Extremal barriers on cones with Phragmèn-Lindelöf theorems and other applications

Large solutions for non-divergence structure equations with singular lower order terms

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabr\'e proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabr\'e's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W 2 , 2 W^{2,2} -regularity for p-harmonic functions in the Heisenberg group H 1 {\mathbb H}^1 for the range 17 − 1 2 ≤ p > 5 + 5 2 \frac {\sqrt {17}-1}{2} \leq p > \frac {5+\sqrt {5}}{2} .

On every right circular cone and for every uniformly elliptic operator in nondivergence form there exists a barrier, that is, a supersolution which is zero at the vertex and positive elsewhere on the closed cone. These barriers are applied to the Dirichlet problem. Other applications are deferred to a later paper on « extremal barriers »

Let  ⇢ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = Xn i, j=1 ai j @i j u + Xn j=1 b j @ j u + cu where a j i = ai j 2 C(¯ ) and b j , c 2 L1(), c  0 . We show that the realization A0 of A in C0() := {u 2 C(¯ ) : u|@ = 0} given by D(A0) := {u 2 C0() W2,n loc () : Au 2 C0()} A0u := Au generates a bounded holomorphic C0-semigroup on C0(). The result is in particular true if  is a Lipschitz domain. So far the best known result seems to be the case where  has C2-boundary [12, Section 3.1.5]. We also study the elliptic problem −Au = f u|@ = g .

We consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form L=Sigma(m)(i,j=1)a(i,j)(z)partial derivative(xixj)+Sigma(m)(i=1)a(i)(z)partial derivative(x ...

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We show that generalized Gaussian estimates for selfadjoint semigroups (e-tA)t∈ R+ onL2implyLpboundedness of Riesz means and other regularizations of the Schrödinger group (eitA)t∈ R. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjöstrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.

On nonlinear elliptic equations with singular lower order term

We give a sufficient condition for maximal regularity of the evolution equation $u'(t) - Au(t) = f(t) ,\ t\ge0 ,\ u(0)=0,$ in $L_p$-spaces. Our condition is a weighted norm estimate for the semigroup $(e^{tA})$ and it is strictly weaker than the assumption that the $e^{tA}$ are integral operators whose kernels satisfy Gaussian estimates. As an application we present new results for the maximal regularity of Schr\"odinger operators with singular potentials, elliptic higher order operators with bounded measurable coefficients, and elliptic second order operators with singular lower order terms. Moreover, we prove a similar result for maximal regularity of the discrete time evolution equation $ u_{n+1} - Tu_n = f_n ,$ $n\in\mathbb N_0 ,$ $u_0=0 $.

Motivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE{Lu=f(u)+h(x)}on bounded smooth domains{\Omega\subseteq\mathbb{R}^{n}}, whereLis a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation,fis a continuous non-decreasing function that satisfies the Keller–Osserman condition, whilehis a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions onfandhthat would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions onfandhfor the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients ofLto be bounded in Ω, but we still allowhto be unbounded on Ω.

The [formula omitted] Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

We consider semilinear second-order parabolic equations whose principal parts may have either divergence or nondivergence form and whose nonlinear terms satisfy conditions of Bernstein-Dini type. We study the qualitative properties of the classical solutions of nondivergence equations and generalized solutions of equations with divergent principal parts: the behavior of solutions in various unbounded domains and near the boundaries of domains, removability of singularities of solutions, vanishing of solutions in unbounded domains, in particular solutions of compact support and uniqueness and continuous dependence on the boundary conditions for solutions of the exterior initial/boundary problem.

We construct the fundamental solutions for the sub-elliptic operators in non-divergence form∑i,jai,j(x,t)XiXj−∂t{\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _tand∑i,jai,j(x)XiXj{\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j, where theXiX_i’s form a stratified system of Hörmander vector fields andai,ja_{i,j}are Hölder continuous functions belonging to a suitable class of ellipticity.