Alexander I. Nazarov

On Fractional Laplacians

We compare two natural types of fractional Laplacians (− Δ) s , namely, the “Navier” and the “Dirichlet” ones. We show that for 0 < s < 1 their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.

A survey of results on nonlinear Venttsel problems

We review the recent results for boundary value problems with boundary conditions given by second-order integral-differential operators. Particular attention has been paid to nonlinear problems (without integral terms in the boundary conditions) for elliptic and parabolic equations. For these problems we formulate some statements concerning a priori estimates and the existence theorems in Sobolev and Hölder spaces.

Dirichlet and Neumann problems to critical Emden---Fowler type equations

We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to critical Emden–Fowler equations.

Non-critical dimensions for critical problems involving fractional Laplacians

Let f be a meromorphic function with bounded set of singular values and for which infinity is a logarithmic singularity. Then we show that f has infinitely many repelling periodic points for any minimal period n ≥ 1, using a much simpler argument than the more general results for arbitrary entire transcendental functions.We study the Brezis–Nirenberg effect in two families of non-compact boundary value problems involving Dirichlet–Laplacian of arbitrary real order m∈(0,n/2).

A counterexample to the Hopf–Oleinik lemma (elliptic case)

We construct a new counterexample confirming the sharpness of the Dini-type condition for the boundary of

Tail behavior of anisotropic norms for Gaussian random fields

\Omega

On fractional Laplacians – 2

. In particular, we show that for convex domains the Dini-type assumption is the necessary and sufficient condition which guarantees the Hopf-Oleinik type estimates.

Weighted Sobolev-type embedding theorems for functions with symmetries

We investigate the logarithmic large deviation asymptotics for anisotropic norms of Gaussian random functions of two variables. The problem is solved by the evaluation of the anisotropic norms of corresponding integral covariance operators. We find the exact values of such norms for some important classes of Gaussian fields. To cite this article: M. Lifshits et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

ON THE EXISTENCE OF EXTREMAL FUNCTIONS IN SOBOLEV EMBEDDING THEOREMS WITH CRITICAL EXPONENTS

Abstract For s > − 1 we compare two natural types of fractional Laplacians ( − Δ ) s , namely, the “Navier” and the “Dirichlet” ones.

Quasilinear Two-Phase Venttsel Problems

It is well known that Sobolev embeddings can be refined in the presence of symmetries. Hebey and Vaugon (1997) studied this phenomena in the context of an arbitrary Riemannian manifold M and a compact group of isometries G. They showed that the limit Sobolev exponent increases if there are no points in M with discrete orbits under the action of G. In the paper, the situation where M contains points with discrete orbits is considered. It is shown that the limit Sobolev exponent for W 1 p (M) increases in the case of embeddings into weighted spaces Lq(M, w) instead of the usual Lq spaces, where the weight function w(x) is a positive power of the distance from x to the set of points with discrete orbits. Also, embeddings of W 1 p (M) into weighted Hölder and Orlicz spaces are treated. Introduction It is well known that Sobolev embeddings can be refined if we deal with subspaces of functions invariant under some group of symmetries. This phenomenon was used in some particular cases to prove the existence of solutions of various boundary value problems (see, e.g., [1]–[5]; a similar effect for trace embeddings was employed, e.g., in the recent paper [6]). In [7], this problem was studied in the more general context of an arbitrary Riemannian manifold and a compact group of isometries. The critical embedding exponent increases if the manifold contains no points with orbits of dimension zero, due to reduction of the effective manifold dimension. Our goal is to consider the case where the manifold contains points with zero-dimensional orbits. The conventional embedding theorem cannot be refined in this case. For example, if p < n, then, in general, even a radially symmetric function in W 1 p (B R) may fail to be integrable in a power exceeding p∗ = np n−p (here and in the sequel, B n R(X) stands for the ball of radius R in R centered at X, B R = B n R(0)). However, the origin is the only possible singular point for radial functions, and their properties improve after multiplication by a positive power of r = |x|. In [8, Theorem 2.5], it was shown that, for q ≥ p∗ and α > np − n q − 1, the set of radially symmetric functions f ∈ W 1 p (B 1 ) is compactly embedded in Lq(B 1 ; r ). 2000 Mathematics Subject Classification. Primary 46E35; Secondary 58D99.