We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics th...
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We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds. (C) 2011 Elsevier Masson SAS. All rights reserved.
We provide a sufficient condition for the existence of a positive solution to -Delta u + V(vertical bar x vertical bar)u = u(p) in B-1, when p is large enough. Here B-1 is the unit ball of R-n, n >= 2, and we deal ...
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We provide a sufficient condition for the existence of a positive solution to -Delta u + V(vertical bar x vertical bar)u = u(p) in B-1, when p is large enough. Here B-1 is the unit ball of R-n, n >= 2, and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case V(vertical bar x vertical bar) >= 0, V not equivalent to 0 is smooth and p is sufficiently large, and the Neumann problem always admits a solution.
The article offers information on the study regarding deformation that occurs in the boundary of the growth of a solid. It mentions the use of theory of stratifications of differentiated varieties. It says that the us...
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The article offers information on the study regarding deformation that occurs in the boundary of the growth of a solid. It mentions the use of theory of stratifications of differentiated varieties. It says that the use of mathematical theory of growing solid contribute in the formulation of the boundary-value problem. It states the determination of a tension as a result of substitution of the relations in the boundary condition.
We consider the first boundary-value problem for a third-order equation with multiple characteristics u (xxx) -aEuro parts per thousand u (yy) = f(x, y) in a domain D = {(x, y):0 < x < p, 0 < y < l}. The u...
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We consider the first boundary-value problem for a third-order equation with multiple characteristics u (xxx) -aEuro parts per thousand u (yy) = f(x, y) in a domain D = {(x, y):0 < x < p, 0 < y < l}. The unique solvability of the problem is proved by the method of energy integrals and its explicit solution is constructed by the method of Green functions.
The existence and multiplicity of positive solutions are established for second-order periodic boundaryvalue problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defin...
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The existence and multiplicity of positive solutions are established for second-order periodic boundaryvalue problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defined on cones due to Cremins. Our approach is different in essence from other papers and the main results of this paper are also new.
We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and no...
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We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Omega subset of R-2 with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L);we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H-m(Omega), m >= 4:
We consider the eigenvalue and eigenfunction problem for a one-dimensional secondorder quasilinear differential equation. We analyze a number of versions of the function f specifying the nonlinearity for which the pro...
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We consider the eigenvalue and eigenfunction problem for a one-dimensional secondorder quasilinear differential equation. We analyze a number of versions of the function f specifying the nonlinearity for which the problem has multiple eigenvalues.
We consider the equation -Delta u = wu(3) on a square domain in R-2, with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equa...
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We consider the equation -Delta u = wu(3) on a square domain in R-2, with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equation is shown to have a solution u, with Morse index 2, that is neither symmetric nor antisymmetric with respect to any nontrivial symmetry of the square. Part of our proof is computer-assisted. An analogous result is proved for index 1. (C) 2011 Elsevier Inc. All rights reserved.
A theory of the nonlinear optical response of an atom interacting with a superposition of arbitrarily polarized fields is developed. The theory is based on the analytical solution of the boundary-value problem for an ...
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A theory of the nonlinear optical response of an atom interacting with a superposition of arbitrarily polarized fields is developed. The theory is based on the analytical solution of the boundary-value problem for an electron moving in a spherically symmetric intraatomic field and in the field of an external electromagnetic field. By means of the example of an argon atom interacting with a bichromatic field formed by the first and second harmonics of a Ti:sapphire laser, it is shown that, when an atom interacts with the field of two polarized pulses the polarization directions of which are not collinear, the response spectrum significantly depends on the laser radiation parameters-the duration and intensity of pulses, the time of delay between them, and the angle between the directions of polarization vectors. Generation of THz radiation is shown to be possible in the ionization-free regime due to intraatomic nonlinearity. DOI: 10.1134/S0030400X12030022
We study the existence of multiple nonnegative solutions for the doubly singular three-point boundaryvalue problem with derivative dependent data function -(p(t)y'(t))' = q(t) f(t, y(t), p(t)y'(t)), 0 0 ...
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We study the existence of multiple nonnegative solutions for the doubly singular three-point boundaryvalue problem with derivative dependent data function -(p(t)y'(t))' = q(t) f(t, y(t), p(t)y'(t)), 0 < t < 1, y(0) = 0, y(1) = alpha(1)y(eta). Here, p epsilon C[0, 1] boolean AND C-1(0, 1] with p(t) > 0 on (0, 1] and q(t) is allowed to be discontinuous at t = 0. The fixed point theory in a cone is applied to achieve new and more general results for existence of multiple nonnegative solutions of the problem. The results are illustrated through examples.
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