In the rest of this paper, we will show some applications of the above results to regularity of solutions to the higher-order elliptic equations with a potential.

We use techniques from Time frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operators.

9 we then gather a series of regularity results that follow as a corollary of the theory presented.

The results of calculations and regularities obtained on their bases are given.

The challenges are particularly accentuated for hyperbolic and convection-dominated PDEs as the discontinuities in physical space such as shocks can propagate into stochastic space resulting in a loss of regularity of the underlying solution with respect to the random parameters.

The other set of results gives local regularity of solutions to a class of elliptic systems with fairly rough coefficients.

One set of results gives microlocal regularity of such a function away from the conormal bundle of a foliation.