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Singular quasilinear convective systems involving variable exponents

Moussaoui Abdelkrim, Nabab Dany, Vélin Jean

Opuscula Mathematica

2024

Vol. 44, no. 1

105--134

The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.

Regularity of fundamental solutions for Lévy-type operators

Szczypkowski Karol

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 2

169--191

For a class of non-symmetric non-local Lévy-type operators Lκ, which include those of the form Lκf(x) := Rd(f(x + z) − f(x) − 1|z|<1⟨z, ∇f(x)⟩)κ(x, z)J(z) dz, e prove regularity of the fundamental solution pκ to the equation ∂t = Lκ.

Some gradient estimates on covering manifolds

Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

437-443

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.