Abstract. We prove the existence of positive solutions with optimal local regularity to the homogeneous equation of Schrödinger type, −div(A∇u) − σu = 0 in Ω, under only a form boundedness assumption on σ ∈ D ′ (Ω) and ellipticity assumption on A ∈ L ∞ (Ω) n×n , for an arbitrary open set Ω ⊆ R n .We demonstrate that there is a two way correspondence between the form boundedness and the existence of positive solutions to this equation, as well as weak solutions to the equation with quadratic nonlinearity in the gradient,As a consequence, we obtain necessary and sufficient conditions for both the form-boundedness (with a sharp upper form bound) and the positivity of the quadratic form of the Schrödinger type operator H = −div(A∇·) − σ with arbitrary distributional potential σ ∈ D ′ (Ω), and give examples clarifying the relationship between these two properties.

We introduce a class of weak solutions to the quasilinear equation −∆ p u = σ|u| p−2 u in an open set Ω ⊂ R n with p > 1, where ∆ p u = ∇ · (|∇u| p−2 ∇u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients σ which satisfy the inequalityAs we shall demonstrate, these conditions on λ are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on σ.We also study weak solutions of the closely related equation −∆ p v = (p − 1)|∇v| p + σ, under the same conditions on σ. Our results for this latter equation will allow us to characterize the class of σ satisfying the above inequality for positive λ and Λ, thereby extending earlier results on the form boundedness problem for the Schrödinger operator to p = 2.2000 Mathematics Subject Classification. Primary 35J60, 42B37. Secondary 31C45, 35J92.

Let f be a zero mean continuous stationary Gaussian process on R whose spectral measure vanishes in a δ-neighborhood of the origin. Then the probability that f stays non-negative on an interval of length L is at most e −cδ 2 L 2 with some absolute c > 0 and the result is sharp without additional assumptions.1991 Mathematics Subject Classification. 60G10, 60G15.

In this paper we study the Dirichlet problemwhere σ and ω are nonnegative Borel measures, and p u = ∇ · (∇u |∇u| p−2 ) is the p-Laplacian. Here ⊆ R n is either a bounded domain, or the entire space. Our main estimates concern optimal pointwise bounds of solutions in terms of two local Wolff's potentials, under minimal regularity assumed on σ and ω. In addition, analogous results for equations modeled by the k-Hessian in place of the p-Laplacian will be discussed.

No abstract