Page 523 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 523
NONLOCAL OPERATORS AND RELATED TOPICS (MS-55)
according to the gradient flow of the renormalized energy. More precisely, we prove that the
gradient flow of the renormalized energy emerges as the Γ-convergence of gradient flows (in
the sense of E. Sandier & S. Serfaty) limit of the gradient flow of the Ginzburg-Landau energy.
This is a joint work with G. Canevari (Verona).
Global Harnack principle for a class of fast diffusion equations
Nikita Simonov, simonov.nikita@gmail.com
Universidad Autónoma de Madrid, Spain
Coauthors: Matteo Bonforte, Diana Stan
We study global properties of non-negative, integrable solutions to the Cauchy problem of the
weighted fast diffusion equation ut = |x|γdiv(|x|−β∇um) with (d − 2 − β)/(d − γ) < m < 1.
The weights |x|γ and |x|−β, with γ < d and γ − 2 < β ≤ γ(d − 2)/d can be both degenerate
and singular and need not belong to the class A2, this range of parameters is optimal for the
validity of a class of Caffarelli-Kohn-Nirenberg inequalities.
We characterize the largest class of data for which the so-called Global Harnack Principle
(GHP) holds (a global lower and upper bound in terms of suitable Barenblatt solutions). As
a consequence of the GHP, we prove convergence of the uniform relative error, namely |(u −
B)/B| → 0 as t → ∞ uniformly in Rd, where B is a suitable Barenblatt solution. In the case
with no weights (γ = β = 0) and for a special class of data, we give (almost) sharp rates of
convergence to the Barenblatt profile in the L1 and the L∞ topologies, in the radial case we give
sharp rates.
We extend some of the results to non-negative, integrable solutions to the Cauchy problem
of the p-Laplace evolution equation ut = ∆pu, where ∆pw := div(|∇w|p−2∇w), with 2d/(d +
1) < p < 2.
The above results were obtained in collaboration with Prof. M. Bonforte and D. Stan.
Nonlocal minimal graphs in the plane are generically sticky
Enrico Valdinoci, enrico.valdinoci@uwa.edu.au
University of Western Australia, Australia
We discuss some recent boundary regularity results for nonlocal minimal surfaces in the plane.
In particular, we show that nonlocal minimal graphs in the plane exhibit generically stickiness
effects and boundary discontinuities. More precisely, if a nonlocal minimal graph in a slab is
continuous up to the boundary, then arbitrarily small perturbations of the far-away data neces-
sarily produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontin-
uous at the boundary, or a small perturbation of the prescribed conditions produces boundary
discontinuities. The proof relies on a sliding method combined with a fine boundary regularity
analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal
minimal graphs are either discontinuous at the boundary or their derivative is Hölder continuous
up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal
graphs in the plane "jumps" from discontinuous to differentiable, with no intermediate possibili-
ties allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up
to the boundary. As a byproduct of our analysis, one describes the "switch" between the regime
of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and
521
according to the gradient flow of the renormalized energy. More precisely, we prove that the
gradient flow of the renormalized energy emerges as the Γ-convergence of gradient flows (in
the sense of E. Sandier & S. Serfaty) limit of the gradient flow of the Ginzburg-Landau energy.
This is a joint work with G. Canevari (Verona).
Global Harnack principle for a class of fast diffusion equations
Nikita Simonov, simonov.nikita@gmail.com
Universidad Autónoma de Madrid, Spain
Coauthors: Matteo Bonforte, Diana Stan
We study global properties of non-negative, integrable solutions to the Cauchy problem of the
weighted fast diffusion equation ut = |x|γdiv(|x|−β∇um) with (d − 2 − β)/(d − γ) < m < 1.
The weights |x|γ and |x|−β, with γ < d and γ − 2 < β ≤ γ(d − 2)/d can be both degenerate
and singular and need not belong to the class A2, this range of parameters is optimal for the
validity of a class of Caffarelli-Kohn-Nirenberg inequalities.
We characterize the largest class of data for which the so-called Global Harnack Principle
(GHP) holds (a global lower and upper bound in terms of suitable Barenblatt solutions). As
a consequence of the GHP, we prove convergence of the uniform relative error, namely |(u −
B)/B| → 0 as t → ∞ uniformly in Rd, where B is a suitable Barenblatt solution. In the case
with no weights (γ = β = 0) and for a special class of data, we give (almost) sharp rates of
convergence to the Barenblatt profile in the L1 and the L∞ topologies, in the radial case we give
sharp rates.
We extend some of the results to non-negative, integrable solutions to the Cauchy problem
of the p-Laplace evolution equation ut = ∆pu, where ∆pw := div(|∇w|p−2∇w), with 2d/(d +
1) < p < 2.
The above results were obtained in collaboration with Prof. M. Bonforte and D. Stan.
Nonlocal minimal graphs in the plane are generically sticky
Enrico Valdinoci, enrico.valdinoci@uwa.edu.au
University of Western Australia, Australia
We discuss some recent boundary regularity results for nonlocal minimal surfaces in the plane.
In particular, we show that nonlocal minimal graphs in the plane exhibit generically stickiness
effects and boundary discontinuities. More precisely, if a nonlocal minimal graph in a slab is
continuous up to the boundary, then arbitrarily small perturbations of the far-away data neces-
sarily produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontin-
uous at the boundary, or a small perturbation of the prescribed conditions produces boundary
discontinuities. The proof relies on a sliding method combined with a fine boundary regularity
analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal
minimal graphs are either discontinuous at the boundary or their derivative is Hölder continuous
up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal
graphs in the plane "jumps" from discontinuous to differentiable, with no intermediate possibili-
ties allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up
to the boundary. As a byproduct of our analysis, one describes the "switch" between the regime
of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and
521
