Nonlocal Harnack inequalities
We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.
math.AP↗
arXiv subjects
Tuomo Kuusi
Publications and source records attributed to Tuomo Kuusi.
A quantitative modulus of continuity for the two-phase Stefan problem
We derive the quantitative modulus of continuity $$ ω(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-α(n,p)}, $$ which we conjecture to be optimal, for solutions of the $p$-degenerate two-phase Stefan problem. Even in the classical case $p=2$, this represents a twofold improvement with respect to the 1984 state-of-the-art result by DiBenedetto and Friedman [J. reine angew. Math., 1984], in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent $α(n,p)$.
math.AP↗
Riesz potentials and nonlinear parabolic equations
The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case
math.AP↗
Borderline estimates for fully nonlinear elliptic equations
We prove new borderline regularity results for solutions to fully nonlinear elliptic equations together with pointwise gradient potential estimates.
math.AP↗
Hölder continuity for Trudinger's equation in measure spaces
We complete the study of the regularity for Trudinger's equation by proving that weak solutions are Hölder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.
math.AP↗
Local Hölder continuity for doubly nonlinear parabolic equations
We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.
math.AP↗