521 research outputs found
Rigidity of optimal bases for signal spaces
No full textWe discuss optimal L2-approximations of functions controlled in the H1-norm. We prove that the basis of eigenfunctions of the Laplace operator with Dirichlet boundary condition is the only orthonormal basis (bi) of L2 that provides an optimal approximation in the sense of ‖f−∑i=1n(f,bi)bi‖L22≤[Formula presented]∀f∈H01(Ω),∀n≥1. This solves an open problem raised by Y. Aflalo, H. Brezis, A. Bruckstein, R. Kimmel, and N. Sochen (Best bases for signal spaces, C. R. Acad. Sci. Paris, Ser. I 354 (12) (2016) 1155–1167)
Preface: International Symposium on Variational Methods and Nonlinear Differential Equations, to honor Antonio Ambrosetti on his 60th birthday
No full textA Brezis-Nirenberg problem on hyperbolic spaces
No full textWe consider a Brezis-Nirenberg problem on the hyperbolic space .
By using the stereographic projection, the problem becomes a singular
problem on the boundary of the open ball .
Thanks to the Hardy inequality, in a version due to Brezis-Marcus,
the difficulty involving singularities can be overcame.
We use the mountain pass theorem due to Ambrosetti-Rabinowitz and
Brezis-Nirenberg arguments to obtain a nontrivial solution
The BBM formula revisited
No full textIn this paper, we revise the BBM formula due to J. Bourgain, H. Brezis, and P. Mironescu in [1]CAM
On the distributional Jacobian of maps from S N into S N in fractional Sobolev and Hölder spaces
No full textAbstract H. Brezis and L. Nirenberg proved that if (g In the same spirit, we consider the quantity J(g, ψ) := S N ψ det(∇g) dσ, for all ψ ∈ C 1 (S N , R) and study the convergence of J(g k , ψ). In particular, we prove . Surprisingly, this result is "optimal" when N > 1. In the case N = 1 we prove that if g k → g almost everywhere and lim sup k→∞ |g k − g|BMO is sufficiently small, then J(g k , ψ) → J(g, ψ) for any ψ ∈ C 1 (S 1 , R). We also establish bounds for J(g, ψ) which are motivated by the works of J. Bourgain, H. Brezis, and H.-M. Nguyen and H.-M. Nguyen. We pay special attention to the case N = 1
Brezis–Marcus Problem and its Generalizations
No full textCertain Hardy inequalities in domains of Euclidean space contain sharp but unreachable constants. V. G. Maz’ya and other authors used this fact to improve the corresponding inequalities by adding new integral terms. In this paper, a survey of results in this direction initiated by H. Brezis and M. Marcus is presented. Also, we give some generalizations of Brezis–Marcus-type inequalities to the case of Rellich-type inequalities with weights that are powers of the distance from a point to the boundary of the domain. Generalizations to the case of conformally invariant integral inequalities in simply connected and doubly connected planar hyperbolic domains are discussed
Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis)
No full textAbstractHaïm Brezis and Augusto Ponce introduced and studied several extensions of Kato's inequality, in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a W1,1 function in a smooth domain [H. Brezis, A.C. Ponce, Kato's inequality when Δu is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599–604; H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241]. Using potential theoretic methods we answer here some questions raised in [H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241] about the relations between the normal derivative of a function u and the normal derivative of its positive part u+. The results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given. In the final appendix, H. Brezis solves an old question of J. Serrin about pathological solutions of certain elliptic equations [J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385–387]. This is used in the paper to extend the first version of our main result
On some conjectures proposed by Haïm Brezis
No full textDruet (Ann. Inst. H. Poincaré Anal. Non Linèaire 19(2) (2002) 125) solved two conjectures proposed by Haim Brezis (Comm. Pure Appl. Math. 39 (1986) 17) about “low”-dimension
phenomena for some elliptic problem with critical Sobolev exponent. In Druet (Ann. Inst.
H. Poincaré Anal. Non Linèaire 19(2) (2002) 125), the proof of one of the two conjectures is reduced to an asymptotic analysis which is carried over with very general techniques involving
pointwise estimates. We propose here a different and simpler approach in the blow-up analysis based on integral estimates and on a careful expansion of the energy functional
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