Page 519 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 519
NONLOCAL OPERATORS AND RELATED TOPICS (MS-55)
the operator L. We present a Fredholm alternative as λ tends to the eigenspace and characterise
the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test
function.
We then extend the results to singular boundary data and study the so-called large solutions,
which blow up at the boundary. For that problem we show that, for any regular value λ, there
exist “large eigenfunctions” that are singular on the boundary and regular inside. We are also
able to present a Fredholm alternative in this setting, as λ approaches the values of the spectrum.
We also obtain a maximum principle for weighted L1 solutions when the operator is L2-
positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from
below.
Finally, we recover the classical Dirichlet problem as the fractional exponent approaches
one under mild assumptions on the Green’s functions. Thus “large eigenfunctions” represent a
purely nonlocal phenomenon.
Fractional dissipations in fluid dynamics: the surface quasigeostrophic
equation
Maria Colombo, maria.colombo@epfl.ch
EPFL Lausanne, Switzerland
The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges
in meteorology and shares many of the essential difficulties of 3d fluid dynamics. In the su-
percritical regime for instance, where dissipation is modelled by a fractional Laplacian of order
less than 1/2, it is not known whether or not smooth solutions blow-up in finite time.
The goal of the talk is to show that every L2 initial datum admits an a.e. smooth solution
of the dissipative surface quasigeostrophic equation (SGQ); more precisely, we prove that those
solutions are smooth outside a compact set (away from t=0) of quantifiable Hausdorff dimen-
sion. We draw analogies between SQG and other PDEs in fluid dynamics in several aspects,
including the partial regularity results, and underline some extra structure that SQG enjoys.
This is a joint work with Silja Haffter (EPFL).
Local asymptotics and unique continuation from boundary points for
fractional equations
Veronica Felli, veronica.felli@gmail.com
Università di Milano - Bicocca, Italy
Coauthors: Alessandra De Luca, Stefano Vita
In this talk I will present some results in collaboration with A. De Luca and S. Vita on unique
continuation and local asymptotics of solutions to fractional elliptic equations at boundary
points, under some outer homogeneous Dirichlet boundary conditions. I will describe a blow-up
procedure which involves an Almgren type monotonicity formula and provides a classification
of all possible homogeneity degrees of limiting entire profiles. The Caffarelli-Silvestre ex-
tension provides an equivalent formulation of the fractional equation as a local degenerate or
singular problem in one dimension more, with mixed Dirichlet and Neumann boundary con-
ditions. In the development of a monotonicity argument, the mixed boundary condition raises
delicate regularity issues, which turn out to be quite difficult in dimension N ≥ 2 due to the
517
the operator L. We present a Fredholm alternative as λ tends to the eigenspace and characterise
the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test
function.
We then extend the results to singular boundary data and study the so-called large solutions,
which blow up at the boundary. For that problem we show that, for any regular value λ, there
exist “large eigenfunctions” that are singular on the boundary and regular inside. We are also
able to present a Fredholm alternative in this setting, as λ approaches the values of the spectrum.
We also obtain a maximum principle for weighted L1 solutions when the operator is L2-
positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from
below.
Finally, we recover the classical Dirichlet problem as the fractional exponent approaches
one under mild assumptions on the Green’s functions. Thus “large eigenfunctions” represent a
purely nonlocal phenomenon.
Fractional dissipations in fluid dynamics: the surface quasigeostrophic
equation
Maria Colombo, maria.colombo@epfl.ch
EPFL Lausanne, Switzerland
The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges
in meteorology and shares many of the essential difficulties of 3d fluid dynamics. In the su-
percritical regime for instance, where dissipation is modelled by a fractional Laplacian of order
less than 1/2, it is not known whether or not smooth solutions blow-up in finite time.
The goal of the talk is to show that every L2 initial datum admits an a.e. smooth solution
of the dissipative surface quasigeostrophic equation (SGQ); more precisely, we prove that those
solutions are smooth outside a compact set (away from t=0) of quantifiable Hausdorff dimen-
sion. We draw analogies between SQG and other PDEs in fluid dynamics in several aspects,
including the partial regularity results, and underline some extra structure that SQG enjoys.
This is a joint work with Silja Haffter (EPFL).
Local asymptotics and unique continuation from boundary points for
fractional equations
Veronica Felli, veronica.felli@gmail.com
Università di Milano - Bicocca, Italy
Coauthors: Alessandra De Luca, Stefano Vita
In this talk I will present some results in collaboration with A. De Luca and S. Vita on unique
continuation and local asymptotics of solutions to fractional elliptic equations at boundary
points, under some outer homogeneous Dirichlet boundary conditions. I will describe a blow-up
procedure which involves an Almgren type monotonicity formula and provides a classification
of all possible homogeneity degrees of limiting entire profiles. The Caffarelli-Silvestre ex-
tension provides an equivalent formulation of the fractional equation as a local degenerate or
singular problem in one dimension more, with mixed Dirichlet and Neumann boundary con-
ditions. In the development of a monotonicity argument, the mixed boundary condition raises
delicate regularity issues, which turn out to be quite difficult in dimension N ≥ 2 due to the
517
