Related papers: Integro-Differential Elliptic Equations
We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide…
We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to…
The focus of this book is on the analysis of regularization methods for solving \emph{nonlinear inverse problems}. Specifically, we place a strong emphasis on techniques that incorporate supervised or unsupervised data derived from prior…
We consider an elliptic equation with the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$ in the dissipative term, a singular integral operator ${\bf A}(\cdot)$ in the nonlinear term, and an external source $f$. The key example…
The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions…
We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
The exact and approximate solutions of singular integro-differential equations relating to the problems of interaction of an elastic thin finite or infinite non-homogeneous patch with a plate are considered, provided that the materials of…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for…
Recently, various systems of nonlinear difference equations, of different forms, were studied. In this existing work, two earlier published papers, due respectively to Bayram and Das. [Appl. Math. Sci. (Ruse), 4(7) (2010) pp. 817-821] and…
This thesis pertains to the study of elliptic and parabolic partial differential equations on "thin" structures. The first main objective is to establish the strong and weak low-dimensional counterparts of the parabolic Neumann problem. The…
This thesis is divided into five chapters. The aim is the study of the effectiveness of a chemical as defined by R. Aris for semilinear elliptic equations. The first chapter focuses on homogenization on quasilinear diffusion-reaction…
In this paper, the continuity of solutions for elliptic equations in divergence form with distributional coefficients is considered. Inspired by the discussion on necessary and sufficient conditions for the form boundedness of elliptic…
This paper aims to establish counterparts of fundamental regularity statements for solutions to elliptic equations in the setting of low-dimensional structures such as, for instance, glued manifolds or CW-complexes. The main result proves…
The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{align*}…
This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence form as follows \begin{align*}…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
This course is intended as an introduction to the analysis of elliptic partial differential equations. The objective is to provide a large overview of the different aspects of elliptic partial differential equations and their modern…