Results 11  20
of
48
Steiner Symmetrization Is Continuous In W 1,P
, 1997
"... We study the continuity, smoothing, and convergence properties of Steiner symmetrization in higher space dimensions. Our main result is that Steiner symmetrization is continuous in W 1;p (1 p ! 1) in all dimensions. This implies that spherical symmetrization cannot be approximated in W 1;p by s ..."
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We study the continuity, smoothing, and convergence properties of Steiner symmetrization in higher space dimensions. Our main result is that Steiner symmetrization is continuous in W 1;p (1 p ! 1) in all dimensions. This implies that spherical symmetrization cannot be approximated in W 1;p by sequences of Steiner symmetrizations. We also give a quantitative version of the standard energy inequalities for spherical symmetrization.
MINIMIZATION OF THE GROUND STATE FOR TWO PHASE CONDUCTORS IN LOW CONTRAST REGIME
, 2011
"... ABSTRACT. In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial ..."
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ABSTRACT. In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial, and it was conjectured that the optimal distribution of the materials consists in putting the material with the highest conductivity in a ball around the center. We show that this conjecture is not true in general. For this, we consider the particular case where the two conductivities are close to each other (low contrast regime) and we perform an asymptotic expansion with respect to the difference of conductivities. We find that the optimal solution is the union of a ball and an outer ring when the amount of the material with the higher density is large enough. 1.
ELLIPTIC EQUATIONS AND SYSTEMS WITH CRITICAL TRUDINGERMOSER NONLINEARITIES
"... Dedicated to Louis Nirenberg on the occasion of his 85th birthday Abstract. In this article we give first a survey on recent results on some TrudingerMoser type inequalities, and their importance in the study of nonlinear elliptic equations with nonlinearities which have critical growth in the sens ..."
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Dedicated to Louis Nirenberg on the occasion of his 85th birthday Abstract. In this article we give first a survey on recent results on some TrudingerMoser type inequalities, and their importance in the study of nonlinear elliptic equations with nonlinearities which have critical growth in the sense of TrudingerMoser. Furthermore, recent results concerning systems of such equations will be discussed. 1. Introduction. Elliptic
Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal
"... Abstract In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is is small enough and c ∈ L N p−1 ,r (Ω), with r < +∞. We also prove a stability result for renormalized solutions to a class of noncoercive equations whose prot ..."
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Abstract In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is is small enough and c ∈ L N p−1 ,r (Ω), with r < +∞. We also prove a stability result for renormalized solutions to a class of noncoercive equations whose prototype is (P ) with b ≡ 0.
Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems
"... Abstract. In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue µ1(Ω) for the pLaplace operator in a Lipschitz, bounded domain Ω in Rn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. 1. ..."
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Abstract. In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue µ1(Ω) for the pLaplace operator in a Lipschitz, bounded domain Ω in Rn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. 1.
On Hardy inequalities with a remainder term
, 2009
"... In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of f ..."
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In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.
Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems
, 2011
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A Symmetrization Result for Nonlinear Elliptic Equations
"... We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x, u) + f, where the principal term is a LerayLions operator defined onW 1,p 0 (Ω). The function g(x, u) satisfies suitable growth assumptions, but no sign hypothesis on i ..."
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We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x, u) + f, where the principal term is a LerayLions operator defined onW 1,p 0 (Ω). The function g(x, u) satisfies suitable growth assumptions, but no sign hypothesis on it is assumed. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.
THE REARRANGEMENTINVARIANT HULL OF A BESOV SPACE
, 2008
"... Let X be a rearrangementinvariant Banach function space over a domain Ω ⊂ Rn. We characterize the Kfunctionals for the pairs (X,V 1X) and (X,SX), where V 1X is the reduced Sobolev space built upon X and SX is the class of measurable functions on Ω such that ‖t− 1 n (f∗∗(t) − f∗(t))‖X < ∞, an ..."
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Let X be a rearrangementinvariant Banach function space over a domain Ω ⊂ Rn. We characterize the Kfunctionals for the pairs (X,V 1X) and (X,SX), where V 1X is the reduced Sobolev space built upon X and SX is the class of measurable functions on Ω such that ‖t− 1 n (f∗∗(t) − f∗(t))‖X < ∞, and X is the representation space of X. Using this result, we obtain a sharp estimate of rearrangements of a function in terms of moduli of continuity. We apply it to sharp embeddings of Besov spaces into Lorentz spaces and to the characterization of the rearrangementinvariant hull of a generalized Besov space.