On symmetry of nonnegative solutions of elliptic equations

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H  . Moreover, we prove that if u?0$u?0$, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.     

Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries

We consider nonnegative solutions of a parabolic equation in a cylinder D×(0,T), where D is a noncompact domain of a Riemannian manifold. Under the assumption [IU] (i.e., the associated heat kernel is intrinsically ultracontractive), we establish an integral representation theorem: any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the associated elliptic operator. We apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions. We also show that [IU] implies the condition [SP] (i.e., the constant function 1 is a small perturbation of the elliptic operator on D).     

Bounds for the fundamental solutions of elliptic and parabolic equations

It is shown that any elloptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.     

On the existence of nonnegative continuous solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations.     

On tracking solutions of parabolic equations

We consider a control problem for a parabolic equation. It consists in constructing an algorithm for finding a feedback control such that a solution of a given equation should track a solution of another equation generated by an unknown right-hand side. We propose two noise-resistant solution algorithms for the indicated problem. They are based on the method of extremal shift well-known in the guaranteed control theory. The first algorithm is applicable in the case of “continuous” measurements of phase states, whereas the second one implies discrete measurements.     

On the existence of nonnegative continuous solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations.     

Nonlinear elliptic and parabolic equations

Results of recent years are presented on the theory of nonlinear elliptic and parabolic equations of any order including equations of infinite order.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 9, pp. 5–130, 1976.     

椭圆方程组径向非负解的存在性

应用锥不动点定理证明了圆环中半线性椭圆方程径向非负解的存在性.     

Hölder regularity of solutions of degenerate elliptic and parabolic equations

We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, and the Hölder continuity of solutions to degenerate parabolic equations of the non-divergence form

(∗)     

Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains

Nonnegativity of weak solutions of parabolic and elliptic equations on nonsmooth domains is established. Strong positivity of weak solutions to elliptic equations is proved via a boundary weak Harnack inequality.