Journal ArticleDOI
On Positive Solutions of Elliptic Equations
TLDR
In this paper, the authors studied weak solutions of elliptic equations of the form in a bounded domain, where it is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts.Abstract:
In this paper the authors study weak solutions of elliptic equations of the form in a bounded domain . It is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts. It is assumed moreover that an estimate on the -norm of the solution holds on some subdomain . Summability of such solutions with a weight function that vanishes at the boundary is established, and with the use of the results of Ja. A. Roĭtberg integral representations are given in terms of the Green's function for the Dirichlet problem.Bibliography: 8 titles.read more
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Functional Analysis, Sobolev Spaces and Partial Differential Equations
TL;DR: In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.
Posted Content
Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result
Yannick Sire,Enrico Valdinoci +1 more
TL;DR: In this paper, the authors studied the symmetry properties of solutions of nonlocal boundary reaction equations of the type where the Dirichlet-to-Neumann operator is the so-called fractional Laplacian.
Journal ArticleDOI
Bernstein and De Giorgi type problems: new results via a geometric approach
TL;DR: In this article, a Poincare type formula and level set analysis are used to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div a(|∇u(x)|)∇ u(x) + f(u(X)) = 0.
Journal ArticleDOI
Schrödinger operators in the twentieth century
TL;DR: In this paper, a review of the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics is presented, with a focus on nonlinearity.
Journal ArticleDOI
Internal lifshits tails for random perturbations of periodic schrodinger operators
TL;DR: In this article, the authors considered the problem of periodic Schrödinger operators and showed that the lower bound of the energy consumption of the periodic approximations to the problem can be reduced to a discrete problem.
References
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Journal ArticleDOI
On boundary values of generalized solutions of elliptic equations
TL;DR: In this article, it was proved that similar properties are also possessed by generalized solutions of elliptic equations in a bounded region of n-dimensional space, and the main results were announced earlier (see MR 41 #3968).
Journal ArticleDOI
Properties of solutions of linear evolutionary systems with elliptic space part
V A Kondrat'ev,S D Èĭdel'man +1 more
TL;DR: In this article, the authors studied the growth of the solution of Cauchy's problem in a convex cone of the space in a band, in a halfspace, and in the entire space.
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