Zeev Sobol

Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations

We develop a new method to uniquely solve a large class of heat equations, so-called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock-Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier-Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators −∇⋅(a∇)+b1⋅∇+∇⋅b2+V with real measurable (singular) coefficients, on an open set Ω⊂RN. The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on Lp(Ω) for p only from an interval in [1,∞), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale Lp(Ω). We show however that the semigroup is ultracontractive in the scale of weighted spaces Lp(Ω,ϕ2 dx) with a suitable weight ϕ and derive an upper and lower bound on its integral kernel.

Second-order semilinear elliptic inequalities in exterior domains

Abstract We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients − ∇ ·a· ∇ u+Vu−Wu p ⩾0 in exterior domains in R N , N⩾3 . For W(x)≍|x|−σ (σ∈ R ) at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.

Positive Solutions to Semi-Linear and Quasi-Linear Elliptic Equations on Unbounded Domains

Abstract In this survey we describe recent results on the existence and nonexistence of positive solution to semi-linear and quasi-linear second-order elliptic equations. A typical example is the equation –Δ u = | x | –σ u q in an exterior of the ball in ℝ N or in a cone-like domain in ℝ N . The equations of this type exhibit a phenomenon of presence of critical exponents in the range of the parameter q ∈ ℝ, which separate the zones of the existence from the zones of the nonexistence. The values of the critical exponents depends on the geometry of the domain, the type of the operator in the main part (divergent or nondivergent), the behaviour of the coefficients in lower order terms at infinity. We investigate these dependencies mostly in the cases of the exterior and cone-like domains. The proofs are often based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the corresponding second-order elliptic operator, comparison principles and Hardys inequality in exterior domains. To construct the barriers in the cases of equations with non-smooth coefficients we obtain detailed estimates at infinity of small and large solutions to the corresponding linear equations. Some of the results for the equations with first order term are new and have not been published before. In discussions we list some open problems in this area.

Gradient estimates for degenerate quasi-linear parabolic equations

For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain

SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS

L^q

On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficients

-estimates for the gradients of solutions, and for the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.

Estimates for fundamental solutions of parabolic equations with application to a nonlinear problem

We study the existence and nonexistence of singular solutions to the equation , p > 1, in ℝN × [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and . The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if .

On the Lp-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

We study Lp-theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of analytic C0-semigroups under very general conditions on the coefficients, related to the notion of form-boundedness. We determine an interval J in the Lp-scale, not necessarily containing p=2, in which one obtains a consistent family of quasi-contractive semigroups. This interval is close to optimal, as shown by several examples. In the case of uniform ellipticity we construct a family of semigroups in an extended range of Lp-spaces, and we prove p-independence of the analyticity sector and of the spectrum of the generators.

Stability of Solutions to Generalized Forchheimer Equations of any Degree

Gaussian estimates for fundamental solutions for a class of degenerate parabolic equations are discussed. Applications to exterior problems for semi-linear elliptic equations are given.