The structure of non-trivial non-negative solutions to singularly perturbed semilinear Dirichlet problems of the form −ε2Δu = f(u) in Ω, u = 0 on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied as ε → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and . It is shown that there are many non-trivial non-negative solutions and they are spike-layer solutions. Moreover, the measure of each spike layer is estimated as ε → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0, ∞). Uniqueness of a large positive solution and many positive intermediate spike-layer solutions are obtained for ε sufficiently small.

In a planar domain, to each function w ∈ W2,2 satisfying a suitable condition we associate a non-divergence elliptic differential operator 𝔏 such that 𝔏w = 0. For a given converging sequence {wk}, we study G-convergence of the corresponding sequence {𝔏k} of operators.

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

The generalized Bendixson's criterion for non-existence of non-constant periodic solutions of an autonomous system dx/dt = f(x), x ∈ D, is further extended to systems when the condition f ∈ C1(D, RN) is violated. In particular, the criterion can be applied to region-wise linear systems in RN.

Suppose f = (f1, …, fm) is a solution of a non-hyperbolic quasi-linear system of the form with fk = φk on ∂Ω, where each fi is a bounded function in C2(Ω) ∩ C0(Ω̄). For a system of equations that can have a slightly more general form than above, when Ω is an unbounded open subset of a slab SM and as |X| → ∞ in a specified manner and when certain other conditions are satisfied, a Phragmèn–Lindelöf theorem that yields the limits at infinity of the functions fk(X), k = 1, …, m, is proven.

Let B1 be a ball of radius R1 in RN with centre at the origin and let B0 be a smaller ball of radius R0 contained inside it. Let u be the solution of the problem −Δu = 1 in B1\B0 vanishing on the boundary. It is shown that is minimal if and only if the balls are concentric. It is also shown that the first (Dirichlet) eigenvalue of the Laplacian in B1\B0 is maximal if and only if the balls are concentric. Generalizations are indicated.

Let Ω be a bounded C2,α domain in R2. We prove that the boundary-value problem Δυ = 0 in Ω, ∂υ/∂n = λsinh(υ) on ∂Ω, has infinitely many (classical) solutions for any given λ > 0. These solutions are constructed by means of a variational principle. We also investigate the limiting behaviour as λ → 0+; indeed, we prove that each of our solutions, as λ → 0+, after passing to a subsequence, develops a finite number of singularities located on ∂Ω.

We study the unique solvability of density-dependent incompressible Navier-Stokes equations in the whole space RN (N ≥ 2). The celebrated results by Fujita and Kato devoted to the constant density case are generalized to the case when the initial density is close to a constant: we find local well posedness for large initial velocity, and global well posedness for initial velocity small with respect to the viscosity. Our functional setting is very close to the one used by Fujita and Kato.

We study the one-dimensional Gross-Pitaevskii-Schrödinger equation with a potential U moving at velocity v. For a fixed v less than the sound velocity, it is proved that there exist two time-independent solutions if the potential is not too big.

The existence of solitons in non-commutative scalar field theories is proved for large values of the non-commutativity parameter using functional analysis. In the case of even phase-space dimensions greater than or equal to four, the solitons include some that are not rotationally symmetric.

Let D be a domain in C and let f be a transcendental meromorphic function on C such that C* \f(C) = ∅, {∞} or {α, β}, where α and β are two distinct values in C* = C ∪ {∞}. Then the family of functions g that are analytic on D and such that f ∘ g has no fixpoints on D is normal.

We consider random perturbations of two-dimensional Navier–Stokes equations. Under some natural conditions on random forces, we study asymptotic properties of solutions and stationary measures.

In this paper we study the problem of the axial symmetry of solutions of some semilinear elliptic equations in unbounded domains. Assuming that the solutions have Morse index one and that the nonlinearity is strictly convex in the second variable, we are able to prove several symmetry results in Rn and in the exterior of a ball. The case of some bounded domains is also discussed.

For a bounded solution to the mean curvature equation in Rn, we shall obtain its Hölder estimates, with exponent α, which are in linear proportion to the number obtained by dividing its global oscillation by the αth power of the distance of a point to the boundary of the domain after a suitable rescaling.

We propose a notion of weak solution for certain nonlinear contractive equations: ‘dissipative solution’. We show that this notion coincides with the viscosity solutions for Hamilton–Jacobi equations, as introduced by Crandall and Lions, and with the entropy solutions for conservation laws introduced by Kružkov. The proposed notion is based on the properties of accretive operators.

A line congruence is a two-parameter family of lines in R3. In this paper we study singularities of line congruences. We show that generic singularities of general line congruences are the same as those of stable mappings between three-dimensional manifolds. Moreover, we also study singularities of normal congruences and equiaffine normal congruences from the viewpoint of the theory of Lagrangian singularities.

We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.

In regular state stabilization of a class of linear parabolic systems, the number of the sensors wk and the actuators hk are required to be greater than or equal to the maximum of the multiplicities of the unstable eigenvalues. In this paper, we ask what control theoretic results we can establish with smaller numbers of wk and hk. The enhancement of stability of output or the output stabilization, a concept weaker than regular state stabilization, gives an answer to the question. Conditions of a fairly different nature than those for regular state stabilization appear; these are the rank conditions of the products of the observability and controllability matrices. The result is applied to a parabolic system with boundary observation/control via two different approaches.

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

We derive new formulae that provide equivalent quasi-norms on Lorentz spaces.