1,721,030 research outputs found
Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems.
No full textSimplified method to implant chronic stimulating electrode in the gasserian ganglion. Technical note.
No full textEstimates for solutions to nonlinear degenerate elliptic equations with lower order terms
No full textWe consider a class of Dirichlet boundary problems for nonlinear degenerate elliptic equations with lower order terms. We prove, using symmetrization techniques, pointwise estimates for the rearrangement of the gradient of a solution u and integral estimates. As consequence, we get apriori estimates which show how the summability of the gradient of a solution increases when the summability of the datum increases, also taking into account the presence of a zero order term which can have a regularizing effect
On the luzin N-property and the uncertainty principle for Sobolev mappings
Full text linkWe say that a mapping v from Rn to Rd satisfies the (tau, sigma) -N-property if H^sigma( v (E)) = 0 whenever H^tau (E) = 0, where H^tau means the Hausdorff measure. We prove that every mapping v of Sobolev class W_p^k (Rn,Rd ) with kp > n satisfies the (tau, sigma)-N-property for every 0 1 and for the critical value tau= tau* the corresponding (tau, sigma)-N-property fails in general. Nevertheless, this (tau, sigma)-N-property holds for tau = tau* if we assume in addition that the highest derivatives (Nabla^k)v belong to the Lorentz space L_p,1(Rn ) instead of L_p. We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for N-properties and discuss their applications to the Morse-Sard theorem and its recent extensions
From the theorems of morse, sard, dubovitskii and federer to the luzin N-property: The story so far
No full textIn this survey we demonstrate the universal synthesis of all the above mentioned analytic phenomena for continuous mappings of Holder and Sobolev classes. This concludes the long time research started with our previous joint papers with Jean Bourgain (2013, 2015)
The Morse–Sard Theorem and Luzin N-Property: A New Synthesis for Smooth and Sobolev Mappings
Full text linkConsidering regular mappings of Euclidean spaces, we study the distortion of the Hausdorff dimension of a given set under restrictions on the rank of the gradient on the set. This problem was solved for the classical cases of k-smooth and Holder mappings by Dubovitskii, Bates, and Moreira. We solve the problem for Sobolev and fractional Sobolev classes as well. Here we study the Sobolev case under minimal integrability assumptions that guarantee in general only the continuity of a mapping (rather than differentiability everywhere). Some new facts are found out in the classical smooth case. The proofs are mostly based on our previous joint papers with Bourgain and Kristensen (2013, 2015)
Simplified method to implant chronic stimulating electrode in the gasserian ganglion. Technical note.
No full textSome properties for eigenvalues and eigenfunctions of a class of linear weighted problems
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