Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization

Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a--priori estimates is used. The extension of the problem to the $L^p$-setting is discussed. As a direct application essential self--adjointness and strong uniqueness in $L^p$ is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume $\Lambda \subset \R^2$

Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,\sigma)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmen--Lindel\"of type comparison arguments and an improved version of Hardy's inequality in cone--like domains.Comment: 30 pages, 1 figur

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 $L^q$-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

A critical phenomenon for sublinear elliptic equations in cone-like domains

We study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such that equation $(*)$ has a positive supersolution if and only if $-\infty<p<p_*$. The value of $p_*$ is determined explicitly by $s$ and the geometry of the cone.Comment: 6 pages, 2 figure

Existence, stability and oscillation properties of slow decay positive solutions of supercritical elliptic equations with Hardy potential

We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem

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

A work in Perturbation Theory, with a purpose to consider well-posedness of elliptic and parabolic PDE with singular complex coefficient