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Augusto C. Ponce 《Comptes Rendus Mathematique》2003,337(4):253-257
We show that if , is a bounded Lipschitz domain and is a sequence of nonnegative radial functions weakly converging to δ0 then there exist C>0 and n0?1 such that The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu (in: Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). As n→∞ in (1) we recover Poincaré's inequality. We also extend a compactness result of Bourgain, Brezis and Mironescu. To cite this article: A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
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M.Francesca Betta Friedman Brock Anna Mercaldo M.Rosaria Posteraro 《Comptes Rendus Mathematique》2002,334(6):451-456
In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type where is an open set of (n?2), ?(x)=(2π)?n/2exp(?|x|2/2), aij(x) are measurable functions such that aij(x)ξiξj??(x)|ξ|2 a.e. , and f(x) is a measurable function taken in order to guarantee the existence of a solution of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456. 相似文献
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We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like n(x)→n∞(x/|x|) as |x|→∞. We prove that the Sommerfeld condition at infinity still holds true under the weaker form Our approach consists in proving this estimate in the framework of the limiting absorbtion principle. We use Morrey–Campanato type of estimates and a new inequality on the energy decay, namely It is a striking feature that the index n∞ appears in this formula and not the phase gradient, in apparent contradiction with existing literature. To cite this article: B. Perthame, L. Vega, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
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《Nonlinear Analysis: Theory, Methods & Applications》2003,54(1):9-37
In this paper, we study the problemin the setting of the weighted sobolev space . The main novelty of our work is L∞ estimates on the solutions, and the existence of a weak and renormalized solution. 相似文献
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M.Francesca Betta Anna Mercaldo François Murat M.Michaela Porzio 《Comptes Rendus Mathematique》2002,334(9):757-762
In this Note we consider a class of noncoercive nonlinear problems whose prototype is where is a bounded open subset of (N?2), △p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on or a function in , λ?0 and b belongs to the Lorentz space or to the Lebesgue space . We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762. 相似文献
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《Nonlinear Analysis: Theory, Methods & Applications》2003,52(5):1535-1552
In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponentwhere is a bounded domain of with a smooth boundary . 相似文献
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《European Journal of Operational Research》2002,143(2):365-376
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We consider the following singularly perturbed semilinear elliptic problem: where is a bounded smooth domain in , ε>0 is a small constant and p is a subcritical exponent. Let be its energy functional, where . Ni and Takagi proved that for a single boundary spike solution uε, the following asymptotic expansion holds where c1>0 is a generic constant, Pε is the unique local maximum point of uε and H(Pε) is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of Jε[uε]: where c2, c3 are generic constants and R(Pε) is the Ricci scalar curvature at Pε. In particular c3>0. Applications of this expansion will be given. To cite this article: J. Wei, M. Winter, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献