Related papers: A Note on Linear Elliptic Systems on $\R^d$
The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients…
The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity…
We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain…
In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body…
We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive…
This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in $L^2$ for the mixed boundary value problems with bounded measurable…
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some…
We consider the problem of existence and uniqueness of strong a.e. solutions $u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE system \[\label{1} \tag{1} F(\cdot,D^2u ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \]…
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case,…
We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for…
We are concerned with the problem of determining the nonlinear term in a semilinear elliptic equation by boundary measurements. Precisely, we improve [5, Theorem 1.3], where a logarithmic type stability estimate was proved. We show actually…
We determine the general form of the potential of the problem of motion of a rigid body about a fixed point, which allows the angular velocity to remain permanently in a principal plane of inertia of the body. Explicit solution of the…
In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both…
We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this…
We survey recent progress in a program aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space ${\mathbb{R}}^n_{+}$ for second-order, homogeneous,…
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…
We consider inverse problems for non-linear hyperbolic and elliptic equations and give an introduction to the method based on the multiple linearization, or on the construction of artificial sources, to solve these problems. The method is…
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…
The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested…
In this paper we establish a new class of weighted Hardy-Sobolev type inequalities under mild monotonicity assumptions on the weight function. As a consequence, we derive the corresponding weighted Sobolev and trace-type inequalities. These…