Related papers: Variational formulae for the capacity induced by s…
We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles…
We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order…
This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.
This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…
A special class of doubly stochastic (Markov) operators is constructed. These operators come from measure preserving transformations and inherit some of their properties, namely ergodicity and positivity of entropy, yet they may have no…
We obtain a derivative formula for various notions of capacity. Namely we identify the second order term in the asymptotic expansion of the capacity of a union of two sets, as their distance goes to infinity. Our result applies to the usual…
The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
We investigate the second time scale of the metastable behavior of the reversible inclusion process in an extension of the study by [Bianchi, Dommers, and Giardin\`a, Electronic Journal of Probability, 22: 1-34, 2017], which presented the…
The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell…
We consider small perturbations of a dynamical system on the one-dimensional torus. We derive sharp estimates for the pre-factor of the stationary state, we examine the asymptotic behavior of the solutions of the Hamilton-Jacobi equation…
For arbitrary second-order differential operators, the existence conditions and the construction of intertwining transmutation operators are shown. In an explicit form found hyperbolic equations with two independent variables and their…
We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special…
The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian manifolds, and to point out a number of…
A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are…
We consider second-order evolution equations in an abstract setting with intermittently delayed/ not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalising our previous results…
We prove the validity of regularizing properties of a double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces by…
We proposed in \cite{bl2} a new approach to prove the metastable behavior of reversible dynamics based on potential theory and local ergodicity. In this article we extend this theory to nonreversible dynamics based on the Dirichlet…
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the…
Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density $\vartheta$ to the $L_2$-adjoints of these operators evaluated at the density $\vartheta'$ are…