Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

<p style='text-indent:20px;'>We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" <i>Analysis &amp; PDE</i>, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.</p>

Lack of smoothing for bounded solutions of a semilinear parabolic equation

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all t > 0. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as t → ∞.

Local Hölder regularity for a doubly singular PDE

We establish the local Hölder continuity for the nonnegative bounded weak solutions of a certain doubly singular parabolic equation. The proof involves the method of intrinsic scaling and the parabolic version of De Giorgi’s iteration method.

Hölder Regularity for Singular Parabolic Obstacle Problems of Porous Medium Type

We study the regularity of weak solutions to parabolic obstacle problems related to equations of singular porous medium type that are modeled after the nonlinear equation $$\partial_{t} u - \Delta u^{m} = 0.$$For the range of exponents 0 &lt; m &lt; 1, we prove that locally bounded weak solutions are locally Hölder continuous, provided the obstacle function is. Moreover, in the case $\frac{(n-2)_{+}}{n+2} &lt; m &lt; 1$ we show that every weak solution is locally bounded and therefore Hölder continuous.