In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operator D(α)=D₀−αV1 acting in L₂(R³; C⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.

We begin by giving some results of continuity with respect to the domain for the Dirichlet problem (without any assumption of regularity on the domains). Then, following an idea of A. Beurling, a technique of subsolutions and supersolutions for the so-called quadrature surface free boundary problem is presented. This technique would apply to many free boundary problems in RN, N≥2, which have overdetermined Cauchy data on the free boundary. Some applications to concrete examples are also given.

We consider systems of partial differential equations of the first order in t and of order 2s in the x variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid in Rⁿ×(0, T] for s≥2.

For second order systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle, to hold in the layer Rⁿ×(0, T] and in the cylinder μ×(0, T], where μ is a bounded subdomain of Rⁿ. If the coefficients of the system do not depend on t, these conditions coincide. The necessary and sufficient condition in this case is that the principal part of the system is scalar and that the coefficients of the system satisfy a certain algebraic inequality. We show by an example that the scalar character of the principal part of the system everywhere in the domain is not necessary for validity of the classical maximum modulus principle when the coefficients depend both on x and t.

We prove the almost everywhere convergence of the inverse spherical transform of L_p bi-K-invariant functions on the group SL(2, R), 4/3< p≤2. The result appears to be sharp.

A new proof is given for a theorem by V. G. Maz'ya. It gives a necessary and sufficient condition on the open set Ω in R^N for the functions in W ${}_{0}^{m,p}$ (Ω) to have the ordinary norm equivalent to the norm obtained when including only the highest order derivatives in the definition. The proof is based on a kind of polynomial capacities, Maz'ya capacities.