The Fredholm property and well-posedness of a general differential boundary-value problem for a general improperly elliptic equation are analyzed in a two-dimensional bounded domain with smooth boundary.
The Fredholm property and well-posedness of a general differential boundary-value problem for a general improperly elliptic equation are analyzed in a two-dimensional bounded domain with smooth boundary.
We consider a shape optimization problem for Maxwell s equations with a strictly dissipative boundary condition In order to characterize the shape derivative as a solution to a boundaryvalue problem sharp regularity ...
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We consider a shape optimization problem for Maxwell s equations with a strictly dissipative boundary condition In order to characterize the shape derivative as a solution to a boundaryvalue problem sharp regularity of the boundary traces is critical This Note establishes the Frechet differentiability of a shape functional (C) 2010 Academie des sciences Published by Elsevier Masson SAS All rights reserved
In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus (T) over bar of R-3, when data are...
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In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus (T) over bar of R-3, when data are invariant under the group G = 0(2) x I subset of 0(3). The model problems of interest are stated below: Delta v + y = f(x)e(v), v>0 on T, v vertical bar partial derivative T=0 (P-1) and Delta v + a + f e(v) = 0, v > 0 on T, partial derivative v/partial derivative n + b + ge(v) = 0 on partial derivative T. (P-2) We prove that exist solutions which are G-invariant and these exhibit no radial symmetries. In order to solve the above problems we need to find the best constants in the Sobolev inequalities in the exceptional case. (C) 2011 Elsevier Inc. All rights reserved.
The purpose of this Note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained by the authors (preprint: http://***/abs/...
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The purpose of this Note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained by the authors (preprint: http://***/abs/math/0308214) in dimension 2. We also give some related trilinear estimates. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.
We study the uniqueness of the solution of a mixed boundaryvalue problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of that problem has a finite Di...
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We study the uniqueness of the solution of a mixed boundaryvalue problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of that problem has a finite Dirichlet integral with weight |x| (a) . Depending on the value of the parameter a, we prove uniqueness theorems and obtain exact formulas for the dimension of the solution space of the mixed boundaryvalue problem.
We consider a quasilinear parabolic boundaryvalue problem of the third kind on an interval. The coefficients of the partial differential equation and the right-hand sides in the boundary conditions and the evolution ...
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We consider a quasilinear parabolic boundaryvalue problem of the third kind on an interval. The coefficients of the partial differential equation and the right-hand sides in the boundary conditions and the evolution equation for the state vector nonlinearly depend on time, the point, the state vector, and the values of the solution at the endpoints. This problem generalizes a number of models of formation and decomposition of metal hydrides. For the simplest finite-difference scheme, we prove the uniform convergence to a continuous generalized solution of the boundaryvalue problem. A sample model is given.
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.
We construct and justify the asymptotics of the solution of a boundaryvalue problem for a singularly perturbed system of two second-order ordinary differential equations that contain distinct powers of a small parame...
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We construct and justify the asymptotics of the solution of a boundaryvalue problem for a singularly perturbed system of two second-order ordinary differential equations that contain distinct powers of a small parameter multiplying second-order derivatives for the case of a multiple root of the degenerate equation. The root multiplicity results in changes in the structure of the asymptotics of the boundary layer solution as compared with the case of a simple root, in particular, in changes in the scale of the boundary layer variables.
The Stokes operator E(2) governs the irrotational axisymmetric Stokes flow and its square governs the corresponding rotational flow. In spheroidal coordinates the elements of the solution space ker E(2) enjoy a spectr...
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The Stokes operator E(2) governs the irrotational axisymmetric Stokes flow and its square governs the corresponding rotational flow. In spheroidal coordinates the elements of the solution space ker E(2) enjoy a spectral decomposition into separable eignefunction, while the elements of the ker E(4) accept a spectral decomposition in terms of semiseparable eigensolutions involving 3D-by-3D eigenfunctions of the Gegenbauer operator. These spectral characteristics are utilized to construct the fundamental solutions for both the E 2 and the E 4 operators in spheroidal geometry. The fundamental solution for E(2) is expressed in terms of the elements of the irrotational space ker E(2), while the fundamental solution for E(4) is expressed in terms of the corresponding generalized eigenfunctions alone.
We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the standard Kigami Laplacian. Our main task is to find a class of potentials analogous to 1/2 (x -...
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We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the standard Kigami Laplacian. Our main task is to find a class of potentials analogous to 1/2 (x - x(0))(2) on the line. We describe a class of potentials u with the properties Delta u = 1, u attains a minimum value zero, and u. 8 at infinity. We show how to construct such potentials attaining the minimum value at any prescribed point, and we show how to parameterize the class of potentials by a certain surface in R-3. We obtain estimates for the growth rate of the eigenvalue counting function for -Delta + u. We obtain numerical approximations to the eigenfunctions, and in particular observe that the ground-state eigenfunction resembles a Gaussian function.
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