Phragmén-Lindelöf principles on algebraic varieties

Estimates of Phragmén-Lindelöf (PL) type for plurisubharmonic functions on algebraic varieties in have been of interest for a number of years because of their equivalence with certain properties of constant coefficient partial differential operators; e.g. surjectivity, continuation properties of solutions and existence of continuous linear right inverses. Besides intrinsic interest, their importance lies in the fact that, in many cases, verification of the relevant PL-condition is the only method to check whether a given operator has the property in question.

In the present paper the property which characterizes the existence of continuous linear right inverses is investigated. It is also the one closest in spirit to the classical Phragmén-Lindelöf Theorem as various equivalent formulations for homogeneous varieties show. These also clarify the relation between and the PL-condition used by Hörmander to characterize the surjectivity of differential operators on real-analytic functions.

We prove the property for an algebraic variety implies that , the tangent cone of at infinity, also has this property. The converse implication fails in general. However, if is a manifold outside the origin, then satisfies if and only if the real points in have maximal dimension and if the distance of to is bounded by as tends to infinity. In the general case, no geometric characterization of the algebraic varieties which satisfy is known, nor any of the other PL-conditions alluded to above.

Besides these main results the paper contains several auxiliary necessary conditions and sufficient conditions which make it possible to treat interesting examples completely. Since it was submitted they have been applied by several authors to achieve further progress on questions left open here.

     

A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

     

On the maximum principle for complete second-order elliptic operators in general domains

This paper is concerned with the maximum principle for second-order linear elliptic equations in a wide generality. By means of a geometric condition previously stressed by Berestycki-Nirenberg-Varadhan, Cabré was very able to improve the classical ABP estimate obtaining the maximum principle also in unbounded domains, such as infinite strips and open connected cones with closure different from the whole space. Now we introduce a new geometric condition that extends the result to a more general class of domains including the complements of hypersurfaces, as for instance the cut plane. The methods developed here allow us to deal with complete second-order equations, where the admissible first-order term, forced to be zero in a preceding result with Cafagna, depends on the geometry of the domain.     

Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle

Second-order elliptic operators are transformed into second-order elliptic operators of a higher dimensionality acting on differences of functions. Applying the maximum principle to the resulting operators yields various a-priori pointwise estimates to difference-quotients of solutions of elliptic differential, as well as finite-difference, equations. We derive Schauder estimates, estimates for equations with discontinuous coefficients, and other estimates.     

Nonlinear nonuniformly elliptic second-order equations

One establishes a priori estimates for the first and second derivatives of the solutions of nonuniformly elliptic equations of the form F(x,u,Du,D^zu) without the assumptions of the convexity of the function F(x,z,p,r) with respect to r. These estimates allow us to extend the results of N. V. Krylov, L. C. Evans, and N. S. Trudinger on the classical solvability of the Dirichlet problem for essential nonlinear uniformly elliptic equations, convex with respect to D², to wider classes of nonlinear equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 35–64, 1984.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

Solvability of the Dirichlet problem for second-order elliptic equations

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n?1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ? ? _n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W ₂ ¹ (Q) under the additional requirement that the boundary function belongs to the space W ₂ ^1/2 (?Q).     

On positivity of solutions of degenerate boundary value problems for second-order elliptic equations

In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B _u =gon Ω∂Г where ω is a domain in ℝ ⁿ ,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary. The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue, the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet boundary value problem, where Γ=∂Ω, were examined intensively by many authors.     

On holomorphic solutions of some boundary-value problems for second-order elliptic operator differential equations

In the class of holomorphic vector functions, we determine the conditions of solvability of the boundaryvalue problem for a class of second-order operator differential equations expressed in terms of the operator coefficients appearing both in the equation and in the boundary condition.     

Mappings by the solutions of second-order elliptic equations

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions.     

Partial Schauder estimates for second-order elliptic and parabolic equations

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H?lder semi-norms not with respect to all, but only with respect to some of the independent variables.