Page 518 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 518
NONLOCAL OPERATORS AND RELATED TOPICS (MS-55)
Boundary value problems for the loaded equation with integro-differential
operator
Umida Baltaeva, umida baltayeva@mail.ru
Khorezm Mamun Academy, Urgench State University, Uzbekistan
It is well known that fractional derivatives (e.g., Riemann-Liouville and Caputo) were widely
used to model the complex phenomenon in science and engineering practice. In this connec-
tion, linear fractional partial differential equations models are commonly encountered in applied
mathematics and engineering.
On the other hand, in recent years it has become increasingly important to investigate a
new class of equations, known as loaded equations, as a direct result of issues with the optimal
control of the agro economical system, long-term forecasting and regulating the level of ground
water and soil moisture. However, we would like to note that boundary value problems for the
loaded equations of a mixed types with the integro-differential operator are not well studied.
Hence, the main aim of the work is to establish unique solvability boundary value problems for
the loaded integro-differential equations associated with non-local problems, for the classical
partial differential equations.
The Bernstein technique for integro-differential equations
Xavier Cabré, xavier.cabre@upc.edu
ICREA and Universitat Politecnica de Catalunya, Spain
In this talk I will present a joint work with S. Dipierro and E. Valdinoci in which we extend the
classical Bernstein technique to the setting of integro-differential operators. As a consequence,
we provide first and one-sided second derivative estimates for solutions to fractional equations,
including some convex fully nonlinear equations of order smaller than two, for which we prove
uniform estimates as their order approaches two. Our method is new even in the linear integro-
differential case. We will also raise some intriguing open questions, one of them concerning the
"pure" linear fractional Laplacian.
Blow-up phenomena in nonlocal eigenvalue problems: when theories of L1
and L2 meet
Hardy Chan, hardy.chan@math.ethz.ch
ETH Zürich, Switzerland
Coauthors: David Gómez-Castro, Juan Luis Vázquez
We develop a linear theory of very weak solutions for nonlocal eigenvalue problems Lu =
λu + f involving integro-differential operators posed in bounded domains with homogeneous
Dirichlet exterior condition, with and without singular boundary data. We consider mild hy-
potheses on the Green’s function and the standard eigenbasis of the operator. The main exam-
ples in mind are the fractional Laplacian operators.
Without singular boundary datum and when λ is not an eigenvalue of the operator, we
construct an L2-projected theory of solutions, which we extend to the optimal space of data for
516
Boundary value problems for the loaded equation with integro-differential
operator
Umida Baltaeva, umida baltayeva@mail.ru
Khorezm Mamun Academy, Urgench State University, Uzbekistan
It is well known that fractional derivatives (e.g., Riemann-Liouville and Caputo) were widely
used to model the complex phenomenon in science and engineering practice. In this connec-
tion, linear fractional partial differential equations models are commonly encountered in applied
mathematics and engineering.
On the other hand, in recent years it has become increasingly important to investigate a
new class of equations, known as loaded equations, as a direct result of issues with the optimal
control of the agro economical system, long-term forecasting and regulating the level of ground
water and soil moisture. However, we would like to note that boundary value problems for the
loaded equations of a mixed types with the integro-differential operator are not well studied.
Hence, the main aim of the work is to establish unique solvability boundary value problems for
the loaded integro-differential equations associated with non-local problems, for the classical
partial differential equations.
The Bernstein technique for integro-differential equations
Xavier Cabré, xavier.cabre@upc.edu
ICREA and Universitat Politecnica de Catalunya, Spain
In this talk I will present a joint work with S. Dipierro and E. Valdinoci in which we extend the
classical Bernstein technique to the setting of integro-differential operators. As a consequence,
we provide first and one-sided second derivative estimates for solutions to fractional equations,
including some convex fully nonlinear equations of order smaller than two, for which we prove
uniform estimates as their order approaches two. Our method is new even in the linear integro-
differential case. We will also raise some intriguing open questions, one of them concerning the
"pure" linear fractional Laplacian.
Blow-up phenomena in nonlocal eigenvalue problems: when theories of L1
and L2 meet
Hardy Chan, hardy.chan@math.ethz.ch
ETH Zürich, Switzerland
Coauthors: David Gómez-Castro, Juan Luis Vázquez
We develop a linear theory of very weak solutions for nonlocal eigenvalue problems Lu =
λu + f involving integro-differential operators posed in bounded domains with homogeneous
Dirichlet exterior condition, with and without singular boundary data. We consider mild hy-
potheses on the Green’s function and the standard eigenbasis of the operator. The main exam-
ples in mind are the fractional Laplacian operators.
Without singular boundary datum and when λ is not an eigenvalue of the operator, we
construct an L2-projected theory of solutions, which we extend to the optimal space of data for
516
