Abstract. Let X t be the relativistic α-stable process in R d , α ∈ (0, 2), d > α, with infinitesimal generator HWe study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup T t for this process with generator HWe prove that if lim |x|→∞ V (x) = ∞, then for every t > 0 the operator T t is compact. We consider the class V of potentials V such that V ≥ 0, lim |x|→∞ V (x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup T t is IU if and only if lim |x|→∞ V (x)/|x| = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction φ 1 for T t . In particular, when V (x) = |x| β , β > 0, then the semigroup T t is IU if and only if β > 1.

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It is shown that the second term in the asymptotic expansion as t → 0 of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0 < α < 2, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.

Vladimir Andreevich Steklov, an outstanding Russian mathematician whose 150th anniversary is celebrated this year, played an important role in the history of mathematics. Largely due to Steklov's efforts, the Russian mathematical school that gave the world such giants as N. Lobachevsky, P. Chebyshev, and A. Lyapunov, survived the revolution and continued to flourish despite political hardships. Steklov was the driving force behind the creation of the Physical-Mathematical Institute in starving Petrograd in 1921, while the civil war was still raging in the newly Soviet Russia. This institute was the predecessor of the now famous mathematical institutes in Moscow and St. Petersburg bearing Steklov's name. Steklov's own mathematical achievements, albeit less widely known, are no less remarkable than his contributions to the development of science. The Steklov eigenvalue problem, the Poincaré-Steklov operator, the Steklov function-there exist probably a dozen mathematical notions associated with Steklov. The present article highlights some of

Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of Pólya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-Pólya-Weinberger conjecture proved by Ashbaugh and Benguria.2000 Mathematics Subject Classification. Primary 35P15.